%!PS-Adobe-2.0 %%Creator: dvips(k) 5.95a Copyright 2005 Radical Eye Software %%Title: bulletin4.dvi %%Pages: 22 %%PageOrder: Ascend %%BoundingBox: 0 0 595 842 %%DocumentFonts: Times-Bold Times-Roman txsy Times-Italic Helvetica %%+ rtxmi rtxr txsya txsyc rtxi txex txtt %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips bulletin4.dvi -o bulletin4.ps %DVIPSParameters: dpi=600 %DVIPSSource: TeX output 2010.01.08:1434 %%BeginProcSet: tex.pro 0 0 %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/CharBuilder{save 3 1 roll S A/base get 2 index get S /BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]{Ci}imagemask restore}B/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: 8r.enc 0 0 % File 8r.enc TeX Base 1 Encoding Revision 2.0 2002-10-30 % % @@psencodingfile@{ % author = "S. Rahtz, P. MacKay, Alan Jeffrey, B. Horn, K. Berry, % W. Schmidt, P. Lehman", % version = "2.0", % date = "30 October 2002", % filename = "8r.enc", % email = "tex-fonts@@tug.org", % docstring = "This is the encoding vector for Type1 and TrueType % fonts to be used with TeX. This file is part of the % PSNFSS bundle, version 9" % @} % % The idea is to have all the characters normally included in Type 1 fonts % available for typesetting. This is effectively the characters in Adobe % Standard encoding, ISO Latin 1, Windows ANSI including the euro symbol, % MacRoman, and some extra characters from Lucida. % % Character code assignments were made as follows: % % (1) the Windows ANSI characters are almost all in their Windows ANSI % positions, because some Windows users cannot easily reencode the % fonts, and it makes no difference on other systems. The only Windows % ANSI characters not available are those that make no sense for % typesetting -- rubout (127 decimal), nobreakspace (160), softhyphen % (173). quotesingle and grave are moved just because it's such an % irritation not having them in TeX positions. % % (2) Remaining characters are assigned arbitrarily to the lower part % of the range, avoiding 0, 10 and 13 in case we meet dumb software. % % (3) Y&Y Lucida Bright includes some extra text characters; in the % hopes that other PostScript fonts, perhaps created for public % consumption, will include them, they are included starting at 0x12. % These are /dotlessj /ff /ffi /ffl. % % (4) hyphen appears twice for compatibility with both ASCII and Windows. % % (5) /Euro was assigned to 128, as in Windows ANSI % % (6) Missing characters from MacRoman encoding incorporated as follows: % % PostScript MacRoman TeXBase1 % -------------- -------------- -------------- % /notequal 173 0x16 % /infinity 176 0x17 % /lessequal 178 0x18 % /greaterequal 179 0x19 % /partialdiff 182 0x1A % /summation 183 0x1B % /product 184 0x1C % /pi 185 0x1D % /integral 186 0x81 % /Omega 189 0x8D % /radical 195 0x8E % /approxequal 197 0x8F % /Delta 198 0x9D % /lozenge 215 0x9E % /TeXBase1Encoding [ % 0x00 /.notdef /dotaccent /fi /fl /fraction /hungarumlaut /Lslash /lslash /ogonek /ring /.notdef /breve /minus /.notdef /Zcaron /zcaron % 0x10 /caron /dotlessi /dotlessj /ff /ffi /ffl /notequal /infinity /lessequal /greaterequal /partialdiff /summation /product /pi /grave /quotesingle % 0x20 /space /exclam /quotedbl /numbersign /dollar /percent /ampersand /quoteright /parenleft /parenright /asterisk /plus /comma /hyphen /period /slash % 0x30 /zero /one /two /three /four /five /six /seven /eight /nine /colon /semicolon /less /equal /greater /question % 0x40 /at /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O % 0x50 /P /Q /R /S /T /U /V /W /X /Y /Z /bracketleft /backslash /bracketright /asciicircum /underscore % 0x60 /quoteleft /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o % 0x70 /p /q /r /s /t /u /v /w /x /y /z /braceleft /bar /braceright /asciitilde /.notdef % 0x80 /Euro /integral /quotesinglbase /florin /quotedblbase /ellipsis /dagger /daggerdbl /circumflex /perthousand /Scaron /guilsinglleft /OE /Omega /radical /approxequal % 0x90 /.notdef /.notdef /.notdef /quotedblleft /quotedblright /bullet /endash /emdash /tilde /trademark /scaron /guilsinglright /oe /Delta /lozenge /Ydieresis % 0xA0 /.notdef /exclamdown /cent /sterling /currency /yen /brokenbar /section /dieresis /copyright /ordfeminine /guillemotleft /logicalnot /hyphen /registered /macron % 0xD0 /degree /plusminus /twosuperior /threesuperior /acute /mu /paragraph /periodcentered /cedilla /onesuperior /ordmasculine /guillemotright /onequarter /onehalf /threequarters /questiondown % 0xC0 /Agrave /Aacute /Acircumflex /Atilde /Adieresis /Aring /AE /Ccedilla /Egrave /Eacute /Ecircumflex /Edieresis /Igrave /Iacute /Icircumflex /Idieresis % 0xD0 /Eth /Ntilde /Ograve /Oacute /Ocircumflex /Otilde /Odieresis /multiply /Oslash /Ugrave /Uacute /Ucircumflex /Udieresis /Yacute /Thorn /germandbls % 0xE0 /agrave /aacute /acircumflex /atilde /adieresis /aring /ae /ccedilla /egrave /eacute /ecircumflex /edieresis /igrave /iacute /icircumflex /idieresis % 0xF0 /eth /ntilde /ograve /oacute /ocircumflex /otilde /odieresis /divide /oslash /ugrave /uacute /ucircumflex /udieresis /yacute /thorn /ydieresis ] def %%EndProcSet %%BeginProcSet: texps.pro 0 0 %! TeXDict begin/rf{findfont dup length 1 add dict begin{1 index/FID ne 2 index/UniqueID ne and{def}{pop pop}ifelse}forall[1 index 0 6 -1 roll exec 0 exch 5 -1 roll VResolution Resolution div mul neg 0 0]FontType 0 ne{/Metrics exch def dict begin Encoding{exch dup type/integertype ne{ pop pop 1 sub dup 0 le{pop}{[}ifelse}{FontMatrix 0 get div Metrics 0 get div def}ifelse}forall Metrics/Metrics currentdict end def}{{1 index type /nametype eq{exit}if exch pop}loop}ifelse[2 index currentdict end definefont 3 -1 roll makefont/setfont cvx]cvx def}def/ObliqueSlant{dup sin S cos div neg}B/SlantFont{4 index mul add}def/ExtendFont{3 -1 roll mul exch}def/ReEncodeFont{CharStrings rcheck{/Encoding false def dup[ exch{dup CharStrings exch known not{pop/.notdef/Encoding true def}if} forall Encoding{]exch pop}{cleartomark}ifelse}if/Encoding exch def}def end %%EndProcSet %%BeginProcSet: special.pro 0 0 %! TeXDict begin/SDict 200 dict N SDict begin/@SpecialDefaults{/hs 612 N /vs 792 N/ho 0 N/vo 0 N/hsc 1 N/vsc 1 N/ang 0 N/CLIP 0 N/rwiSeen false N /rhiSeen false N/letter{}N/note{}N/a4{}N/legal{}N}B/@scaleunit 100 N /@hscale{@scaleunit div/hsc X}B/@vscale{@scaleunit div/vsc X}B/@hsize{ /hs X/CLIP 1 N}B/@vsize{/vs X/CLIP 1 N}B/@clip{/CLIP 2 N}B/@hoffset{/ho X}B/@voffset{/vo X}B/@angle{/ang X}B/@rwi{10 div/rwi X/rwiSeen true N}B /@rhi{10 div/rhi X/rhiSeen true N}B/@llx{/llx X}B/@lly{/lly X}B/@urx{ /urx X}B/@ury{/ury X}B/magscale true def end/@MacSetUp{userdict/md known {userdict/md get type/dicttype eq{userdict begin md length 10 add md maxlength ge{/md md dup length 20 add dict copy def}if end md begin /letter{}N/note{}N/legal{}N/od{txpose 1 0 mtx defaultmatrix dtransform S atan/pa X newpath clippath mark{transform{itransform moveto}}{transform{ itransform lineto}}{6 -2 roll transform 6 -2 roll transform 6 -2 roll transform{itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll curveto}}{{closepath}}pathforall newpath counttomark array astore/gc xdf pop ct 39 0 put 10 fz 0 fs 2 F/|______Courier fnt invertflag{PaintBlack} if}N/txpose{pxs pys scale ppr aload pop por{noflips{pop S neg S TR pop 1 -1 scale}if xflip yflip and{pop S neg S TR 180 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{pop S neg S TR pop 180 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{ppr 1 get neg ppr 0 get neg TR}if}{ noflips{TR pop pop 270 rotate 1 -1 scale}if xflip yflip and{TR pop pop 90 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{TR pop pop 90 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{TR pop pop 270 rotate ppr 2 get ppr 0 get neg sub neg 0 S TR}if}ifelse scaleby96{ppr aload pop 4 -1 roll add 2 div 3 1 roll add 2 div 2 copy TR .96 dup scale neg S neg S TR}if}N/cp{pop pop showpage pm restore}N end}if}if}N/normalscale{ Resolution 72 div VResolution 72 div neg scale magscale{DVImag dup scale }if 0 setgray}N/psfts{S 65781.76 div N}N/startTexFig{/psf$SavedState save N userdict maxlength dict begin/magscale true def normalscale currentpoint TR/psf$ury psfts/psf$urx psfts/psf$lly psfts/psf$llx psfts /psf$y psfts/psf$x psfts currentpoint/psf$cy X/psf$cx X/psf$sx psf$x psf$urx psf$llx sub div N/psf$sy psf$y psf$ury psf$lly sub div N psf$sx psf$sy scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub TR/showpage{}N/erasepage{}N/setpagedevice{pop}N/copypage{}N/p 3 def @MacSetUp}N/doclip{psf$llx psf$lly psf$urx psf$ury currentpoint 6 2 roll newpath 4 copy 4 2 roll moveto 6 -1 roll S lineto S lineto S lineto closepath clip newpath moveto}N/endTexFig{end psf$SavedState restore}N /@beginspecial{SDict begin/SpecialSave save N gsave normalscale currentpoint TR @SpecialDefaults count/ocount X/dcount countdictstack N} N/@setspecial{CLIP 1 eq{newpath 0 0 moveto hs 0 rlineto 0 vs rlineto hs neg 0 rlineto closepath clip}if ho vo TR hsc vsc scale ang rotate rwiSeen{rwi urx llx sub div rhiSeen{rhi ury lly sub div}{dup}ifelse scale llx neg lly neg TR}{rhiSeen{rhi ury lly sub div dup scale llx neg lly neg TR}if}ifelse CLIP 2 eq{newpath llx lly moveto urx lly lineto urx ury lineto llx ury lineto closepath clip}if/showpage{}N/erasepage{}N /setpagedevice{pop}N/copypage{}N newpath}N/@endspecial{count ocount sub{ pop}repeat countdictstack dcount sub{end}repeat grestore SpecialSave restore end}N/@defspecial{SDict begin}N/@fedspecial{end}B/li{lineto}B /rl{rlineto}B/rc{rcurveto}B/np{/SaveX currentpoint/SaveY X N 1 setlinecap newpath}N/st{stroke SaveX SaveY moveto}N/fil{fill SaveX SaveY moveto}N/ellipse{/endangle X/startangle X/yrad X/xrad X/savematrix matrix currentmatrix N TR xrad yrad scale 0 0 1 startangle endangle arc savematrix setmatrix}N end %%EndProcSet %%BeginProcSet: color.pro 0 0 %! TeXDict begin/setcmykcolor where{pop}{/setcmykcolor{dup 10 eq{pop setrgbcolor}{1 sub 4 1 roll 3{3 index add neg dup 0 lt{pop 0}if 3 1 roll }repeat setrgbcolor pop}ifelse}B}ifelse/TeXcolorcmyk{setcmykcolor}def /TeXcolorrgb{setrgbcolor}def/TeXcolorgrey{setgray}def/TeXcolorgray{ setgray}def/TeXcolorhsb{sethsbcolor}def/currentcmykcolor where{pop}{ /currentcmykcolor{currentrgbcolor 10}B}ifelse/DC{exch dup userdict exch known{pop pop}{X}ifelse}B/GreenYellow{0.15 0 0.69 0 setcmykcolor}DC /Yellow{0 0 1 0 setcmykcolor}DC/Goldenrod{0 0.10 0.84 0 setcmykcolor}DC /Dandelion{0 0.29 0.84 0 setcmykcolor}DC/Apricot{0 0.32 0.52 0 setcmykcolor}DC/Peach{0 0.50 0.70 0 setcmykcolor}DC/Melon{0 0.46 0.50 0 setcmykcolor}DC/YellowOrange{0 0.42 1 0 setcmykcolor}DC/Orange{0 0.61 0.87 0 setcmykcolor}DC/BurntOrange{0 0.51 1 0 setcmykcolor}DC /Bittersweet{0 0.75 1 0.24 setcmykcolor}DC/RedOrange{0 0.77 0.87 0 setcmykcolor}DC/Mahogany{0 0.85 0.87 0.35 setcmykcolor}DC/Maroon{0 0.87 0.68 0.32 setcmykcolor}DC/BrickRed{0 0.89 0.94 0.28 setcmykcolor}DC/Red{ 0 1 1 0 setcmykcolor}DC/OrangeRed{0 1 0.50 0 setcmykcolor}DC/RubineRed{ 0 1 0.13 0 setcmykcolor}DC/WildStrawberry{0 0.96 0.39 0 setcmykcolor}DC /Salmon{0 0.53 0.38 0 setcmykcolor}DC/CarnationPink{0 0.63 0 0 setcmykcolor}DC/Magenta{0 1 0 0 setcmykcolor}DC/VioletRed{0 0.81 0 0 setcmykcolor}DC/Rhodamine{0 0.82 0 0 setcmykcolor}DC/Mulberry{0.34 0.90 0 0.02 setcmykcolor}DC/RedViolet{0.07 0.90 0 0.34 setcmykcolor}DC /Fuchsia{0.47 0.91 0 0.08 setcmykcolor}DC/Lavender{0 0.48 0 0 setcmykcolor}DC/Thistle{0.12 0.59 0 0 setcmykcolor}DC/Orchid{0.32 0.64 0 0 setcmykcolor}DC/DarkOrchid{0.40 0.80 0.20 0 setcmykcolor}DC/Purple{ 0.45 0.86 0 0 setcmykcolor}DC/Plum{0.50 1 0 0 setcmykcolor}DC/Violet{ 0.79 0.88 0 0 setcmykcolor}DC/RoyalPurple{0.75 0.90 0 0 setcmykcolor}DC /BlueViolet{0.86 0.91 0 0.04 setcmykcolor}DC/Periwinkle{0.57 0.55 0 0 setcmykcolor}DC/CadetBlue{0.62 0.57 0.23 0 setcmykcolor}DC /CornflowerBlue{0.65 0.13 0 0 setcmykcolor}DC/MidnightBlue{0.98 0.13 0 0.43 setcmykcolor}DC/NavyBlue{0.94 0.54 0 0 setcmykcolor}DC/RoyalBlue{1 0.50 0 0 setcmykcolor}DC/Blue{1 1 0 0 setcmykcolor}DC/Cerulean{0.94 0.11 0 0 setcmykcolor}DC/Cyan{1 0 0 0 setcmykcolor}DC/ProcessBlue{0.96 0 0 0 setcmykcolor}DC/SkyBlue{0.62 0 0.12 0 setcmykcolor}DC/Turquoise{0.85 0 0.20 0 setcmykcolor}DC/TealBlue{0.86 0 0.34 0.02 setcmykcolor}DC /Aquamarine{0.82 0 0.30 0 setcmykcolor}DC/BlueGreen{0.85 0 0.33 0 setcmykcolor}DC/Emerald{1 0 0.50 0 setcmykcolor}DC/JungleGreen{0.99 0 0.52 0 setcmykcolor}DC/SeaGreen{0.69 0 0.50 0 setcmykcolor}DC/Green{1 0 1 0 setcmykcolor}DC/ForestGreen{0.91 0 0.88 0.12 setcmykcolor}DC /PineGreen{0.92 0 0.59 0.25 setcmykcolor}DC/LimeGreen{0.50 0 1 0 setcmykcolor}DC/YellowGreen{0.44 0 0.74 0 setcmykcolor}DC/SpringGreen{ 0.26 0 0.76 0 setcmykcolor}DC/OliveGreen{0.64 0 0.95 0.40 setcmykcolor} DC/RawSienna{0 0.72 1 0.45 setcmykcolor}DC/Sepia{0 0.83 1 0.70 setcmykcolor}DC/Brown{0 0.81 1 0.60 setcmykcolor}DC/Tan{0.14 0.42 0.56 0 setcmykcolor}DC/Gray{0 0 0 0.50 setcmykcolor}DC/Black{0 0 0 1 setcmykcolor}DC/White{0 0 0 0 setcmykcolor}DC end %%EndProcSet TeXDict begin @defspecial /DvipsToPDF { 72.27 mul Resolution div } def /PDFToDvips { 72.27 div Resolution mul } def /HyperBorder { 1 PDFToDvips } def /H.V {pdf@hoff pdf@voff null} def /H.B {/Rect[pdf@llx pdf@lly pdf@urx pdf@ury]} def /H.S { currentpoint HyperBorder add /pdf@lly exch def dup DvipsToPDF /pdf@hoff exch def HyperBorder sub /pdf@llx exch def } def /H.L { 2 sub dup /HyperBasePt exch def PDFToDvips /HyperBaseDvips exch def currentpoint HyperBaseDvips sub /pdf@ury exch def /pdf@urx exch def } def /H.A { H.L currentpoint exch pop vsize 72 sub exch DvipsToPDF HyperBasePt sub sub /pdf@voff exch def } def /H.R { currentpoint HyperBorder sub /pdf@ury exch def HyperBorder add /pdf@urx exch def currentpoint exch pop vsize 72 sub exch DvipsToPDF sub /pdf@voff exch def } def systemdict /pdfmark known not {userdict /pdfmark systemdict /cleartomark get put} if @fedspecial end %%BeginFont: rtxmi %!PS-AdobeFont-1.0: rtxmi 3.1 %%CreationDate: 12/15/2000 at 12:00 PM %%VMusage: 1024 18707 20 dict begin /FontInfo 16 dict dup begin /version (3.1) readonly def /FullName (rtxmi) readonly def /FamilyName (rtxmi) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def /Notice (Version 3.1, GPL) readonly def /em 1000 def /ascent 800 def /descent 200 def end readonly def /FontName /rtxmi def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 26 /rho put dup 28 /tau put dup 39 /phi1 put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 103 /g put dup 118 /v put dup 119 /w put readonly def /PaintType 0 def /FontType 1 def /StrokeWidth 0 def /FontMatrix[0.001 0 0 0.001 0 0]readonly def /FontBBox{-124 -214 902 762}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: txtt %!PS-AdobeFont-1.0: txtt 3.0 %%CreationDate: 12/14/2000 at 12:00 PM %%VMusage: 1024 21990 20 dict begin /FontInfo 16 dict dup begin /version (3.0) readonly def /FullName (txtt) readonly def /FamilyName (txtt) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch true def /UnderlinePosition -100 def /UnderlineThickness 50 def /Notice (Version 3.0, GPL) readonly def /em 1000 def /ascent 800 def /descent 200 def end readonly def /FontName /txtt def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 66 /B put dup 84 /T put readonly def /PaintType 0 def /FontType 1 def /StrokeWidth 0 def /FontMatrix[0.001 0 0 0.001 0 0]readonly def /FontBBox{-5 -183 542 746}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: txsy %!PS-AdobeFont-1.0: txsy 3.0 %%CreationDate: 12/14/2000 at 12:00 PM %%VMusage: 1024 24296 20 dict begin /FontInfo 16 dict dup begin /version (3.0) readonly def /FullName (txsy) readonly def /FamilyName (txsy) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def /Notice (Version 3.0, GPL) readonly def /em 1000 def /ascent 800 def /descent 200 def end readonly def /FontName /txsy def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 3 /asteriskmath put dup 14 /openbullet put dup 15 /bullet put dup 18 /reflexsubset put dup 32 /arrowleft put dup 33 /arrowright put dup 48 /prime put dup 50 /element put dup 71 /G put dup 83 /S put dup 85 /U put dup 92 /intersection put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put readonly def /PaintType 0 def /FontType 1 def /StrokeWidth 0 def /FontMatrix[0.001 0 0 0.001 0 0]readonly def /FontBBox{-22 -944 1227 866}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: txex %!PS-AdobeFont-1.0: txex 3.0 %%CreationDate: 12/14/2000 at 12:00 PM %%VMusage: 1024 21483 20 dict begin /FontInfo 16 dict dup begin /version (3.0) readonly def /FullName (txex) readonly def /FamilyName (txex) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def /Notice (Version 3.0, GPL) readonly def /em 1000 def /ascent 800 def /descent 200 def end readonly def /FontName /txex def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 32 /parenleftBigg put dup 33 /parenrightBigg put dup 40 /braceleftBigg put dup 41 /bracerightBigg put dup 98 /hatwide put readonly def /PaintType 0 def /FontType 1 def /StrokeWidth 0 def /FontMatrix[0.001 0 0 0.001 0 0]readonly def /FontBBox{-5 -2960 1609 752}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: txsyc %!PS-AdobeFont-1.0: txsyc 3.0 %%CreationDate: 12/14/2000 at 12:00 PM %%VMusage: 1024 31202 20 dict begin /FontInfo 16 dict dup begin /version (3.0) readonly def /FullName (txsyc) readonly def /FamilyName (txsyc) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def /Notice (Version 3.0, GPL) readonly def /em 1000 def /ascent 800 def /descent 200 def end readonly def /FontName /txsyc def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 27 /simequal put dup 60 /nelement put readonly def /PaintType 0 def /FontType 1 def /StrokeWidth 0 def /FontMatrix[0.001 0 0 0.001 0 0]readonly def /FontBBox{-170 -330 1336 843}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: txsya %!PS-AdobeFont-1.0: txsya 3.0 %%CreationDate: 12/14/2000 at 12:00 PM %%VMusage: 1024 22875 20 dict begin /FontInfo 16 dict dup begin /version (3.0) readonly def /FullName (txsya) readonly def /FamilyName (txsya) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def /Notice (Version 3.0, GPL) readonly def /em 1000 def /ascent 800 def /descent 200 def end readonly def /FontName /txsya def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 54 /lessorequalslant put dup 62 /greaterorequalslant put readonly def /PaintType 0 def /FontType 1 def /StrokeWidth 0 def /FontMatrix[0.001 0 0 0.001 0 0]readonly def /FontBBox{-53 -275 1163 861}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: rtxi %!PS-AdobeFont-1.0: rtxi 3.0 %%CreationDate: 12/14/2000 at 12:00 PM %%VMusage: 1024 12681 20 dict begin /FontInfo 16 dict dup begin /version (3.0) readonly def /FullName (rtxi) readonly def /FamilyName (rtxi) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def /Notice (Version 3.0, GPL) readonly def /em 1000 def /ascent 800 def /descent 200 def end readonly def /FontName /rtxi def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put readonly def /PaintType 0 def /FontType 1 def /StrokeWidth 0 def /FontMatrix[0.001 0 0 0.001 0 0]readonly def /FontBBox{-141 -207 808 682}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: rtxr %!PS-AdobeFont-1.0: rtxr 3.0 %%CreationDate: 12/14/2000 at 12:00 PM %%VMusage: 1024 12232 20 dict begin /FontInfo 16 dict dup begin /version (3.0) readonly def /FullName (rtxr) readonly def /FamilyName (rtxr) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def /Notice (Version 3.0, GPL) readonly def /em 1000 def /ascent 800 def /descent 200 def end readonly def /FontName /rtxr def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 11 /ff put dup 14 /ffi put dup 43 /plus put dup 61 /equal put readonly def /PaintType 0 def /FontType 1 def /StrokeWidth 0 def /FontMatrix[0.001 0 0 0.001 0 0]readonly def /FontBBox{-70 -218 792 683}readonly def currentdict end currentfile eexec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cleartomark %%EndFont TeXDict begin 39158280 55380996 1000 600 600 (bulletin4.dvi) @start /Fa 158[42 97[{TeXBase1Encoding ReEncodeFont}1 83.022 /Times-Italic rf /Fc 137[36 118[{}1 90.9091 /rtxmi rf /Fd 147[14 22 14 14 105[{TeXBase1Encoding ReEncodeFont}4 49.8132 /Times-Italic rf /Ff 209[52 46[{TeXBase1Encoding ReEncodeFont}1 206.559 /Times-Roman rf /Fg 171[35 17[35 66[{}2 66.4176 /txtt rf /Fh 207[17 47[32{}2 49.8132 /txsy rf /Fi 157[70 56[56 56 6[38 38 32[{}5 99.6264 /txex rf /Fj 195[53 32[63 27[{}2 99.6264 /txsyc rf /Fk 144[35 11[32 13[46 8[35 76[{TeXBase1Encoding ReEncodeFont}4 63.0965 /Helvetica rf /Fl 204[25 25 25 49[{TeXBase1Encoding ReEncodeFont}3 49.8132 /Times-Roman rf /Fm 163[78 92[{}1 119.552 /txsy rf /Fn 144[63 11[57 13[82 8[63 76[{TeXBase1Encoding ReEncodeFont}4 113.574 /Helvetica rf /Fo 204[72 51[{TeXBase1Encoding ReEncodeFont}1 143.462 /Times-Roman rf /Fp 134[50 50 1[50 55 33 39 44 1[55 50 55 83 28 55 1[28 55 50 33 44 55 44 55 50 12[66 55 72 2[78 1[94 9[72 8[33 5[50 50 50 50 2[25 33 45[{ TeXBase1Encoding ReEncodeFont}35 99.6264 /Times-Bold rf /Fq 193[63 7[63 54[{}2 99.6264 /txsya rf /Fr 138[33 18 26 2[33 1[33 1[18 29 18 18 33 2[29 33 1[33 33 12[37 33 10[48 48 3[44 41 66[{TeXBase1Encoding ReEncodeFont}20 66.4176 /Times-Italic rf /Fs 137[26 14[33 40[42 1[42 17 59[{}5 66.4176 /rtxmi rf /Ft 133[53 60 60 1[60 66 40 47 53 66 66 60 66 100 33 2[33 66 60 40 53 66 53 66 60 11[86 80 3[73 3[80 8[86 1[86 12[60 60 60 60 2[30 46[{ TeXBase1Encoding ReEncodeFont}34 119.552 /Times-Bold rf /Fu 244[53 11[{}1 99.6264 /rtxi rf /Fv 149[20 2[33 33 9[65 6[89 1[68 11[70 20[53 16[96 14[63 2[50 50 12[25 63{}14 99.6264 /txsy rf /Fw 163[43 43[23 14[64 64 28[31 2[42{}6 66.4176 /txsy rf /Fx 3[42 23[37 108[60 42 42 23 32 28 2[42 42 65 23 42 1[23 42 42 28 37 42 37 42 37 11[60 51 10[28 8[76 17[21 28 21 44[{TeXBase1Encoding ReEncodeFont}29 83.022 /Times-Roman rf /Fy 252[27 3[{}1 58.1154 /txsy rf /Fz 194[42 17[42 43[{}2 66.4176 /rtxr rf /FA 203[33 33 33 33 33 6[22 22 40[{TeXBase1Encoding ReEncodeFont}7 66.4176 /Times-Roman rf /FB 133[39 44 44 66 44 50 28 39 39 50 50 50 50 72 28 44 28 28 50 50 28 44 50 44 50 50 3[39 1[39 4[61 1[55 50 3[72 66 1[55 2[33 72 72 1[61 1[66 61 61 7[50 50 1[50 50 1[50 1[50 2[25 33 25 44[{ TeXBase1Encoding ReEncodeFont}50 99.6264 /Times-Italic rf /FC 136[62 39 55[63 44 63 25 25 18[57 10[47 1[51 26[{}10 99.6264 /rtxmi rf /FD 144[53 9[53 53 47 13[68 58 63 2[63 1[68 1[53 5[58 2[68 1[63 65[{TeXBase1Encoding ReEncodeFont}13 94.6448 /Helvetica rf /FE 194[63 17[63 28[82 2[60 10[55{}5 99.6264 /rtxr rf /FF 9[50 95[50 27[44 50 50 72 50 50 28 39 33 50 50 50 50 78 28 50 28 28 50 50 33 44 50 44 50 44 33 2[33 1[33 3[94 72 1[61 55 66 1[55 72 72 89 61 72 1[33 72 72 55 61 72 66 66 72 6[28 50 50 50 50 50 50 50 50 50 50 1[25 33 25 2[33 33 33 3[50 32[55 2[{ TeXBase1Encoding ReEncodeFont}70 99.6264 /Times-Roman rf /FG 137[72 80 48 56 64 1[80 72 80 120 40 2[40 80 1[48 64 80 64 1[72 13[80 104 1[88 6[56 1[112 18[72 72 72 72 3[48 45[{TeXBase1Encoding ReEncodeFont}27 143.462 /Times-Bold rf /FH 179[48 76[{TeXBase1Encoding ReEncodeFont}1 86.3632 /Helvetica rf /FI 134[40 40 61 40 45 25 35 35 45 45 45 45 66 25 40 1[25 45 45 25 40 45 40 45 45 12[51 45 56 1[56 66 1[76 51 1[40 30 66 66 56 56 66 61 1[56 6[30 45 45 45 1[45 45 45 45 45 45 1[23 30 23 2[30 30 30 39[{ TeXBase1Encoding ReEncodeFont}56 90.9091 /Times-Italic rf /FJ 241[75 14[{}1 90.9091 /rtxr rf /FK 9[45 20[40 74[45 27[40 45 45 66 45 45 25 35 30 1[45 45 45 71 25 45 25 25 45 45 30 40 45 40 45 40 3[30 1[30 56 1[66 86 66 66 56 51 61 1[51 66 66 81 56 66 35 30 66 66 51 56 66 61 61 66 6[25 45 45 45 45 45 45 45 45 45 45 1[23 30 23 2[30 30 36[51 3[{TeXBase1Encoding ReEncodeFont}71 90.9091 /Times-Roman rf /FL 139[30 35 40 14[40 51 45 31[66 65[{TeXBase1Encoding ReEncodeFont}7 90.9091 /Times-Bold rf /FM 252[39 3[{}1 83.022 /txsy rf /FN 30[53 110[40 2[60 60 4[33 1[60 1[53 60 53 60 53 9[113 2[73 9[47 3[66 70[{TeXBase1Encoding ReEncodeFont}15 119.552 /Times-Roman rf /FO 134[86 86 3[57 67 76 1[96 86 96 143 48 2[48 96 1[57 76 3[86 12[115 3[105 6[67 1[134 3[124 67[{ TeXBase1Encoding ReEncodeFont}20 172.188 /Times-Bold rf end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%BeginPaperSize: a4 a4 %%EndPaperSize end %%EndSetup %%Page: 1 1 TeXDict begin 1 0 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 0 a SDict begin [ /Title () /Subject () /Creator (LaTeX with hyperref package) /Author () /Producer (dvips + Distiller) /Keywords () /DOCINFO pdfmark end 0 0 a Black 257 266 a SDict begin H.S end 257 266 a Black Black 257 266 a SDict begin H.R end 257 266 a 257 266 a SDict begin [ /View [/XYZ H.V] /Dest (page.1) cvn H.B /DEST pdfmark end 257 266 a Black 257 473 a SDict begin [ /Count -0 /Dest (section.1) cvn /Title (Introduction) /OUT pdfmark end 257 473 a 257 473 a SDict begin [ /Count -3 /Dest (section.2) cvn /Title (Some previous results) /OUT pdfmark end 257 473 a 257 473 a SDict begin [ /Count -0 /Dest (subsection.2.1) cvn /Title (Tree isomorphism in L) /OUT pdfmark end 257 473 a 257 473 a SDict begin [ /Count -0 /Dest (subsection.2.2) cvn /Title (Planarity testing and distance computation) /OUT pdfmark end 257 473 a 257 473 a SDict begin [ /Count -0 /Dest (subsection.2.3) cvn /Title (Universal exploration sequences) /OUT pdfmark end 257 473 a 257 473 a SDict begin [ /Count -1 /Dest (section.3) cvn /Title (Planar 3-connected Graph Isomorphism) /OUT pdfmark end 257 473 a 257 473 a SDict begin [ /Count -0 /Dest (subsection.3.1) cvn /Title (An isomorphism algorithm in ULcoUL) /OUT pdfmark end 257 473 a 257 473 a SDict begin [ /Count -2 /Dest (section.4) cvn /Title (Planar Graph Isomorphism) /OUT pdfmark end 257 473 a 257 473 a SDict begin [ /Count -0 /Dest (subsection.4.1) cvn /Title (Canonization of biconnected planar graphs) /OUT pdfmark end 257 473 a 257 473 a SDict begin [ /Count -0 /Dest (subsection.4.2) cvn /Title (Canonization of planar graphs) /OUT pdfmark end 257 473 a 257 473 a SDict begin [ /Page 1 /View [ /Fit ] /PageMode /UseOutlines /DOCVIEW pdfmark end 257 473 a 257 473 a SDict begin [ {Catalog} << >> /PUT pdfmark end 257 473 a 257 473 a SDict begin H.S end 257 473 a 257 473 a SDict begin 14.5 H.A end 257 473 a 257 473 a SDict begin [ /View [/XYZ H.V] /Dest (Doc-Start) cvn H.B /DEST pdfmark end 257 473 a Black Black 657 982 a FO(The)43 b(Complexity)h(of)g(Planar)f(Graph)1389 1189 y(Isomor)n(phism)1080 1491 y FN(Jacobo)29 b(T)-10 b(or\341n)30 b(and)f(F)n(abian)i(W)-10 b(agner)2673 1491 y SDict begin H.S end 2673 1491 a -44 x FM(\003)2673 1491 y SDict begin 0 H.L end 2673 1491 a 2673 1491 a SDict begin [ /Subtype /Link /Dest (Hfootnote.1) cvn /H /I /Border [0 0 12] /Color [1 0 0] H.B /ANN pdfmark end 2673 1491 a Black Black 1707 1851 a FL(Abstract)p Black Black 638 2042 a FK(The)31 b(Graph)g(Isomorphism)j(problem)f (restricted)g(to)f FI(planar)h FK(graphs)g(has)f(been)501 2155 y(kno)n(wn)25 b(to)f(be)h(solv)n(able)h(in)f(polynomial)h(time)f (man)o(y)f(years)h(ago.)32 b(In)25 b(terms)f(of)h(com-)501 2268 y(ple)o(xity)d(classes)f(ho)n(we)n(v)o(er)l(,)g(the)f(e)o(xact)h (comple)o(xity)h(of)e(the)g(problem)h(has)f(been)h(estab-)501 2381 y(lished)g(only)g(v)o(ery)f(recently)-6 b(.)29 b(It)19 b(w)o(as)h(pro)o(v)o(ed)g(in)g([)2055 2382 y SDict begin H.S end 2055 2382 a Black -1 x FK(6)p Black 2101 2320 a SDict begin H.R end 2101 2320 a 2101 2381 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.DLNTW08) cvn H.B /ANN pdfmark end 2101 2381 a FK(])f(that)h(planar)h(graph)g(isomorphism)501 2494 y(can)27 b(be)f(computed)i(within)e(logarithmic)i(space.)38 b(Since)26 b(there)h(is)f(a)g(matching)i(hard-)501 2607 y(ness)33 b(result)g([)952 2607 y SDict begin H.S end 952 2607 a Black FK(12)p Black 1043 2545 a SDict begin H.R end 1043 2545 a 1043 2607 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.JKMT03) cvn H.B /ANN pdfmark end 1043 2607 a FK(],)g(this)g(sho)n(ws)f(that)g(the)g(problem)h(is)f(complete)h(for)f FH(L)p FK(.)53 b(Although)501 2720 y(this)22 b(could)h(be)f(considered) i(as)e(a)f(result)h(in)g(algorithmics)i(its)e(proof)g(relies)h(on)f(se) n(v)o(eral)501 2833 y(important)37 b(ne)n(w)e(de)n(v)o(elopments)i(in)e (the)h(area)f(of)g(logarithmic)i(space)g(comple)o(xity)501 2945 y(classes)26 b(and)e(re\003ects)g(the)g(close)h(connections)i (between)e(algorithms)h(and)e(comple)o(x-)501 3058 y(ity)h(theory)-6 b(.)32 b(In)25 b(this)g(column)g(we)e(gi)n(v)o(e)i(an)f(o)o(v)o(ervie)n (w)h(of)f(this)h(result)h(mentioning)g(the)501 3171 y(de)n(v)o (elopments)g(that)e(led)g(to)g(it.)257 3360 y SDict begin H.S end 257 3360 a 257 3360 a SDict begin 14.5 H.A end 257 3360 a 257 3360 a SDict begin [ /View [/XYZ H.V] /Dest (section.1) cvn H.B /DEST pdfmark end 257 3360 a 165 x FG(1)144 b(Intr)m(oduction)257 3754 y FF(The)29 b(Graph)f(Isomorphism)e(problem) h(asks)h(whether)g(tw)o(o)g(gi)n(v)o(en)e(graphs)i(are)h(isomorphic)257 3874 y(or)22 b(in)f(other)g(w)o(ords)g(whether)g(there)g(is)g(a)h (bijection)e(between)h(the)h(nodes)e(of)i(the)f(tw)o(o)g(graphs,)257 3994 y(preserving)i(the)g(adjacenc)o(y)g(relation.)30 b(Graph)23 b(Isomorphism)e(is)h(one)h(of)g(the)g(fe)n(w)g(problems)257 4115 y(in)k FD(NP)g FF(that)f(is)g(not)g(kno)n(wn)g(to)g(be)h(in)f FD(P)h FF(or)f FD(NP)q FF(-complete.)35 b(On)27 b(the)f(other)h(hand,)f (for)h(man)o(y)257 4235 y(restricted)f(graph)f(classes)g(lik)o(e)g (trees,)h(graphs)f(of)g(bounded)g(de)o(gree)g(or)h(partial)f FB(k)r FF(-trees,)h(ef-)257 4355 y(\002cient)32 b(algorithms)e(for)i (the)f(isomorphism)e(problem)i(are)h(kno)n(wn.)50 b(W)-8 b(e)32 b(consider)f(in)g(this)257 4476 y(column)k(the)g(class)h(of)f (planar)h(graphs)f(and)h(for)g(simplicity)d(restrict)i(ourselv)o(es)g (to)g(undi-)257 4596 y(rected)e(simple)e(graphs.)54 b(A)32 b(graph)g(is)g(planar)h(if)f(it)g(can)h(be)f(dra)o(wn)g(in)g(the)g (plane)h(without)257 4717 y(an)o(y)25 b(crossing)g(edges.)32 b(A)25 b(special)g(class)g(of)h(planar)f(graphs)g(is)g(that)g(of)g (3-connected)g(planar)257 4837 y(graphs.)45 b(A)30 b(graph)f(is)g FB(k)r FF(-connected)h(if)f(it)g(remains)g(connected)h(after)g (deleting)f FB(k)i FF(arbitrary)257 4957 y(v)o(ertices.)48 b(It)30 b(w)o(as)h(sho)n(wn)e(in)h(1933)g(that)g(3-connected)h(planar)f (graphs)h(ha)n(v)o(e)f(e)o(xactly)g(tw)o(o)257 5078 y(planar)23 b(embeddings)f([)1067 5078 y SDict begin H.S end 1067 5078 a Black FF(22)p Black 1167 5010 a SDict begin H.R end 1167 5010 a 1167 5078 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.Whi33) cvn H.B /ANN pdfmark end 1167 5078 a FF(].)30 b(This)22 b(f)o(act)i(w)o(as)f(used)g(by)f(W)-8 b(einber)n(g)23 b(in)g(1966)f(to)h(gi)n(v)o(e)f(an)h FB(O)p FF(\()p FB(n)3425 5042 y FA(2)3462 5078 y FF(\))257 5198 y(algorithm)30 b(for)i(testing)e(isomorphism)f(of)i(3-connected)g (planar)h(graphs)f([)2916 5198 y SDict begin H.S end 2916 5198 a Black FF(21)p Black 3016 5131 a SDict begin H.R end 3016 5131 a 3016 5198 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.Wei66) cvn H.B /ANN pdfmark end 3016 5198 a FF(])g(\(althrough)p Black 257 5297 1296 4 v 375 5358 a Fy(\003)407 5388 y Fx(Uni)n(v)o(ersit\344t)19 b(Ulm,)34 b(Institut)20 b(f\374r)g(Theoretische)e(Informatik,)32 b(f)o(abian.w)o(agner@uni-ulm.de)p Black Black Black eop end %%Page: 2 2 TeXDict begin 2 1 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 257 266 a SDict begin H.S end 257 266 a Black Black 257 266 a SDict begin H.R end 257 266 a 257 266 a SDict begin [ /View [/XYZ H.V] /Dest (page.2) cvn H.B /DEST pdfmark end 257 266 a Black 307 x FF(the)29 b(paper)l(,)h FB(n)f FF(is)f(the)h(number)f(of) h(v)o(ertices)f(of)h(the)f(input)g(graphs\).)42 b(The)29 b(idea)g(of)g(the)f(algo-)257 693 y(rithm)i(is)h(simple,)g(the)g(tw)o (o)f(gi)n(v)o(en)g(graphs)g(are)i(embedded)f(in)f(the)h(plane)g(and)g (it)f(is)h(tested)257 814 y(whether)25 b(the)g(embedding)f(of)h(the)f (\002rst)h(graph)g(is)g(isomorphic)e(to)i(one)f(of)h(the)g(tw)o(o)g (possible)257 934 y(embeddings)e(of)g(the)h(second)f(graph.)31 b(It)23 b(su)p FE(\016)p FF(ces)h(to)g(\002x)g(an)g(arbitrary)f(edge)h FB(e)g FF(in)g(one)f(graph)257 1054 y(and)g(run)f(through)g(all)g(the)h (edges)f FB(e)1472 1018 y Fw(0)1522 1054 y FF(of)h(the)f(other)h (graph,)g(i.e.)30 b(test)22 b(whether)g(there)h(e)o(xists)e(an)257 1175 y(automorphism)26 b(which)i(maps)f FB(e)i FF(onto)e FB(e)1693 1139 y Fw(0)1720 1175 y FF(.)41 b(W)l(ith)27 b(this)g(method)g(it)h(is)f(also)h(possible)f(to)g(e)p FE(\016)p FF(-)257 1295 y(ciently)e(assign)f(codes)g(to)h(3-connected)g (planar)g(graphs)f(as)h(a)g(w)o(ay)g(to)g(identify)f(them.)30 b(This)257 1416 y(means)f(that)g(there)g(is)g(an)g(e)p FE(\016)p FF(ciently)g(computable)g(function)f(mapping)g(graphs)h(to)g (strings)257 1536 y(or)g(codes,)g(so)f(that)h(tw)o(o)f(graphs)g(are)h (mapped)f(to)g(the)h(same)f(code)h(if)f(and)h(only)e(if)i(the)o(y)f (are)257 1656 y(isomorphic.)66 b(Hopcroft)37 b(and)g(T)-8 b(arjan)37 b(e)o(xtended)f(W)-8 b(einber)n(g')j(s)37 b(algorithm)e(and)i(ga)n(v)o(e)f(the)257 1777 y(\002rst)25 b(polynomial)e(time)g(algorithm)g(for)i(the)g(isomorphism)d(of)i (planar)h(graphs)f([)3066 1778 y SDict begin H.S end 3066 1778 a Black -1 x FF(8)p Black 3116 1709 a SDict begin H.R end 3116 1709 a 3116 1777 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.HoTa71) cvn H.B /ANN pdfmark end 3116 1777 a FF(].)31 b(In)25 b(their)257 1897 y(algorithm)j(the)i(graphs)f (are)h(di)n(vided)e(\002rst)i(in)f(connected)g(components,)h(these)f (are)h(subdi-)257 2017 y(vided)c(in)f(biconnected)h(components)e(and)i (\002nally)g(these)g(components)e(are)j(partitioned)e(in)257 2138 y(triconnected)g(components.)30 b(An)25 b(initial)f(connected)h (component)f(has)h(articulation)f(points)257 2258 y(separating)f(its)g (biconnected)g(components)f(from)h(each)i(other)-5 b(.)29 b(These)24 b(initial)e(components)257 2379 y(are)30 b(represented)f(by) g(a)g(tree-lik)o(e)h(structure)e(containing)g(v)o(ertices)g(for)i (articulation)e(points)257 2499 y(and)e(v)o(ertices)e(for)i (biconnected)f(components.)30 b(In)c(a)f(similar)g(w)o(ay)-6 b(,)25 b(the)g(biconnected)f(com-)257 2619 y(ponents)j(are)i (represented)f(by)f(tree)i(structures)e(containing)f(v)o(ertices)i(for) g(its)f(triconnected)257 2740 y(components)36 b(and)h(v)o(ertices)f (for)h(pairs)g(of)g(nodes)f(\(called)h(separating)f(pairs\))h(which)g (are)257 2860 y(separating)30 b(these)f(components.)45 b(A)30 b(triconnected)f(component)g(is)g(either)h(a)g(3-connected)257 2980 y(graph,)35 b(a)d(c)o(ycle)h(or)f(a)h(3-bond.)53 b(Each)33 b(v)o(erte)o(x)f(representing)g(a)g(triconnected)g(component) 257 3101 y(is)d(then)g(labeled)g(with)f(the)h(code)h(of)f(the)g (component)f(and)h(the)g(whole)g(structure)g(can)h(then)257 3221 y(be)36 b(tested)g(for)g(isomorphism)d(in)i(a)h(similar)f(w)o(ay)h (as)g(it)f(is)g(done)h(for)g(tree)g(isomorphism.)257 3342 y(The)j(original)e(algorithm)g(had)i(a)f(running)g(time)g(of)g FB(O)p FF(\()p FB(n)2331 3305 y FA(2)2368 3342 y FF(\).)72 b(This)38 b(w)o(as)g(then)g(impro)o(v)o(ed)257 3462 y(to)33 b FB(O)p FF(\()p FB(n)17 b FF(log)e FB(n)p FF(\))33 b(in)g([)943 3463 y SDict begin H.S end 943 3463 a Black -1 x FF(10)p Black 1043 3395 a SDict begin H.R end 1043 3395 a 1043 3462 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.HoTa74) cvn H.B /ANN pdfmark end 1043 3462 a FF(])g(and)g(\002nally)f (Hopcroft)h(and)g(W)-8 b(ang)33 b([)2439 3462 y SDict begin H.S end 2439 3462 a Black FF(11)p Black 2539 3395 a SDict begin H.R end 2539 3395 a 2539 3462 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.HW74) cvn H.B /ANN pdfmark end 2539 3462 a FF(])g(obtained)f(a)h(linear)g(time)257 3582 y(algorithm)e(for)i(the)g(isomorphism)d(of)i(planar)h(graphs.)54 b(Recently)32 b(K)o(ukluk,)h(Holder)l(,)h(and)257 3703 y(Cook)29 b([)535 3704 y SDict begin H.S end 535 3704 a Black -1 x FF(13)p Black 634 3636 a SDict begin H.R end 634 3636 a 634 3703 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.KHC04) cvn H.B /ANN pdfmark end 634 3703 a FF(])g(ga)n(v)o(e)e (an)i FB(O)p FF(\()p FB(n)1186 3667 y FA(2)1223 3703 y FF(\))f(algorithm)f(for)i(planar)f(graph)g(isomorphism,)f(which)g(is) h(suit-)257 3823 y(able)d(for)g(practical)g(applications.)404 3944 y(Re)o(garding)k(the)h(parallel)g(comple)o(xity)e(of)j(the)f (problem,)g(Miller)g(and)g(Reif)h([)3148 3945 y SDict begin H.S end 3148 3945 a Black -1 x FF(16)p Black 3248 3876 a SDict begin H.R end 3248 3876 a 3248 3944 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.MR91) cvn H.B /ANN pdfmark end 3248 3944 a FF(])f(ga)n(v)o(e)257 4064 y(the)38 b(\002rst)g FD(NC)g FF(algorithm)e(for)i(planar)g(graph)g (isomorphism.)67 b(Their)37 b(algorithm)f(runs)i(in)257 4184 y(time)k FB(O)p FF(\(log)16 b FB(n)p FF(\))43 b(with)f(a)h (polynomial)e(number)h(of)h(processors)f(in)g(the)h(CRCW)h(PRAM)257 4305 y(model.)62 b(This)34 b(corresponds)h(to)g(the)g(comple)o(xity)e (class)i FD(A)m(C)2433 4261 y FA(1)2506 4305 y FF(of)g(problems)f (computable)257 4425 y(by)k(unbounded)e(f)o(an-in)i(circuits)f(of)g (polynomial)f(size)i(and)f(logarithmic)f(depth.)68 b(More)257 4545 y(recently)-6 b(,)31 b(using)d(a)i(completely)f(di)p FE(\013)p FF(erent)g(approach)h(based)g(on)f(descripti)n(v)o(e)f (comple)o(xity)-6 b(,)257 4666 y(Grohe)22 b(and)f(V)-11 b(erbitsk)o(y)19 b([)1127 4667 y SDict begin H.S end 1127 4667 a Black -1 x FF(7)p Black 1177 4599 a SDict begin H.R end 1177 4599 a 1177 4666 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.GrVe06) cvn H.B /ANN pdfmark end 1177 4666 a FF(])j(pro)o(vided)e(a)h(ne)n(w)g(method)f(for)h(testing)f (isomorphism)f(of)i(planar)257 4786 y(graphs)28 b(also)g(within)f FD(A)m(C)1151 4743 y FA(1)1189 4786 y FF(.)41 b(The)o(y)27 b(pro)o(v)o(ed)g(that)h(for)g(a)h(class)f Fv(G)g FF(of)g(graphs,)h(if)f (e)n(v)o(ery)f(graph)257 4907 y(in)j(the)h(class)f(is)g(de\002nable)g (in)g(\002rst)h(order)f(logic)g(with)g(a)g(\002nite)h(number)e(of)i(v)n (ariables)e(and)257 5027 y(logarithmic)24 b(quanti\002er)h(depth,)g (then)g(the)h(isomorphism)d(problem)h(for)i Fv(G)g FF(is)f(in)g FD(A)m(C)3231 4984 y FA(1)3268 5027 y FF(.)33 b(V)-11 b(er)n(-)257 5147 y(bitsk)o(y)19 b([)553 5148 y SDict begin H.S end 553 5148 a Black -1 x FF(20)p Black 653 5080 a SDict begin H.R end 653 5080 a 653 5147 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.Ve07) cvn H.B /ANN pdfmark end 653 5147 a FF(])i(sho)n(wed)f(that)g(the)g(class)h(of)g(planar)f (3-connected)h(graphs)f(is)h(de\002nable)g(with)f(15)257 5268 y(v)n(ariables)25 b(and)h(logarithmic)d(quanti\002er)j(depth.)32 b(T)-8 b(ogether)25 b(with)g(the)g FD(A)m(C)2837 5224 y FA(1)2900 5268 y FF(reduction)g(from)257 5388 y(planar)g(isomorphism) d(to)i(3-connected)g(planar)h(isomorphism)d(from)i([)2759 5389 y SDict begin H.S end 2759 5389 a Black -1 x FF(16)p Black 2859 5321 a SDict begin H.R end 2859 5321 a 2859 5388 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.MR91) cvn H.B /ANN pdfmark end 2859 5388 a FF(])g(this)g(pro)o(vides)f(a)p Black Black eop end %%Page: 3 3 TeXDict begin 3 2 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 257 266 a SDict begin H.S end 257 266 a Black Black 257 266 a SDict begin H.R end 257 266 a 257 266 a SDict begin [ /View [/XYZ H.V] /Dest (page.3) cvn H.B /DEST pdfmark end 257 266 a Black 307 x FF(di)p FE(\013)p FF(erent)25 b(w)o(ay)g(to)f(sho)n(w)g (that)g(planar)h(isomorphism)d(lies)j(in)f FD(A)m(C)2532 529 y FA(1)2569 573 y FF(.)404 693 y(W)-8 b(e)28 b(describe)g(in)g (this)f(surv)o(e)o(y)g(some)g(of)i(the)e(results)h(leading)f(to)h(the)g (impro)o(v)o(ement)d(of)257 814 y(the)36 b(upper)g(bound)f(from)g FD(A)m(C)1321 770 y FA(1)1394 814 y FF(to)g(logarithmic)f(space.)64 b(Located)35 b(between)h FD(L)g FF(and)f FD(A)m(C)3433 770 y FA(1)3470 814 y FF(,)257 934 y(the)e(comple)o(xity)e(class)h FD(UL)h FF(played)g(an)f(important)g(part)h(in)f(the)h(de)n(v)o (elopment)d(of)j(the)g(re-)257 1054 y(sults.)47 b FD(UL)31 b FF(\(unambiguous)e(logarithmic)g(space\))i(is)f(the)h(class)f(of)h (problems)e(computable)257 1175 y(by)35 b(a)f(logarithmic)f(space)i (nondeterministic)d(machine)i(ha)n(ving)g(at)h(most)e(one)h(accepting) 257 1295 y(computation)23 b(path)g(for)i(each)g(input.)k(The)24 b(relation)g(between)g(the)g(considered)f(comple)o(xity)257 1416 y(classes)i(is)f(as)h(follo)n(ws:)1575 1536 y FD(L)j Fv(\022)g FD(UL)f Fv(\022)h FD(A)m(C)2115 1493 y FA(1)2152 1536 y FC(:)257 1706 y FF(W)-8 b(e)26 b(denote)e(by)h FD(FL)f FF(the)h(class)f(of)h(functions)f(computable)g(within)f (logarithmic)g(space.)404 1826 y(All)f(the)h(isomorphism)d(results)j (described)f(in)h(this)f(o)o(v)o(ervie)n(w)f(can)j(in)e(f)o(act)i(be)f (e)o(xtended)257 1946 y(to)29 b(graph)g(canonization)f(results.)43 b(F)o(or)29 b(a)g(class)g Fv(G)g FF(of)g(graphs)g(we)g(say)g(that)f(a)i (function)43 b FB(f)50 b FF(:)257 2067 y Fv(G)44 b(!)f(f)p FF(0)p FC(;)17 b FF(1)p Fv(g)718 2031 y Fw(\003)786 2067 y FF(computes)32 b(a)i FB(complete)f(in)l(variant)e FF(for)j(the)f (class)g(if)g(for)h(e)n(v)o(ery)d FB(G)r FC(;)19 b FB(H)47 b Fv(2)d(G)257 2187 y FF(it)29 b(holds)f(that)e FB(G)31 b FF(and)g FB(H)j FF(are)c(isomorphic)d(if)i(and)g(only)f(if)44 b FB(f)15 b FF(\()n FB(G)r FF(\))35 b FE(=)51 b FB(f)15 b FF(\()r FB(H)t FF(\).)44 b(If)29 b(moreo)o(v)o(er)43 b FB(f)257 2308 y FF(computes)29 b(for)i(each)d FB(G)k FF(a)e(graph)45 b FB(f)15 b FF(\()n FB(G)r FF(\))30 b(isomorphic)e(to)g FB(G)k FF(then)d(we)i(call)44 b FB(f)h FF(a)30 b FB(canonizing)257 2428 y(function)24 b FF(and)40 b FB(f)15 b FF(\()n FB(G)r FF(\))24 b(a)h FB(canon)p FF(.)404 2548 y(W)-8 b(e)28 b(recall)g(in)f(Section)1243 2548 y SDict begin H.S end 1243 2548 a Black FF(2)p Black 1293 2481 a SDict begin H.R end 1293 2481 a 1293 2548 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (section.2) cvn H.B /ANN pdfmark end 1293 2548 a 28 w FF(se)n(v)o(eral)g(f)o(acts)h(that)f(were)i(used)e(in)g (the)h(proof)g(of)g(the)f(results.)257 2669 y(Similar)k(as)h(in)f(the)h (de)n(v)o(elopments)d(leading)i(to)g(polynomial)f(time)h(test)g(for)h (planar)f(graph)257 2789 y(isomorphism,)19 b(the)i(\002rst)g (logarithmic)d(space)j(isomorphism)e(algorithms)g(w)o(ork)o(ed)h(for)h (trees)257 2910 y(and)31 b(3-connected)g(graphs.)49 b(These)31 b(are)h(o)o(v)o(ervie)n(wed)e(in)g(Sections)2680 2910 y SDict begin H.S end 2680 2910 a Black FF(2)p Black 2730 2842 a SDict begin H.R end 2730 2842 a 2730 2910 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (section.2) cvn H.B /ANN pdfmark end 2730 2910 a 31 w FF(and)2936 2911 y SDict begin H.S end 2936 2911 a Black -1 x FF(3)p Black 2986 2842 a SDict begin H.R end 2986 2842 a 2986 2910 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (section.3) cvn H.B /ANN pdfmark end 2986 2910 a FF(.)49 b(Finally)30 b(the)257 3030 y(logarithmic)j(space)j(algorithm)d(for)j(planar)f(graph)g (isomorphism)d(is)j(e)o(xplained)f(in)g(Sec-)257 3150 y(tion)437 3150 y SDict begin H.S end 437 3150 a Black FF(4)p Black 487 3083 a SDict begin H.R end 487 3083 a 487 3150 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (section.4) cvn H.B /ANN pdfmark end 487 3150 a FF(.)257 3307 y SDict begin H.S end 257 3307 a 257 3307 a SDict begin 14.5 H.A end 257 3307 a 257 3307 a SDict begin [ /View [/XYZ H.V] /Dest (section.2) cvn H.B /DEST pdfmark end 257 3307 a 181 x FG(2)144 b(Some)34 b(pr)m(e)n(vious)g(r)m(esults)257 3621 y SDict begin H.S end 257 3621 a 257 3621 a SDict begin 14.5 H.A end 257 3621 a 257 3621 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.2.1) cvn H.B /DEST pdfmark end 257 3621 a 120 x Ft(2.1)119 b(T)-9 b(r)n(ee)30 b(isomor)o(phism)f(in)i(L) 257 3928 y FF(Lindell)25 b(ga)n(v)o(e)g(in)g([)917 3928 y SDict begin H.S end 917 3928 a Black FF(14)p Black 1017 3861 a SDict begin H.R end 1017 3861 a 1017 3928 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.Lin92) cvn H.B /ANN pdfmark end 1017 3928 a FF(])h(a)g(logarithmic)e(space)i(algorithm)e (for)i(tree)g(isomorphism)d(and)j(can-)257 4049 y(onization.)54 b(Some)33 b(of)g(the)g(ideas)g(in)f(this)g(algorithm)g(are)h(used)g (also)f(in)h(the)g(ne)n(w)f(results.)257 4169 y(Lindell)25 b(de\002ned)i(a)f(canonical)h(ordering)e(between)h(trees.)35 b(F)o(or)27 b(a)f(tree)h FB(T)12 b FF(,)26 b(we)h(represent)f(its)257 4290 y(root)d(by)g FB(t)i FF(and)f(the)f(out-de)o(gree)g(of)g FB(t)i FF(by)e(#)p FB(t)r FF(.)30 b(The)23 b(size)h(of)f(the)h(tree)f (is)g(denoted)g Fv(j)p FB(T)12 b Fv(j)p FF(.)30 b(The)24 b(tree)257 4410 y(isomorphism)19 b(ordering)i FC(<)1221 4425 y Fr(T)1291 4410 y FF(from)g(Lindell)f(is)g(de\002ned)i(in)e(the)h (follo)n(wing)e(w)o(ay:)29 b(Gi)n(v)o(en)20 b(tw)o(o)257 4530 y(trees)25 b FB(S)41 b FF(and)25 b FB(T)12 b FF(,)25 b(we)g(say)f FB(S)44 b FC(<)1302 4545 y Fr(T)1379 4530 y FB(T)37 b FF(if:)p Black Black 306 4747 a FB(i)p FF(\))25 b Fv(j)p FB(S)16 b Fv(j)27 b FC(<)h Fv(j)p FB(T)12 b Fv(j)25 b FF(or)p Black Black 306 4947 a FB(ii)p FF(\))g Fv(j)p FB(S)16 b Fv(j)27 b FE(=)h Fv(j)p FB(T)12 b Fv(j)25 b FF(and)f(#)5 b FB(s)28 b FC(<)g FF(#)p FB(t)r FF(,)c(or)p Black Black 306 5147 a FB(iii)p FF(\))36 b Fv(j)p FB(S)16 b Fv(j)48 b FE(=)h Fv(j)p FB(T)12 b Fv(j)p FF(,)36 b(#)5 b FB(s)48 b FE(=)h FF(#)p FB(t)h FE(=)f FB(k)38 b FF(and)e(\()p FB(S)1747 5162 y FA(1)1784 5147 y FC(;)17 b(:)g(:)g(:)f(;)h FB(S)2058 5162 y Fr(k)2092 5147 y FF(\))48 b FC(<)h FF(\()p FB(T)2377 5162 y FA(1)2415 5147 y FC(;)17 b(:)g(:)g(:)e(;)i FB(T)2682 5162 y Fr(k)2717 5147 y FF(\))37 b(le)o(xicographically)-6 b(,)501 5268 y(were)29 b FB(S)788 5283 y FA(1)859 5268 y Fq(6)922 5283 y Fr(T)1005 5268 y FB(S)1070 5283 y FA(2)1141 5268 y Fq(6)1204 5283 y Fr(T)1287 5268 y FC(:)17 b(:)g(:)33 b Fq(6)1492 5283 y Fr(T)1575 5268 y FB(S)1640 5283 y Fr(k)1675 5268 y FF(,)c(and)f FB(T)1960 5283 y FA(1)2032 5268 y Fq(6)2095 5283 y Fr(T)2178 5268 y FB(T)2237 5283 y FA(2)2309 5268 y Fq(6)2372 5283 y Fr(T)2455 5268 y FC(:)17 b(:)g(:)33 b Fq(6)2660 5283 y Fr(T)2743 5268 y FB(T)2802 5283 y Fr(k)2837 5268 y FF(,)c(are)g(the)f(ordered)501 5388 y(subtrees)d(of)g FB(S)40 b FF(and)25 b FB(T)37 b FF(rooted)24 b(at)h(the)g(children)f(of)30 b FB(s)25 b FF(and)f FB(t)r FF(.)p Black Black eop end %%Page: 4 4 TeXDict begin 4 3 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 257 266 a SDict begin H.S end 257 266 a Black Black 257 266 a SDict begin H.R end 257 266 a 257 266 a SDict begin [ /View [/XYZ H.V] /Dest (page.4) cvn H.B /DEST pdfmark end 257 266 a Black 404 573 a FF(It)36 b(is)g(not)f(hard)i(to)f(see)g(that)g(if)g (neither)g FB(S)64 b FC(<)2031 588 y Fr(T)2130 573 y FB(T)48 b FF(nor)36 b FB(T)61 b FC(<)2581 588 y Fr(S)2678 573 y FB(S)52 b FF(then)36 b FB(S)52 b FF(and)36 b FB(T)49 b FF(are)257 693 y(isomorphic.)37 b(Ob)o(viously)24 b(the)k(\002rst)f (tw)o(o)g(tests)f(in)h(the)g(de\002nition)f(of)h(tree)h(ordering)f(can) g(be)257 814 y(done)i(in)f(logarithmic)e(space.)42 b(Lindell)28 b(pro)o(v)o(ed)f(that)h(this)f(is)h(also)h(true)f(for)h(the)f(third)g (step.)257 934 y(This)c(is)h(done)f(with)g(the)h(follo)n(wing)e(v)n (ariant)h(of)h(depth)f(\002rst)h(search:)p Black Black 306 1196 a Fv(\017)35 b FF(P)o(artition)f(the)h(children)f(into)h (classes,)i(according)e(to)g(their)f(size.)62 b(Find)35 b(the)g(number)501 1316 y(of)g(children)f(of)40 b FB(s)34 b FF(and)h FB(t)h FF(of)e(minimal)f(size.)60 b(If)35 b(these)g(numbers)e(do)i(not)f(coincide)501 1436 y(then)25 b(declare)h(the)f(tree)h(with)e(the)h(lar)n(gest)g(number)g(of)g (minimal)e(size)j(children)e(to)h(be)501 1557 y(the)k(smaller)f(one.)43 b(Otherwise,)29 b(check)g(the)g(ne)o(xt)f(size)g(class)h(until)e(an)i (inequality)f(is)501 1677 y(found)d(or)f(all)h(the)g(children)f(ha)n(v) o(e)h(been)g(considered.)p Black Black 306 1904 a Fv(\017)34 b FF(If)40 b FB(s)34 b FF(and)h FB(t)h FF(ha)n(v)o(e)e(the)g(same)h (number)e(of)i(children)f(of)h(each)g(size,)h(then)f(we)f(compare)501 2025 y(the)g(children)f(in)g(each)h(size)f(class)h(in)f(increasing)g (order)g(of)h(the)f(sizes)h(recursi)n(v)o(ely)501 2145 y(as)40 b(follo)n(ws:)59 b(Let)40 b FB(k)h FF(be)f(the)g(number)f(of)h (children)f(in)h(one)g(size)f(class.)76 b(W)-8 b(e)40 b(can)501 2265 y(suppose)24 b FB(k)30 b FC(>)e FF(0.)p Black Black 550 2527 a(\226)21 b(If)h FB(k)30 b FE(=)e FF(1)21 b(then)h(only)e(one)i(recursi)n(v)o(e)e(call)i(is)f(made)g(and) h(no)f(e)o(xtra)g(space)h(is)f(needed)716 2647 y(for)k(this.)p Black Black 550 2819 a(\226)19 b(If)h FB(k)30 b FC(>)e FF(1)19 b(then)g(for)g(each)h(node)f(in)g(the)h(size)f(class)g(in)g FB(S)35 b FF(we)20 b(compute)e(its)h(order)g(pro-)716 2939 y(\002le.)42 b(This)27 b(consists)h(of)g(three)h(counters)f FB(c)2209 2954 y Fs(<)2255 2939 y FC(;)17 b FB(c)2341 2954 y Fs(>)2416 2939 y FF(and)28 b FB(c)2632 2954 y Fz(=)2708 2939 y FF(indicating)f(the)h(num-)716 3059 y(ber)d(of)g(children)g(in)f(the)h(corresponding)f(size)h(class)g(of)g FB(T)37 b FF(being)25 b(respecti)n(v)o(ely)716 3180 y(smaller)c(than,)i (lar)n(ger)f(than)g(or)g(equal)g(to)g(the)g(node)g(under)g (consideration.)28 b(The)716 3300 y(counters)e(are)h(updated)e(by)h (making)g(cross)g(comparisons.)33 b(W)-8 b(e)27 b(start)f(with)f(the) 716 3421 y(children)34 b(with)g(minimal)f(order)j(pro\002le,)h(those)d (with)g FB(c)2725 3436 y Fs(>)2818 3421 y FE(=)47 b FF(0.)60 b(The)o(y)35 b(form)716 3541 y(an)e(isomorphism)d(class.)55 b(The)33 b(size)g(of)g(this)f(class)h(is)g(compared)f(across)h(the)716 3661 y(trees)28 b(by)g(comparing)f(the)h(v)n(alues)g(of)g(the)g FB(c)2250 3676 y Fz(=)2325 3661 y FF(counters.)40 b(If)28 b(the)o(y)g(match,)g(both)716 3782 y(trees)j(ha)n(v)o(e)g(the)h(same)f (number)f(of)i(minimal)d(children.)50 b(T)-8 b(o)31 b(compare)h(lar)n (ger)716 3902 y(children)22 b(in)h(the)f(same)h(size)f(class,)h(the)g (v)n(alue)f(of)h FB(c)2488 3917 y Fz(=)2557 3902 y FF(in)g(the)f(last)h (step)f(w)o(orks)g(as)716 4022 y(a)29 b(threshold.)41 b(This)28 b(is)g(used)h(to)f(search)h(for)g(the)g(ne)o(xt)f(minimal)e (children)j(of)k FB(s)716 4143 y FF(and)e FB(t)r FF(.)48 b(The)30 b(process)h(is)f(then)g(repeated)i(and)e(the)h(threshold)f(is) g(incremented)716 4263 y(until)g(reaching)g FB(k)r FF(,)j(at)e(which)f (point)g(we)h(proceed)g(to)g(the)f(ne)o(xt)g(size)h(class.)49 b(If)716 4384 y(all)25 b(the)h(size)f(classes)h(are)g(visited)f (without)f(detecting)h(an)h(inequality)e(then)h(the)716 4504 y(trees)g(are)g(isomorphic.)257 4697 y SDict begin H.S end 257 4697 a 257 4697 a SDict begin 14.5 H.A end 257 4697 a 257 4697 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.2.2) cvn H.B /DEST pdfmark end 257 4697 a 132 x Ft(2.2)119 b(Planarity)30 b(testing)g(and)h(distance)f(computation)257 5027 y FF(A)f(graph)g(is)f FB(planar)i FF(if)f(it)f(can)h(be)g(dra)o (wn)g(on)f(the)h(plane)f(so)h(that)f(no)h(edges)f(cross.)43 b(Such)29 b(a)257 5147 y(dra)o(wing)d(is)g(a)h FB(planar)e(embedding)p FF(.)35 b(Allender)26 b(and)h(Mahajan)f([)2517 5148 y SDict begin H.S end 2517 5148 a Black -1 x FF(3)p Black 2567 5080 a SDict begin H.R end 2567 5080 a 2567 5147 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.AM00) cvn H.B /ANN pdfmark end 2567 5147 a FF(])h(sho)n(wed)e(that)h(the)g(prob-) 257 5268 y(lem)g(of)g(determining)f(if)h(a)g(gi)n(v)o(en)f(graph)h(is)g (planar)l(,)g(can)g(be)h(computed)e(in)g(the)h(comple)o(xity)257 5388 y(class)33 b FD(SL)g FF(\(symmetric)g(logarithmic)e(space\).)56 b(Some)33 b(year)h(later)g(Reingold)e([)3055 5389 y SDict begin H.S end 3055 5389 a Black -1 x FF(18)p Black 3155 5321 a SDict begin H.R end 3155 5321 a 3155 5388 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.Rei05) cvn H.B /ANN pdfmark end 3155 5388 a FF(])i(pro)o(v)o(ed)p Black Black eop end %%Page: 5 5 TeXDict begin 5 4 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 257 266 a SDict begin H.S end 257 266 a Black Black 257 266 a SDict begin H.R end 257 266 a 257 266 a SDict begin [ /View [/XYZ H.V] /Dest (page.5) cvn H.B /DEST pdfmark end 257 266 a Black 307 x FF(that)29 b(the)h(classed)f FD(SL)g FF(and)g FD(L)g FF(coincide,)i(thus)d(bringing)g(the)h(planarity)g(recognition)f (prob-)257 693 y(lem)d(to)f FD(L)p FF(.)404 818 y(Let)e FB(E)623 833 y Fs(v)673 818 y FF(be)f(the)f(set)g(of)g(edges)g(\()p FB(u)p FC(;)d(v)p FF(\))j(in)g(the)g(graph)e FB(G)30 b FE(=)e FF(\()p FB(V)-9 b FC(;)19 b FB(E)5 b FF(\))20 b(incident)f(to)h FC(v)p FF(.)29 b(A)21 b FB(r)l(otation)257 938 y(sc)o(heme)30 b FF(for)h(a)g(graph)d FB(G)k FF(is)e(a)h(set)f FC(\032)g FF(of)g(permutations,)g FC(\032)38 b FE(=)g Fv(f)p FC(\032)2518 953 y Fs(v)2585 938 y Fv(j)g FC(v)g Fv(2)g FB(V)8 b Fv(g)p FF(,)31 b(where)g FC(\032)3294 953 y Fs(v)3354 938 y FF(is)f(a)257 1058 y(permutation)24 b(on)k FB(E)956 1073 y Fs(v)1011 1058 y FF(that)e(has)f(only)g(one)g(c) o(ycle)g(\(which)h(is)f(called)g(a)h FB(r)l(otation)p FF(\).)31 b(Let)25 b FC(\032)3296 1022 y Fw(\000)p FA(1)3401 1058 y FF(be)257 1179 y(the)31 b(set)g(of)h(in)l(v)o(erse)e(rotations,) i FC(\032)1438 1142 y Fw(\000)p FA(1)1556 1179 y FE(=)40 b Fv(f)p FC(\032)1743 1142 y Fw(\000)p FA(1)1743 1203 y Fs(v)1861 1179 y Fv(j)g FC(v)f Fv(2)h FB(V)8 b Fv(g)p FF(.)49 b(A)31 b(rotation)g(scheme)g FC(\032)g FF(describes)257 1299 y(an)38 b(embedding)f(of)g(graph)f FB(G)k FF(in)d(the)h(plane.)69 b(If)38 b(the)f(embedding)g(is)g(planar)l(,)k(we)d(call)g FC(\032)257 1419 y FF(a)d FB(planar)e(r)l(otation)f(sc)o(heme)p FF(.)59 b(The)35 b(pair)f(\()n FB(G)r FC(;)17 b(\032)p FF(\))34 b(is)g(called)g(a)h FB(combinatorial)d(embedding)257 1540 y FF(for)23 b FB(G)r FF(.)31 b(Allender)25 b(and)f(Mahajan)h(also) f(sho)n(wed)g(the)h(follo)n(wing)e(useful)h(result:)257 1561 y SDict begin H.S end 257 1561 a 257 1561 a SDict begin 14.5 H.A end 257 1561 a 257 1561 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.2.1) cvn H.B /DEST pdfmark end 257 1561 a Black 202 x Fp(Theor)n(em)j(2.1.)p Black 67 w FB([)924 1764 y SDict begin H.S end 924 1764 a Black -1 x FB(3)p Black 973 1697 a SDict begin H.R end 973 1697 a 973 1763 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.AM00) cvn H.B /ANN pdfmark end 973 1763 a FB(])f(Ther)l(e)g(is)f(a)g(lo)o (garithmic)e(space)i(algorithm)f(that)g(on)h(input)f(a)i(planar)257 1883 y(gr)o(aph)c(G)27 b(pr)l(oduces)c(a)i(planar)f(r)l(otation)e(sc)o (heme)j FC(\032)f FB(for)e(G)r(.)404 2106 y FF(A)40 b(planar)g (3-connected)g(graph)g(has)g(e)o(xactly)g(tw)o(o)f(planar)h(rotation)g (schemes)f([)3337 2106 y SDict begin H.S end 3337 2106 a Black FF(22)p Black 3437 2039 a SDict begin H.R end 3437 2039 a 3437 2106 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.Whi33) cvn H.B /ANN pdfmark end 3437 2106 a FF(],)257 2226 y(some)25 b(rotation)e(scheme)i FC(\032)g FF(and)f(its)g(in)l(v)o (erse)h FC(\032)1873 2190 y Fw(\000)p FA(1)1952 2226 y FF(.)404 2351 y(An)e(important)f(tool)g(in)h(the)g(\002rst)h (isomorphism)d(tests)h(for)i(3-connected)f(planar)h(graphs)257 2471 y(is)34 b(the)f(computation)f(of)i(distances)f(in)g(planar)h (graphs)f(within)g(the)g(class)h FD(UL)p FF(.)57 b(This)33 b(the-)257 2591 y(orem)38 b(b)n(uilds)f(on)h(a)g(series)g(of)g(results) f(dealing)g(with)h(the)f(reachability)h(problem)f(in)h(di-)257 2712 y(rected)k(planar)f(graphs,)k([)1201 2712 y SDict begin H.S end 1201 2712 a Black FF(2)p Black 1251 2644 a SDict begin H.R end 1251 2644 a 1251 2712 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.ADR05) cvn H.B /ANN pdfmark end 1251 2712 a FF(],[)1342 2713 y SDict begin H.S end 1342 2713 a Black -1 x FF(17)p Black 1442 2644 a SDict begin H.R end 1442 2644 a 1442 2712 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.RA00) cvn H.B /ANN pdfmark end 1442 2712 a FF(])d(that)f(lead)g(to)g(the)g(algorithm)f(from)h(Bourk) o(e,)46 b(T)-7 b(e)n(w)o(ari)257 2832 y(and)40 b(V)-6 b(inodchandran)38 b([)1121 2832 y SDict begin H.S end 1121 2832 a Black FF(4)p Black 1171 2765 a SDict begin H.R end 1171 2765 a 1171 2832 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.BTV07) cvn H.B /ANN pdfmark end 1171 2832 a FF(])i(to)f(compute)f(the)h(reachability)g(problem)g(for)g(planar)h (graphs)257 2952 y(within)24 b FD(UL)e Fv(\\)g FD(coUL)p FF(.)257 2954 y SDict begin H.S end 257 2954 a 257 2954 a SDict begin 14.5 H.A end 257 2954 a 257 2954 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.2.2) cvn H.B /DEST pdfmark end 257 2954 a Black 221 x Fp(Theor)n(em)36 b(2.2.)p Black 46 w FB([)912 3177 y SDict begin H.S end 912 3177 a Black -2 x FB(19)p Black 1011 3109 a SDict begin H.R end 1011 3109 a 1011 3175 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.TW08) cvn H.B /ANN pdfmark end 1011 3175 a FB(])e(The)g(distance)f(between)h(two)g(given)g(vertices)f(in)h(a)f (planar)g(gr)o(aph)257 3296 y(can)25 b(be)g(computed)f(in)h FD(UL)d Fv(\\)g FD(coUL)p FB(.)257 3483 y SDict begin H.S end 257 3483 a 257 3483 a SDict begin 14.5 H.A end 257 3483 a 257 3483 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.2.3) cvn H.B /DEST pdfmark end 257 3483 a 133 x Ft(2.3)119 b(Uni)o(v)o(ersal)30 b(exploration)h(sequences)257 3811 y FF(The)h(celebrated)g(result)g(from)f(Reingold)g([)1794 3812 y SDict begin H.S end 1794 3812 a Black -1 x FF(18)p Black 1894 3744 a SDict begin H.R end 1894 3744 a 1894 3811 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.Rei05) cvn H.B /ANN pdfmark end 1894 3811 a FF(])h(sho)n(wing)e(that)h(the)h (reachability)f(problem)257 3932 y(in)25 b(undirected)h(graphs)f(can)h (be)f(computed)g(in)g FD(L)p FF(,)g(has)h(an)f(important)f(consequence) i(for)f(the)257 4052 y(construction)g(of)h(uni)n(v)o(ersal)f(e)o (xploration)f(sequences)i(in)g(logarithmic)e(space.)35 b(This)25 b(f)o(act)i(is)257 4173 y(used)e(in)f(some)h(of)f(the)h (isomorphism)d(algorithms.)404 4297 y(F)o(or)32 b(a)g FB(d)s FF(-re)o(gular)g(graph)e FB(G)r FF(,)k(a)e(numbering)f(of)h(the) g(edges)g(and)h(a)f(starting)f(edge)h FB(e)3354 4312 y FA(0)3392 4297 y FF(,)i(a)257 4417 y(sequence)28 b(\()p FC(\034)730 4432 y FA(1)768 4417 y FC(;)17 b(\034)857 4432 y FA(2)894 4417 y FC(;)g(:)g(:)g(:)e(;)i(\034)1149 4432 y Fr(l)1172 4417 y FF(\))33 b Fv(2)h(f)p FF(0)p FC(;)17 b(:)g(:)g(:)e(;)i FB(d)26 b Fv(\000)f FF(1)p Fv(g)1863 4381 y Fr(l)1913 4417 y FF(de\002nes)j(a)h(w)o(alk)e FC(v)2555 4432 y Fw(\000)p FA(1)2635 4417 y FC(;)17 b(v)2716 4432 y FA(0)2753 4417 y FC(;)g(:)g(:)g(:)e(;)i(v)3000 4432 y Fr(k)3063 4417 y FF(in)25 b FB(G)30 b FF(in)d(the)257 4538 y(follo)n(wing)d(w)o(ay:)31 b(starting)25 b(at)g FB(e)1358 4553 y FA(0)1424 4538 y FE(=)k FF(\()p FC(v)1588 4553 y Fw(\000)p FA(1)1668 4538 y FC(;)17 b(v)1749 4553 y FA(0)1786 4538 y FF(\),)26 b(for)f(each)h FB(i)f FF(if)g(\()p FC(v)2430 4553 y Fr(i)p Fw(\000)p FA(1)2529 4538 y FC(;)17 b(v)2610 4553 y Fr(i)2632 4538 y FF(\))26 b(is)e(the)i FB(k)r FF(-th)f(edge)g(of)g FC(v)3472 4553 y Fr(i)257 4658 y FF(then)g(\()p FC(v)526 4673 y Fr(i)549 4658 y FC(;)17 b(v)630 4673 y Fr(i)p Fz(+)p FA(1)728 4658 y FF(\))25 b(is)f(the)h FB(k)f FE(+)e FC(\034)1224 4673 y Fr(i)1247 4658 y FF(-th)j(edge)g(of)f FC(v)1742 4673 y Fr(i)1790 4658 y FF(modulo)g FB(d)s FF(.)404 4782 y(A)h(sequence)h (\()p FC(\034)972 4797 y FA(1)1010 4782 y FC(;)17 b(\034)1099 4797 y FA(2)1136 4782 y FC(;)g(:)g(:)g(:)e(;)i(\034)1391 4797 y Fr(l)1414 4782 y FF(\))29 b Fv(2)h(f)p FF(0)p FC(;)17 b(:)g(:)g(:)d(;)j FB(d)25 b Fv(\000)e FF(1)p Fv(g)2093 4746 y Fr(l)2141 4782 y FF(is)i(called)h(an)g(\()p FB(n)p FC(;)17 b FB(d)s FF(\))25 b FB(univer)o(sal)f(e)n(xplo-)257 4903 y(r)o(ation)19 b(sequence)i FF(if)g(for)g(e)n(v)o(ery)f(connected) g FB(d)s FF(-re)o(gular)h(graph)f(with)g(at)h(most)e FB(n)i FF(v)o(ertices,)g(an)o(y)257 5023 y(numbering)e(of)i(its)e (edges)i(and)f(an)o(y)g(starting)f(edge,)i(the)f(w)o(alk)g(obtained)g (from)g(the)g(sequence)257 5143 y(e)o(xplores)k(all)h(the)f(v)o (ertices)h(in)f(the)h(graph.)404 5268 y(W)-8 b(e)37 b(use)g(the)g(f)o (act)h(that)e(such)h(uni)n(v)o(ersal)e(e)o(xploration)h(sequences)h(e)o (xist)f(and)h(can)g(be)257 5388 y(computed)24 b(in)h(logarithmic)d (space.)257 5410 y SDict begin H.S end 257 5410 a 257 5410 a SDict begin 14.5 H.A end 257 5410 a 257 5410 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.2.3) cvn H.B /DEST pdfmark end 257 5410 a Black Black eop end %%Page: 6 6 TeXDict begin 6 5 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 257 266 a SDict begin H.S end 257 266 a Black Black 257 266 a SDict begin H.R end 257 266 a 257 266 a SDict begin [ /View [/XYZ H.V] /Dest (page.6) cvn H.B /DEST pdfmark end 257 266 a Black Black 307 x Fp(Theor)n(em)54 b(2.3.)p Black 56 w FB([)940 574 y SDict begin H.S end 940 574 a Black -1 x FB(18)p Black 1040 507 a SDict begin H.R end 1040 507 a 1040 573 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.Rei05) cvn H.B /ANN pdfmark end 1040 573 a FB(])e(Ther)l(e)g(e)n(xists)g(a) g(lo)o(garithmic)e(space)i(algorithm)f(that)g(on)h(in-)257 693 y(put)25 b FF(\(1)493 657 y Fr(n)530 693 y FC(;)17 b FF(1)622 657 y Fr(d)660 693 y FF(\))26 b FB(pr)l(oduces)d(an)i FF(\()p FB(n)p FC(;)17 b FB(d)s FF(\))p FB(-univer)o(sal)23 b(e)n(xplor)o(ation)g(sequence)i(of)f(polynomial)f(size)o(.)257 869 y SDict begin H.S end 257 869 a 257 869 a SDict begin 14.5 H.A end 257 869 a 257 869 a SDict begin [ /View [/XYZ H.V] /Dest (section.3) cvn H.B /DEST pdfmark end 257 869 a 162 x FG(3)144 b(Planar)34 b Fo(3)p FG(-connected)f(Graph)h(Isomor)o (phism)257 1254 y FF(W)-8 b(einber)n(g')j(s)29 b([)769 1254 y SDict begin H.S end 769 1254 a Black FF(21)p Black 869 1187 a SDict begin H.R end 869 1187 a 869 1254 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.Wei66) cvn H.B /ANN pdfmark end 869 1254 a FF(])57 b FB(O)p FF(\()p FB(n)1114 1218 y FA(2)1152 1254 y FF(\))29 b(algorithm)e(for)i(testing) f(isomorphism)e(of)j(planar)f(3-connected)257 1375 y(graphs)h (constructs)f(for)h(a)h(graph)d FB(G)k FF(a)e(code)g(with)g(respect)g (to)f(each)i(one)f(of)g(its)f(edges)h(and)257 1495 y(both)22 b(rotation)f(schemes.)29 b(Of)22 b(all)g(these)g(codes,)g(the)g(le)o (xicographical)f(smallest)f(one)i(is)g(used)257 1615 y(as)28 b(a)g(canonical)f(form)h(for)e FB(G)r FF(.)38 b(W)-8 b(einber)n(g')j(s)28 b(algorithm)e(does)h(not)g(w)o(ork)g (within)g(logspace,)257 1736 y(because)32 b(the)g(v)o(ertices)f(and)h (edges)f(ha)n(v)o(e)h(to)f(be)h(stored.)51 b(Thierauf)32 b(and)f(W)-8 b(agner)32 b(sho)n(wed)257 1856 y(in)e([)398 1858 y SDict begin H.S end 398 1858 a Black -2 x FF(19)p Black 498 1789 a SDict begin H.R end 498 1789 a 498 1856 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.TW08) cvn H.B /ANN pdfmark end 498 1856 a FF(])g(ho)n(w)e(to)i(construct)f(a)h (di)p FE(\013)p FF(erent)f(code)h(in)g FD(UL)o FF(.)46 b(Some)30 b(months)e(later)l(,)j(using)e(Rein-)257 1977 y(golds)36 b(results)g(on)g(logarithmic)f(space)i(uni)n(v)o(ersal)d(e)o (xploration)h(sequences)i([)3057 1978 y SDict begin H.S end 3057 1978 a Black -1 x FF(18)p Black 3157 1909 a SDict begin H.R end 3157 1909 a 3157 1977 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.Rei05) cvn H.B /ANN pdfmark end 3157 1977 a FF(],)j(Datta,)257 2097 y(Limaye)23 b(and)f(Nimbhorkar)g (impro)o(v)o(ed)f(this)h(to)h(an)g(isomorphism)d(algorithm)h(for)i (planar)g(3-)257 2217 y(connected)i(graphs)f(that)g(w)o(orks)g(in)g (logarithmic)e(space)j([)2281 2218 y SDict begin H.S end 2281 2218 a Black -1 x FF(5)p Black 2331 2150 a SDict begin H.R end 2331 2150 a 2331 2217 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.DLN08) cvn H.B /ANN pdfmark end 2331 2217 a FF(].)31 b(W)-8 b(e)25 b(describe)f(both)g(results)g(in)257 2338 y(this)g(section.)257 2483 y SDict begin H.S end 257 2483 a 257 2483 a SDict begin 14.5 H.A end 257 2483 a 257 2483 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.3.1) cvn H.B /DEST pdfmark end 257 2483 a 148 x Ft(3.1)119 b(An)31 b(isomor)o(phism)e(algorithm)h(in)g Fn(UL)d Fm(\\)g Fn(coUL)257 2722 y SDict begin H.S end 257 2722 a 257 2722 a SDict begin 14.5 H.A end 257 2722 a 257 2722 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.3.1) cvn H.B /DEST pdfmark end 257 2722 a Black 96 x Fp(Theor)n(em)k(3.1.)p Black 44 w FB([)905 2820 y SDict begin H.S end 905 2820 a Black -2 x FB(19)p Black 1004 2752 a SDict begin H.R end 1004 2752 a 1004 2818 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.TW08) cvn H.B /ANN pdfmark end 1004 2818 a FB(])f(The)f(isomorphism)f(pr)l (oblem)g(for)h(planar)-11 b(,)28 b FF(3)p FB(-connected)h(gr)o(aphs)f (is)257 2939 y(in)d FD(UL)d Fv(\\)g FD(coUL)p FB(.)404 3131 y FF(Let)34 b(\()5 b FB(s)p FC(;)17 b FB(t)r FF(\))33 b(be)h(a)h(designated)e(edge)i(and)f FC(\032)g FF(be)g(a)h(rotation)e (scheme)h(for)e FB(G)r FF(.)59 b(The)34 b(con-)257 3251 y(struction)e(has)h(three)h(steps:)46 b(First,)35 b(we)e(compute)g(a)g (canonical)g(spanning)f(tree)i FB(T)45 b FF(for)31 b FB(G)r FF(.)257 3372 y(Second,)f(with)d(help)h(of)g(this)f(spanning)g (tree)h(and)g(the)g(rotation)g(function)f FC(\032)h FF(we)g(perform)g (a)257 3492 y(depth-\002rst)h(tra)n(v)o(ersal)g(on)g(the)g(edges)g(of)h (the)f(graph)g(and)g(construct)g(a)g(canonical)h(list)g FB(L)g FF(of)257 3612 y(all)c(edges)f(of)f FB(G)r FF(.)33 b(Finally)-6 b(,)25 b(we)h(rename)g(the)f(v)o(ertices)g(depending)g(on) h(the)f(position)f(of)i(their)257 3733 y(\002rst)f(occurrence)h(in)f (the)f(list)i FB(L)p FF(.)404 3853 y(W)-8 b(e)23 b(will)g(see)g(that)g (the)g(spanning)f(tree)h(in)g(step)g(1)g(can)h(be)f(computed)f(in)h (\(the)g(functional)257 3973 y(v)o(ersion)34 b(of\))h FD(UL)29 b Fv(\\)h FD(coUL)p FF(.)60 b(The)34 b(list)g(and)h(the)f (renaming)g(in)g(step)g(2)h(and)f(step)h(3)f(can)h(be)257 4094 y(computed)24 b(in)h FD(FL)o FF(.)404 4214 y(The)33 b(o)o(v)o(erall)f(algorithm)g(has)i(to)f(decide)h(whether)f(tw)o(o)g (gi)n(v)o(en)f(graphs)f FB(G)36 b FF(and)f FB(H)k FF(are)257 4335 y(isomorphic.)j(T)-8 b(o)29 b(do)g(so,)h(we)g(\002x)f(\()5 b FB(s)p FC(;)17 b FB(t)r FF(\))28 b(and)h FC(\032)g FF(for)f FB(G)j FF(and)e(c)o(ycle)g(through)f(all)h(edges)g(of)i FB(H)257 4455 y FF(as)k(designated)f(edge)h(and)f(the)g(tw)o(o)h (possible)e(embeddings)g(of)k FB(H)t FF(.)61 b(Then)32 b FB(G)37 b FF(and)f FB(H)k FF(are)257 4575 y(isomorphic)23 b(if)i(and)f(only)f(if)i(the)f(canonical)g(forms)g(for)f FB(G)j FF(and)h FB(H)i FF(match.)h(It)24 b(is)g(not)g(hard)h(to)257 4696 y(see)g(that)g(this)f(outer)g(loop)g(w)o(orks)h(in)f(logspace.)257 4960 y Fp(Step)i(1:)31 b(Construction)26 b(of)f(a)g(canonical)g (spanning)g(tr)n(ee)257 5047 y SDict begin H.S end 257 5047 a 257 5047 a SDict begin 14.5 H.A end 257 5047 a 257 5047 a SDict begin [ /View [/XYZ H.V] /Dest (section*.1) cvn H.B /DEST pdfmark end 257 5047 a 100 x FF(W)-8 b(e)40 b(sho)n(w)e(that)h(the)g(follo)n(wing)e(problem)i(can)g(be)h(solv)o(ed) e(in)h(unambiguous)e(logspace.)257 5268 y(Gi)n(v)o(en,)19 b(an)h(undirected)f(graph)f FB(G)29 b FE(=)f FF(\()p FB(V)-9 b FC(;)19 b FB(E)5 b FF(\),)21 b(a)f(rotation)e(scheme)h FC(\032)h FF(for)d FB(G)r FF(,)k(and)e(a)h(designated)257 5388 y(edge)35 b(\()5 b FB(s)p FC(;)17 b FB(t)r FF(\))44 b Fv(2)k FB(E)5 b FF(,)37 b(output)c(a)i(canonical)f(spanning)g(tree)h FB(T)57 b Fv(\022)48 b FB(E)40 b FF(of)32 b FB(G)r FF(,)37 b(which)d(does)g(not)p Black Black eop end %%Page: 7 7 TeXDict begin 7 6 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 257 266 a SDict begin H.S end 257 266 a Black Black 257 266 a SDict begin H.R end 257 266 a 257 266 a SDict begin [ /View [/XYZ H.V] /Dest (page.7) cvn H.B /DEST pdfmark end 257 266 a Black 307 x FF(depend)24 b(on)f(the)g(input)g(representation)g(of)g FC(\032)g FF(for)f FB(G)r FF(,)i(an)o(y)f(representation)g(will)f (result)h(in)g(the)257 693 y(same)i(spanning)f(tree)h FB(T)12 b FF(.)404 815 y(The)35 b(idea)h(to)e(construct)h(the)g (spanning)g(tree)g(is)g(to)g(tra)n(v)o(erse)f FB(G)j FF(with)e(a)g(breadth-\002rst)257 935 y(search)c(starting)e(at)h(node) 35 b FB(s)p FF(.)46 b(The)30 b(neighbors)f(of)h(a)g(node)g(are)g (visited)f(in)h(the)g(order)g(gi)n(v)o(en)257 1056 y(by)22 b(the)h(rotation)e(scheme)h FC(\032)p FF(.)30 b(Since)22 b(the)h(algorithm)d(should)i(w)o(ork)g(in)g(logspace,)g(we)h(cannot)257 1176 y(a)p FE(\013)p FF(ord)29 b(to)e(store)h(all)g(the)g(nodes)g(that) g(we)g(already)h(visited,)f(as)g(in)g(a)g(standard)g(breadth-\002rst) 257 1296 y(search.)k(W)-8 b(e)25 b(get)f(around)h(this)f(problem)g(by)g (w)o(orking)g(with)g(distances)h(between)f(nodes.)404 1418 y(W)-8 b(e)25 b(start)f(with)g(the)h(nodes)f(at)g(distance)h(1)f (from)30 b FB(s)p FF(.)g(That)25 b(is,)f(write)g(\()5 b FB(s)p FC(;)17 b(v)p FF(\))25 b(on)f(the)h(output)257 1538 y(tape,)41 b(for)d(all)g FC(v)52 b Fv(2)g FE(\000)p FF(\()5 b FB(s)p FF(\).)70 b(No)n(w)37 b(let)h FB(d)54 b Fq(>)e FF(2)37 b(and)h(assume)f(that)h(we)g(ha)n(v)o(e)f(already)h (con-)257 1659 y(structed)26 b FB(T)38 b FF(up)25 b(to)h(nodes)f(at)h (distance)f Fq(6)30 b FB(d)25 b Fv(\000)f FF(1)h(from)31 b FB(s)p FF(.)i(Then)26 b(we)g(consider)f(the)h(nodes)f(at)257 1779 y(distance)f FB(d)j FF(from)i FB(s)p FF(.)h(Let)24 b FC(w)h FF(be)f(a)h(node)f(with)f(distance)h FD(d)p FF(\()5 b FB(s)p FC(;)17 b(w)p FF(\))27 b FE(=)h FB(d)s FF(.)i(W)-8 b(e)25 b(connect)f FC(w)h FF(to)e(the)257 1900 y(tree)j(constructed)f(so)g(f)o(ar)h(by)f(computing)f(a)h (shortest)g(path)g(from)30 b FB(s)25 b FF(to)g FC(w)p FF(.)33 b(Ambiguities)23 b(are)257 2020 y(resolv)o(ed)j(by)f(using)g (the)h(\002rst)h(feasible)f(edge)g(according)g(to)f FC(\032)p FF(.)35 b(W)-8 b(e)26 b(start)g(with)f(\()5 b FB(s)p FC(;)17 b FB(t)r FF(\))25 b(as)h(the)257 2140 y(acti)n(v)o(e)e(edge)h (\()p FB(u)p FC(;)17 b(v)p FF(\).)p Black 403 2350 a Fv(\017)p Black 48 w FF(If)38 b FD(d)p FF(\()p FB(u)p FC(;)17 b(w)p FF(\))51 b FC(>)h FD(d)p FF(\()p FC(v;)17 b(w)p FF(\),)41 b(then)c(\()p FB(u)p FC(;)17 b(v)p FF(\))38 b(is)f(the)g(\002rst)h(edge)g(encountered)f(that)h(is)f(on)g(a)501 2471 y(shortest)18 b(path)h(from)g FB(u)g FF(to)g FC(w)p FF(.)29 b(Therefore)20 b(we)f(go)g(from)g FB(u)g FF(to)f FC(v)i FF(and)f(start)g(searching)f(the)501 2591 y(ne)o(xt)28 b(edge)h(from)f FC(v)p FF(.)43 b(As)28 b(starting)g(edge)h(we)g(tak)o (e)f FC(\032)2319 2606 y Fs(v)2349 2591 y FF(\()p FC(v;)17 b FB(u)p FF(\),)30 b(the)e(successor)h(of)f(\()p FC(v;)17 b FB(u)p FF(\).)501 2711 y(This)24 b(is)h(the)f(ne)n(w)h(acti)n(v)o(e)f (edge.)p Black 403 2921 a Fv(\017)p Black 48 w FF(If)30 b FD(d)p FF(\()p FB(u)p FC(;)17 b(w)p FF(\))36 b Fq(6)h FD(d)p FF(\()p FC(v;)17 b(w)p FF(\),)31 b(then)e(\()p FB(u)p FC(;)17 b(v)p FF(\))29 b(is)g(not)g(on)h(a)g(shortest)e(path)h (from)h FB(u)f FF(to)g FC(w)p FF(.)45 b(Then)501 3041 y(we)25 b(proceed)h(with)e FC(\032)1236 3056 y Fr(u)1273 3041 y FF(\()p FB(u)p FC(;)17 b(v)p FF(\))24 b(as)h(the)g(ne)n(w)f (acti)n(v)o(e)g(edge.)257 3251 y(After)33 b FB(d)d Fv(\000)f FF(1)j(steps)g(in)g(direction)g(of)g FC(w)p FF(,)j(the)d(node)h FC(v)f FF(of)h(the)f(acti)n(v)o(e)g(edge)h(\()p FB(u)p FC(;)17 b(v)p FF(\))32 b(is)g(a)h(pre-)257 3372 y(decessor)d(of)f FC(w)g FF(on)g(a)h(shortest)e(path)g(from)34 b FB(s)29 b FF(to)g FC(w)p FF(.)44 b(Then)29 b(we)g(write)g(\()p FC(v;)17 b(w)p FF(\))30 b(on)f(the)g(output)257 3492 y(tape.)404 3614 y(The)c(spanning)f(tree)i FB(T)38 b FF(is)25 b(canonical,)g(because)h(its)f(construction)f(depends)h(only)g (on)g FC(\032)p FF(,)257 3734 y(edge)h(\()5 b FB(s)p FC(;)17 b FB(t)r FF(\),)26 b(and)g(edge)g(set)h FB(E)5 b FF(.)34 b(The)26 b(follo)n(wing)e(\002gure)j(sho)n(ws)d(an)i(e)o (xample)f(of)h(a)g(spanning)257 3854 y(tree)20 b FB(T)31 b FF(for)19 b(a)h(graph)d FB(G)k FF(with)d(rotation)g(function)g FC(\032)h FF(which)g(arranges)g(the)g(edges)g(in)g(clockwise)257 3975 y(order)25 b(around)g(each)g(v)o(erte)o(x)f(of)f FB(G)r FF(.)p Black Black Black 257 5055 a currentpoint currentpoint translate 0.9 0.9 scale neg exch neg exch translate 257 5055 a 1918 4811 a FC(\032)1969 4826 y Fs(v)1995 4836 y Fl(3)1918 4699 y FC(\032)1969 4714 y Fs(v)1995 4724 y Fl(2)1918 4585 y FC(\032)1969 4600 y Fs(v)1995 4610 y Fl(1)1918 4476 y FC(\032)1969 4491 y Fr(t)2069 4366 y FE(=)28 b FF(\()d(\()5 b FB(s)p FC(;)17 b FB(t)r FF(\))24 b(\()5 b FB(s)p FC(;)17 b(v)2582 4381 y FA(1)2619 4366 y FF(\))25 b(\()5 b FB(s)p FC(;)17 b(v)2835 4381 y FA(2)2872 4366 y FF(\))25 b(\))2069 4479 y FE(=)j FF(\()d(\()p FB(t)r FC(;)d FB(s)p FF(\))i(\()p FB(t)r FC(;)17 b(v)2568 4494 y FA(3)2605 4479 y FF(\))25 b(\()p FB(t)r FC(;)17 b(v)2807 4494 y FA(1)2844 4479 y FF(\))25 b(\))2069 4593 y FE(=)j FF(\()d(\()p FC(v)2290 4608 y FA(1)2328 4593 y FC(;)d FB(s)p FF(\))i(\()p FC(v)2543 4608 y FA(1)2581 4593 y FC(;)17 b FB(t)r FF(\))24 b(\()p FC(v)2782 4608 y FA(1)2820 4593 y FC(;)17 b(v)2901 4608 y FA(3)2938 4593 y FF(\))25 b(\()p FC(v)3068 4608 y FA(1)3105 4593 y FC(;)17 b(v)3186 4608 y FA(2)3223 4593 y FF(\))25 b(\))2069 4706 y FE(=)j FF(\()d(\()p FC(v)2290 4721 y FA(2)2328 4706 y FC(;)d FB(s)p FF(\))i(\()p FC(v)2543 4721 y FA(2)2581 4706 y FC(;)17 b(v)2662 4721 y FA(1)2699 4706 y FF(\))25 b(\()p FC(v)2829 4721 y FA(2)2867 4706 y FC(;)17 b(v)2948 4721 y FA(3)2985 4706 y FF(\))25 b(\))2069 4819 y FE(=)j FF(\()d(\()p FC(v)2290 4834 y FA(3)2328 4819 y FC(;)17 b FB(t)r FF(\))24 b(\()p FC(v)2529 4834 y FA(3)2567 4819 y FC(;)17 b(v)2648 4834 y FA(2)2685 4819 y FF(\))25 b(\()p FC(v)2815 4834 y FA(3)2852 4819 y FC(;)17 b(v)2933 4834 y FA(1)2970 4819 y FF(\))25 b(\))1918 4363 y FC(\032)1972 4378 y Fr(s)1918 4212 y FC(\032)2069 4207 y FE(=)j Fv(f)p FC(\032)2247 4222 y Fr(s)2277 4207 y FC(;)17 b(\032)2370 4222 y Fr(t)2392 4207 y FC(;)g(\032)2485 4222 y Fs(v)2511 4232 y Fl(1)2544 4207 y FC(;)g(\032)2637 4222 y Fs(v)2663 4232 y Fl(2)2695 4207 y FC(;)g(\032)2788 4222 y Fs(v)2814 4232 y Fl(3)2847 4207 y Fv(g)p 0 setgray 3 setlinewidth np 861 4988 18 0 360.001 arc fil 0 setgray np 861 4988 18 0 360.001 arc st 3 setlinewidth np 861 4988 18 0 360.001 arc st 0 setgray 3 setlinewidth np 1314 4535 18 0 360.001 arc fil 0 setgray np 1314 4535 18 0 360.001 arc st 3 setlinewidth np 1314 4535 18 0 360.001 arc st 0 setgray 3 setlinewidth np 861 4535 18 0 360.001 arc fil 0 setgray np 861 4535 18 0 360.001 arc st 3 setlinewidth np 861 4535 18 0 360.001 arc st 0 setgray 3 setlinewidth np 408 4535 18 0 360.001 arc fil 0 setgray np 408 4535 18 0 360.001 arc st 3 setlinewidth np 408 4535 18 0 360.001 arc st 0 setgray 3 setlinewidth np 634 4082 18 0 360.001 arc fil 0 setgray np 634 4082 18 0 360.001 arc st 3 setlinewidth np 634 4082 18 0 360.001 arc st 7 setlinewidth np 860.6 4987.6 a 860.6 4534.6 li st 7 setlinewidth np 860.6 4987.6 a 1313.6 4534.6 li st 7 setlinewidth np 860.6 4987.6 a 407.6 4534.6 li st 3 setlinewidth np 633.8 4081.6 a 860.6 4534.6 li st 3 setlinewidth np 633.8 4081.6 a 1313.6 4534.6 li st 7 setlinewidth np 633.8 4081.6 a 407.6 4534.6 li st 3 setlinewidth np 407.6 4534.6 a 860.6 4534.6 li st 3 setlinewidth np 860.6 4534.6 a 1313.6 4534.6 li st 898 4595 a FC(v)937 4610 y FA(1)672 4066 y FC(v)711 4081 y FA(3)332 4609 y FB(t)752 5024 y(s)1352 4595 y FC(v)1391 4610 y FA(2)257 5055 y currentpoint currentpoint translate 1 0.9 div 1 0.9 div scale neg exch neg exch translate 257 5055 a Black 502 5259 a FF(Figure)25 b(1:)896 5259 y SDict begin H.S end 896 5259 a Black Black 896 5259 a SDict begin H.R end 896 5259 a 896 5259 a SDict begin [ /View [/XYZ H.V] /Dest (figure.1) cvn H.B /DEST pdfmark end 896 5259 a FF(A)g(3-connected)f(planar)h(graph)g(and)g(its)f (planar)h(rotation)f(scheme.)p Black Black Black Black eop end %%Page: 8 8 TeXDict begin 8 7 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 257 266 a SDict begin H.S end 257 266 a Black Black 257 266 a SDict begin H.R end 257 266 a 257 266 a SDict begin [ /View [/XYZ H.V] /Dest (page.8) cvn H.B /DEST pdfmark end 257 266 a Black 404 573 a FF(Except)25 b(for)h(the)f(computation)f(of)h(the)g (distances,)g(the)h(algorithm)e(w)o(orks)h(in)g(logspace.)257 693 y(W)-8 b(e)25 b(ha)n(v)o(e)e(to)h(store)g(the)g(v)n(alues)f(of)h FB(d)s FF(,)f FB(k)r FF(,)h FB(u)g FF(and)g FC(v)p FF(,)g(and)g(the)g (position)e(of)i FC(w)p FF(,)h(plus)e(some)h(e)o(xtra)257 814 y(space)29 b(for)g(doing)e(calculations.)41 b(Thierauf)28 b(and)g(W)-8 b(agner)29 b(also)f(pro)o(v)o(ed,)g(that)g(distances)g(in) 257 934 y(planar)h(graphs)g(can)g(be)h(computed)e(in)g FD(UL)d Fv(\\)h FD(coUL)p FF(.)43 b(This)28 b(is)h(based)g(on)g(the)f (results)h(from)257 1054 y(Reinhard)e(and)f(Allender)g([)1227 1055 y SDict begin H.S end 1227 1055 a Black -1 x FF(17)p Black 1327 987 a SDict begin H.R end 1327 987 a 1327 1054 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.RA00) cvn H.B /ANN pdfmark end 1327 1054 a FF(])h(and)f(Bourk)o(e,)h(T)-7 b(e)n(w)o(ari,)27 b(and)f(V)-6 b(inodchandran)26 b([)3063 1054 y SDict begin H.S end 3063 1054 a Black FF(4)p Black 3113 987 a SDict begin H.R end 3113 987 a 3113 1054 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.BTV07) cvn H.B /ANN pdfmark end 3113 1054 a FF(],)h(that)e(the)257 1175 y(reachability)e(problem)f(for)h(planar)g(directed)g(graphs)f(is)h (in)f FD(UL)15 b Fv(\\)g FD(co)q(UL)p FF(.)30 b(Since)23 b FD(L)3129 1139 y Fk(UL)o Fw(\\)p Fk(coUL)3432 1175 y FE(=)257 1295 y FD(UL)f Fv(\\)h FD(coUL)i FF(the)f(canonical)h (spanning)f(tree)h(can)g(be)g(computed)f(in)g FD(UL)e Fv(\\)h FD(coUL)o FF(.)257 1557 y Fp(Step)j(2:)31 b(Computation)26 b(of)f(a)f(canonical)h(list)g(of)g(all)f(edges)257 1645 y SDict begin H.S end 257 1645 a 257 1645 a SDict begin 14.5 H.A end 257 1645 a 257 1645 a SDict begin [ /View [/XYZ H.V] /Dest (section*.2) cvn H.B /DEST pdfmark end 257 1645 a 100 x FF(W)l(ith)33 b FB(G)49 b FE(=)e FF(\()p FB(V)-9 b FC(;)19 b FB(E)5 b FF(\),)38 b(a)d(rotation)g(scheme)g FC(\032)g FF(for)e FB(G)r FF(,)38 b(a)d(spanning)f(tree)i FB(T)59 b Fv(\022)49 b FB(E)41 b FF(of)33 b FB(G)k FF(and)257 1865 y(a)h(designated)f(edge)h(\()5 b FB(s)p FC(;)17 b FB(t)r FF(\))51 b Fv(2)h FB(T)e FF(we)38 b(compute)f(a)h(canonical)f (list)i FB(L)g FF(of)f(all)f(edges)h(in)h FB(E)5 b FF(.)257 1986 y(The)33 b(list)g FB(L)h FF(then)e(still)f(contains)h(the)g (original)g(v)o(erte)o(x)f(names)h(in)e FB(G)r FF(,)35 b(it)c(does)i(not)f(depend)257 2106 y(otherwise)25 b(on)f(the)h(input)f (representation)g(of)h FC(\032)p FF(,)d FB(G)27 b FF(or)e FB(T)12 b FF(.)404 2226 y(The)39 b(idea)h(is)f(to)h(tra)n(v)o(erse)f (the)h(spanning)e(tree)i(in)f(a)h(depth-\002rst)g(manner)-5 b(.)74 b(At)40 b(each)257 2347 y(v)o(erte)o(x)29 b FB(u)h FF(we)g(visit)e(all)h(incident)g(edges)h(of)f FB(u)h FF(in)f(a)h(c)o(yclic)f(manner)h(according)g(to)f FC(\032)3246 2362 y Fr(u)3313 2347 y FF(until)257 2467 y(the)e(ne)o(xt)f(edge)h FB(e)h FF(of)f(the)f(spanning)g(tree)h(is)g(reached.)38 b(W)-8 b(e)27 b(go)g(do)n(wn)f(the)g(tree)i(along)e FB(e)h FF(and)257 2588 y(recursi)n(v)o(ely)i(do)g(the)g(same)g(at)h(the)f (node)g(reached.)45 b(At)29 b(some)g(point)f(we)i(will)f(encounter)g FB(e)257 2708 y FF(again)c(and)f(come)h(back)g(to)f FB(u)p FF(.)31 b(Then)25 b(we)g(continue)f(to)g(output)g(the)g(edges)h (incident)f(to)g FB(u)p FF(.)404 2828 y(More)k(formally)-6 b(,)28 b(we)g(start)g(the)g(tra)n(v)o(ersal)g(with)g(edge)h(\()5 b FB(s)p FC(;)17 b FB(t)r FF(\))27 b(as)h(the)h(acti)n(v)o(e)e(edge)h (\()p FB(u)p FC(;)17 b(v)p FF(\).)257 2949 y(W)-8 b(e)20 b(write)e(\()p FB(u)p FC(;)f(v)p FF(\))i(on)g(the)f(output)g(tape)h (and)g(then)f(compute)g(the)h(ne)o(xt)f(acti)n(v)o(e)g(edge)h(as)g (follo)n(ws:)p Black 403 3130 a Fv(\017)p Black 48 w FF(If)h(\()p FB(u)p FC(;)d(v)p FF(\))27 b Fv(2)h FB(T)k FF(then)19 b(we)g(w)o(alk)g(depth-\002rst)g(in)g FB(T)32 b FF(from)19 b FB(u)g FF(to)g FC(v)p FF(,)h(consider)f(the)g(edge)g(\() p FC(v;)e FB(u)p FF(\))501 3251 y(and)25 b(tak)o(e)g FC(\032)911 3266 y Fs(v)941 3251 y FF(\()p FC(v;)17 b FB(u)p FF(\))24 b(its)h(successor)f(according)h(to)f FC(\032)2259 3266 y Fs(v)2289 3251 y FF(.)p Black 403 3447 a Fv(\017)p Black 48 w FF(If)i(\()p FB(u)p FC(;)17 b(v)p FF(\))27 b Fj(<)h FB(T)37 b FF(then)25 b(we)g(proceed)g (breadth-\002rst)g(with)f FC(\032)2433 3462 y Fr(u)2470 3447 y FF(\()p FB(u)p FC(;)17 b(v)p FF(\).)257 3628 y(This)i(step)f(is) h(repeated)h(until)e(we)h(entirely)g(tra)n(v)o(ersed)h FB(E)25 b FF(and)19 b(the)g(acti)n(v)o(e)f(edge)h(is)g(again)f(\()5 b FB(s)p FC(;)17 b FB(t)r FF(\).)257 3749 y(Ev)o(ery)29 b(undirected)g(edge)g(is)f(encountered)h(e)o(xactly)g(once)g(in)f(each) i(direction.)43 b(According)257 3869 y(to)25 b(Figure)645 3869 y SDict begin H.S end 645 3869 a Black FF(1)p Black 695 3802 a SDict begin H.R end 695 3802 a 695 3869 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (figure.1) cvn H.B /ANN pdfmark end 695 3869 a 25 w FF(the)f(canonical)h(list)h FB(L)g FF(is)e(the)h(follo)n(wing.)456 4104 y FB(L)j FE(=)125 b FF(\()5 b FB(s)p FC(;)17 b FB(t)r FF(\)\()p FB(t)r FC(;)g(v)1053 4119 y FA(3)1089 4104 y FF(\)\()p FC(v)1194 4119 y FA(3)1232 4104 y FC(;)g(v)1313 4119 y FA(2)1350 4104 y FF(\)\()p FC(v)1455 4119 y FA(3)1493 4104 y FC(;)g(v)1574 4119 y FA(1)1611 4104 y FF(\)\()p FC(v)1716 4119 y FA(3)1754 4104 y FC(;)g FB(t)r FF(\)\()p FB(t)r FC(;)g(v)2003 4119 y FA(1)2039 4104 y FF(\)\()p FB(t)r FC(;)22 b FB(s)p FF(\))727 4224 y(\()5 b FB(s)p FC(;)17 b(v)885 4239 y FA(1)922 4224 y FF(\)\()p FC(v)1027 4239 y FA(1)1065 4224 y FC(;)g FB(t)r FF(\)\()p FC(v)1242 4239 y FA(1)1279 4224 y FC(;)g(v)1360 4239 y FA(3)1397 4224 y FF(\)\()p FC(v)1502 4239 y FA(1)1540 4224 y FC(;)g(v)1621 4239 y FA(2)1658 4224 y FF(\)\()p FC(v)1763 4239 y FA(1)1801 4224 y FC(;)22 b FB(s)p FF(\))727 4344 y(\()5 b FB(s)p FC(;)17 b(v)885 4359 y FA(2)922 4344 y FF(\)\()p FC(v)1027 4359 y FA(2)1065 4344 y FC(;)g(v)1146 4359 y FA(1)1183 4344 y FF(\)\()p FC(v)1288 4359 y FA(2)1326 4344 y FC(;)g(v)1407 4359 y FA(3)1444 4344 y FF(\)\()p FC(v)1549 4359 y FA(2)1587 4344 y FC(;)22 b FB(s)p FF(\))257 4599 y Fp(Step)k(3:)31 b(Renaming)26 b(the)f(v)o(ertices)257 4686 y SDict begin H.S end 257 4686 a 257 4686 a SDict begin 14.5 H.A end 257 4686 a 257 4686 a SDict begin [ /View [/XYZ H.V] /Dest (section*.3) cvn H.B /DEST pdfmark end 257 4686 a 100 x FF(The)30 b(last)f(step)f(is)h(to)g(rename)h(the)f(v)o(ertices)g(in)g(the)g(list) h FB(L)h FF(so)e(that)g(the)o(y)f(become)i(indepen-)257 4907 y(dent)f(of)g(the)g(names)g(the)o(y)f(ha)n(v)o(e)h(in)e FB(G)r FF(.)43 b(This)29 b(is)f(achie)n(v)o(ed)h(as)g(follo)n(ws:)38 b(consider)28 b(the)h(\002rst)257 5027 y(occurrence)k(\(from)f(left\))g (of)g(node)g FC(v)g FF(in)i FB(L)p FF(.)53 b(Let)32 b FB(k)d Fv(\000)f FF(1)k(be)g(the)f(number)h(of)g(pairwise)f(dif-)257 5147 y(ferent)e(nodes)f(to)g(the)g(left)h(of)f FC(v)p FF(.)42 b(Then)28 b(all)h(occurrences)g(of)f FC(v)h FF(are)g(replaced)g (by)f FB(k)r FF(.)42 b(Recall,)257 5268 y(that)27 b FB(L)g FF(starts)d(with)h(the)g(edge)g(\()5 b FB(s)p FC(;)17 b FB(t)r FF(\).)31 b(Hence,)26 b(all)f(occurrences)h(of)31 b FB(s)25 b FF(get)g(replaced)g(by)g(1,)h(all)257 5388 y(occurrences)g(of)f FB(t)h FF(by)f(2,)f(and)h(so)g(on.)30 b(Call)25 b(the)g(ne)n(w)f(list)g FD(code)o FF(\()n FB(G)r FC(;)17 b(\032;)22 b FB(s)p FC(;)17 b FB(t)r FF(\).)p Black Black eop end %%Page: 9 9 TeXDict begin 9 8 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 257 266 a SDict begin H.S end 257 266 a Black Black 257 266 a SDict begin H.R end 257 266 a 257 266 a SDict begin [ /View [/XYZ H.V] /Dest (page.9) cvn H.B /DEST pdfmark end 257 266 a Black 404 573 a FF(Gi)n(v)o(en)35 b FB(L)g FF(as)f(input,)i(the)e (list)f FD(code)o FF(\()n FB(G)r FC(;)17 b(\032;)22 b FB(s)p FC(;)17 b FB(t)r FF(\))32 b(can)j(be)f(computed)f(in)h (logspace.)59 b(W)-8 b(e)257 693 y(start)28 b(with)g(the)g(\002rst)g (node)g FC(v)h FF(\(which)f(is)33 b FB(s)p FF(\))28 b(and)g(a)h (counter)f FB(k)r FF(,)h(that)f(counts)f(the)i(number)e(of)257 814 y(di)p FE(\013)p FF(erent)e(nodes)f(we)h(ha)n(v)o(e)g(seen)g(so)f (f)o(ar)-5 b(.)31 b(In)25 b(the)f(be)o(ginning,)f(we)i(set)g FB(k)30 b FE(=)e FF(1.)p Black 403 989 a Fv(\017)p Black 48 w FF(If)e FC(v)f FF(occurs)f(for)i(the)e(\002rst)h(time,)f(then)h (we)g(output)e FB(k)k FF(and)e(increase)g FB(k)i FF(by)d(1.)p Black 403 1183 a Fv(\017)p Black 48 w FF(If)k FC(v)g FF(occurs)g(already)g(to)f(the)g(left)h(of)f(the)h(current)g(position)d (then)j(we)f(ha)n(v)o(e)h(to)f(deter)n(-)501 1304 y(mine)h(the)h (number)l(,)g(that)f FC(v)h FF(got)f(at)h(its)f(\002rst)g(occurrence.) 44 b(T)-8 b(o)28 b(do)h(so,)g(we)g(determine)501 1424 y(the)24 b(\002rst)g(appearance)g(of)g FC(v)g FF(and)g(then)f(count)g (the)h(number)f(of)h(di)p FE(\013)p FF(erent)f(nodes)g(to)g(the)501 1544 y(left)i(of)g FC(v)g FF(at)g(its)f(\002rst)h(appearance.)257 1720 y(Then)g(we)g(go)g(to)f(the)h(ne)o(xt)f(node)g(in)j FB(L)p FF(.)404 1840 y(Consider)h(the)h(e)o(xample)f(from)h(abo)o(v)o (e.)41 b(The)29 b(code)g(constructed)f(from)h(list)h FB(L)g FF(for)d FB(G)k FF(is)257 1961 y(as)25 b(follo)n(ws.)950 2241 y FB(L)k FE(=)107 b FF(\()5 b FB(s)p FC(;)17 b FB(t)r FF(\))129 b(\()p FB(t)r FC(;)17 b(v)1659 2256 y FA(3)1695 2241 y FF(\))107 b(\()p FC(v)1907 2256 y FA(3)1945 2241 y FC(;)17 b(v)2026 2256 y FA(2)2063 2241 y FF(\))83 b(\()p FC(v)2251 2256 y FA(3)2289 2241 y FC(;)17 b(v)2370 2256 y FA(1)2407 2241 y FF(\))90 b(\()p FC(v)2602 2256 y FA(3)2640 2241 y FC(;)17 b FB(t)r FF(\))89 b(\()p FB(t)r FC(;)17 b(v)2978 2256 y FA(1)3015 2241 y FF(\))96 b(\()p FB(t)r FC(;)22 b FB(s)p FF(\))414 2361 y FD(code)o FF(\()n FB(G)r FC(;)17 b(\032;)22 b FB(s)p FC(;)17 b FB(t)r FF(\))26 b FE(=)93 b FF(\(1)p FC(;)17 b FF(2\))120 b(\(2)p FC(;)17 b FF(3\))136 b(\(3)p FC(;)17 b FF(4\))136 b(\(3)p FC(;)17 b FF(5\))119 b(\(3)p FC(;)17 b FF(2\))96 b(\(2)p FC(;)17 b FF(5\))86 b(\(2)p FC(;)17 b FF(1\))652 2526 y(sequel)24 b(of)j FB(L)84 b FF(\()5 b FB(s)p FC(;)17 b(v)1338 2541 y FA(1)1375 2526 y FF(\))107 b(\()p FC(v)1587 2541 y FA(1)1624 2526 y FC(;)17 b FB(t)r FF(\))106 b(\()p FC(v)1907 2541 y FA(1)1945 2526 y FC(;)17 b(v)2026 2541 y FA(3)2063 2526 y FF(\))83 b(\()p FC(v)2251 2541 y FA(1)2289 2526 y FC(;)17 b(v)2370 2541 y FA(2)2407 2526 y FF(\))83 b(\()p FC(v)2595 2541 y FA(1)2633 2526 y FC(;)22 b FB(s)p FF(\))505 2647 y(sequel)i(of)h FD(code)92 b FF(\(1)p FC(;)17 b FF(5\))120 b(\(5)p FC(;)17 b FF(2\))136 b(\(5)p FC(;)17 b FF(3\))136 b(\(5)p FC(;)17 b FF(4\))119 b(\(5)p FC(;)17 b FF(1\))652 2812 y(sequel)24 b(of)j FB(L)84 b FF(\()5 b FB(s)p FC(;)17 b(v)1338 2827 y FA(2)1375 2812 y FF(\))83 b(\()p FC(v)1563 2827 y FA(2)1601 2812 y FC(;)17 b(v)1682 2827 y FA(1)1719 2812 y FF(\))83 b(\()p FC(v)1907 2827 y FA(2)1945 2812 y FC(;)17 b(v)2026 2827 y FA(3)2063 2812 y FF(\))99 b(\()p FC(v)2267 2827 y FA(2)2305 2812 y FC(;)22 b FB(s)p FF(\))505 2932 y(sequel)i(of)h FD(code)92 b FF(\(1)p FC(;)17 b FF(4\))120 b(\(4)p FC(;)17 b FF(5\))136 b(\(4)p FC(;)17 b FF(3\))136 b(\(4)p FC(;)17 b FF(1\))404 3111 y(It)29 b(remains)h(to)f(ar)n(gue)h(that)g(the)f(ne)n(w)g(names)h (of)g(the)f(nodes)h(are)g(independent)f(of)h(their)257 3231 y(names)d(in)e FB(G)r FF(.)38 b(Let)29 b FB(H)j FF(be)c(a)f(graph)h(which)e(is)h(isomorphic)f(to)f FB(G)r FF(,)i(and)h(let)f FC(')g FF(be)g(an)g(isomor)n(-)257 3352 y(phism)k(between)f FB(G)k FF(and)g FB(H)t FF(.)53 b(Note)32 b(that)f FC(\032)c Fv(\016)g FC(')32 b FF(is)f(a)h(rotation)f (scheme)h(for)i FB(H)t FF(.)53 b(Consider)257 3472 y(the)32 b(computation)f(of)h(the)g(code)g(for)g(graph)i FB(H)j FF(with)32 b(rotation)f(scheme)h FC(\032)27 b Fv(\016)g FC(')32 b FF(and)g(desig-)257 3593 y(nated)h(edge)h(\()p FC(')p FF(\()5 b FB(s)p FF(\))p FC(;)17 b(')p FF(\()p FB(t)r FF(\)\).)54 b(The)34 b(spanning)d(tree)j(computed)e(in)h(step)g (1)g(will)f(be)h FC(')p FF(\()p FB(T)12 b FF(\))33 b(and)257 3713 y(the)39 b(list)e(computed)h(in)g(step)g(2)h(will)e(be)i FC(')p FF(\()r FB(L)p FF(\).)73 b(No)n(w)38 b(the)g(abo)o(v)o(e)g (renaming)g(procedure)257 3833 y(will)33 b(gi)n(v)o(e)f(the)h(same)h (number)e(to)h(node)h FC(v)f FF(in)i FB(L)g FF(and)e(to)h(node)f FC(')p FF(\()p FC(v)p FF(\))g(in)g FC(')p FF(\()r FB(L)p FF(\).)58 b(F)o(or)34 b(e)o(xam-)257 3954 y(ple,)39 b(the)c(nodes)h FC(')p FF(\()5 b FB(s)p FF(\))35 b(and)72 b FC(')p FF(\()p FB(t)r FF(\))35 b(will)g(get)h(number)f(1)g(and)h(2,)j(respecti)n(v)o (ely)-6 b(.)61 b(It)36 b(follo)n(ws)257 4074 y(that)25 b FD(code)o FF(\()n FB(G)r FC(;)17 b(\032;)22 b FB(s)p FC(;)17 b FB(t)r FF(\))25 b FE(=)j FD(code)o FF(\()r FB(H)o FC(;)17 b(\032)22 b Fv(\016)g FC(';)17 b(')p FF(\()5 b FB(s)p FF(\))p FC(;)17 b(')p FF(\()p FB(t)r FF(\)\).)29 b(W)-8 b(e)25 b(summarize:)257 4095 y SDict begin H.S end 257 4095 a 257 4095 a SDict begin 14.5 H.A end 257 4095 a 257 4095 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.3.2) cvn H.B /DEST pdfmark end 257 4095 a Black 155 x Fp(Theor)n(em)37 b(3.2.)p Black 46 w FB([)913 4252 y SDict begin H.S end 913 4252 a Black -2 x FB(19)p Black 1013 4183 a SDict begin H.R end 1013 4183 a 1013 4250 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.TW08) cvn H.B /ANN pdfmark end 1013 4250 a FB(])d(Let)g(G)j(and)f(H)k(be)35 b(connected,)i(undir)l (ected)e(gr)o(aphs,)g(let)g FC(\032)3228 4265 y Fr(G)3316 4250 y FB(be)g(a)257 4370 y(r)l(otation)g(sc)o(heme)h(for)e(G)39 b(and)d FF(\()5 b FB(s)p FC(;)17 b FB(t)r FF(\))36 b FB(be)h(an)g(edg)o(e)g(in)d(G)r(.)66 b(Then)36 b(G)i(and)h(H)i(ar)l(e)c (isomor)n(-)257 4490 y(phic)28 b(i)p Fu(\013)g FB(ther)l(e)g(e)n(xists) g(a)g(r)l(otation)d(sc)o(heme)j FC(\032)1838 4505 y Fr(H)1921 4490 y FB(for)h(H)k(and)28 b(an)g(edg)o(e)g FF(\()p FB(u)p FC(;)17 b(v)p FF(\))28 b FB(in)i(H)j(suc)o(h)27 b(that)257 4611 y FD(code)p FF(\()n FB(G)r FC(;)17 b(\032)659 4626 y Fr(G)712 4611 y FC(;)22 b FB(s)p FC(;)17 b FB(t)r FF(\))26 b FE(=)i FD(code)o FF(\()r FB(H)o FC(;)17 b(\032)1426 4626 y Fr(H)1481 4611 y FC(;)g FB(u)p FC(;)g(v)p FF(\))p FB(.)404 4786 y FF(W)l(ith)27 b(a)h(v)o(ery)f(di)p FE(\013)p FF(erent)g(approach,)i(Datta,)f(Limaye)f(and)h(Nimbhorkar)e([)3005 4787 y SDict begin H.S end 3005 4787 a Black -1 x FF(5)p Black 3055 4719 a SDict begin H.R end 3055 4719 a 3055 4786 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.DLN08) cvn H.B /ANN pdfmark end 3055 4786 a FF(])i(impro)o(v)o(ed)257 4907 y(the)k(pre)n(vious)f(result)g(from)h FD(UL)c Fv(\\)f FD(coUL)32 b FF(to)g FD(L)p FF(.)52 b(Their)32 b(method)e(is)i(in)g (some)f(sense)h(much)257 5027 y(easier)24 b(since)g(it)e(a)n(v)n(oids)h (the)g(spanning)g(tree)h(construction)e(eliminating)f(the)i(distance)g (com-)257 5147 y(putations)28 b(\(the)i(part)f(in)58 b FD(UL)26 b Fv(\\)g FD(coUL)p FF(\).)44 b(It)30 b(uses)f(ho)n(we)n(v)o (er)f(the)h(concept)g(of)h(uni)n(v)o(ersal)d(e)o(x-)257 5268 y(ploration)h(sequence)h(and)g(the)g(non-tri)n(vial)e(f)o(act)j (that)e(such)h(sequences)g(can)g(be)g(computed)257 5388 y(in)c FD(L)p FF(.)257 5389 y SDict begin H.S end 257 5389 a 257 5389 a SDict begin 14.5 H.A end 257 5389 a 257 5389 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.3.3) cvn H.B /DEST pdfmark end 257 5389 a Black Black eop end %%Page: 10 10 TeXDict begin 10 9 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 257 266 a SDict begin H.S end 257 266 a Black Black 257 266 a SDict begin H.R end 257 266 a 257 266 a SDict begin [ /View [/XYZ H.V] /Dest (page.10) cvn H.B /DEST pdfmark end 257 266 a Black Black 307 x Fp(Theor)n(em)37 b(3.3.)p Black 46 w FB([)913 574 y SDict begin H.S end 913 574 a Black -1 x FB(5)p Black 963 507 a SDict begin H.R end 963 507 a 963 573 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.DLN08) cvn H.B /ANN pdfmark end 963 573 a FB(])d(The)h(isomorphism)e(pr)l (oblem)g(for)h(planar)-11 b(,)33 b FF(3)p FB(-connected)i(gr)o(aphs)e (is)257 693 y(in)25 b FD(L)p FB(.)404 917 y FF(The)j(idea)g(of)g(the)g (algorithm)f(is)g(to)h(use)g(a)g(uni)n(v)o(ersal)f(sequence)h([)2707 918 y SDict begin H.S end 2707 918 a Black -1 x FF(18)p Black 2807 850 a SDict begin H.R end 2807 850 a 2807 917 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.Rei05) cvn H.B /ANN pdfmark end 2807 917 a FF(])g(in)g(order)g(to)g(con-)257 1037 y(struct)22 b(a)g(canonical)g(code)g(for)h(a)f(gi)n(v)o(en)e (planar)i(3-connected)g(graph)e FB(G)r FF(.)30 b(Since)23 b(Reingolds')-5 b(s)257 1158 y(construction)32 b(requires)h(the)g (graph)h(to)e(ha)n(v)o(e)h(constant)g(de)o(gree,)i(there)e(is)g(a)h (preprocesing)257 1278 y(step)22 b(in)g(which)e FB(G)k FF(is)e(transformed)g(into)f(a)h(3-re)o(gular)g(colored)h(graph)d FB(G)2765 1242 y Fw(0)2814 1278 y FF(with)i(the)g(property)257 1398 y(that)e(tw)o(o)g(graphs)g(are)h(isomorphic)d(if)i(and)g(only)g (if)g(their)g(transformations)f(are)i(also)e(isomor)n(-)257 1519 y(phic)27 b(\(with)f(a)i(color)f(preserving)f(isomorphism\).)35 b(In)27 b(a)h(second)f(step)f(a)i(canonical)f(code)g(is)257 1639 y(computed.)55 b(The)33 b(code)g(is)f(speci\002c)i(to)e(the)h (choice)g(of)g(a)g(planar)g(embedding)f FC(\032)h FF(for)e FB(G)r FF(,)k(a)257 1760 y(starting)c(node)h(and)g(a)g(starting)f (edge.)52 b(Since)32 b(there)g(are)g(only)g(polynomially)d(man)o(y)i (pos-)257 1880 y(sible)f(choices)g(for)h(these)f(parameters,)h(for)g (tw)o(o)f(gi)n(v)o(en)f(graphs)f FB(G)k FF(and)g FB(H)t FF(,)h(a)d(logarithmic)257 2000 y(space)e(procedure)f(can)g(c)o(ycle)g (through)f(all)g(the)h(possibilities)d(and)j(decide)g(that)f(the)h (graphs)257 2121 y(are)f(isomorphic)d(if)i(and)g(only)f(if)h(the)f (canonical)h(codes)g(match)f(for)h(an)o(y)f(of)h(the)g(choices.)257 2413 y Fp(Step)h(1:)31 b(Making)25 b(the)h(graph)g(3-r)n(egular)257 2505 y SDict begin H.S end 257 2505 a 257 2505 a SDict begin 14.5 H.A end 257 2505 a 257 2505 a SDict begin [ /View [/XYZ H.V] /Dest (section*.4) cvn H.B /DEST pdfmark end 257 2505 a 104 x FF(Gi)n(v)o(en)h(a)h(3-connected)g(planar)g(graph)e FB(G)36 b FE(=)e FF(\()p FB(V)-9 b FC(;)19 b FB(E)5 b FF(\))28 b(and)g(a)g(planar)g(embedding)f FC(\032)h FF(we)g(con-)257 2729 y(struct)39 b(a)h(3-re)o(gular)g(planar)f(graph)f FB(G)1633 2693 y Fw(0)1700 2729 y FF(with)h(the)g(edges)h(colored)f (with)g(tw)o(o)g(colors.)e FB(G)3468 2693 y Fw(0)257 2850 y FF(might)30 b(not)g(be)h(3-connected,)h(ho)n(we)n(v)o(er)d(the)i (planar)g(embedding)e(from)g FB(G)k FF(will)c(be)i(inher)n(-)257 2970 y(ited)g(to)d FB(G)618 2934 y Fw(0)645 2970 y FF(.)49 b(Ev)o(ery)31 b(v)o(erte)o(x)f FC(v)h FF(of)e FB(G)k FF(is)d(replaced)i(in)c FB(G)2197 2934 y Fw(0)2255 2970 y FF(by)j(a)g(c)o(ycle)g Fv(f)p FC(v)2773 2985 y FA(1)2810 2970 y FC(;)17 b(:)g(:)g(:)e(;)i(v)3057 2985 y Fr(d)3096 2970 y Fv(g)30 b FF(\()p FB(d)k FF(is)c(the)257 3090 y(de)o(gree)36 b(of)g FC(v)p FF(\).)65 b(The)36 b FB(d)i FF(edges)e FB(e)1424 3105 y FA(1)1462 3090 y FC(;)17 b(:)g(:)g(:)e(;)i FB(e)1714 3105 y Fr(d)1788 3090 y FF(incident)35 b(with)h FC(v)g FF(in)d FB(G)38 b FF(are)f(no)n(w)e(respecti)n(v)o(ely) 257 3211 y(incident)g(to)g Fv(f)p FC(v)799 3226 y FA(1)836 3211 y FC(;)17 b(:)g(:)g(:)e(;)i(v)1083 3226 y Fr(d)1122 3211 y Fv(g)35 b FF(in)e FB(G)1375 3175 y Fw(0)1402 3211 y FF(.)63 b(The)35 b(edges)g(of)h(the)f(c)o(ycles)g(are)h(colored)g (with)e(color)h(1)257 3331 y(and)d(the)f(edges)h FB(e)889 3346 y FA(1)926 3331 y FC(;)17 b(:)g(:)g(:)e(;)i FB(e)1178 3346 y Fr(d)1249 3331 y FF(with)31 b(color)g(2.)51 b(The)32 b(obtained)f(graph)e FB(G)2712 3295 y Fw(0)2771 3331 y FF(is)i(3-re)o(gular)g(and)h(it)257 3451 y(is)k(not)f(hard)h(to)f (see)h(that)f(tw)o(o)g(graphs)f FB(G)k FF(and)f FB(H)k FF(are)36 b(isomorphic)f(if)g(and)h(only)f(if)h(their)257 3572 y(transformation)22 b FB(G)941 3536 y Fw(0)993 3572 y FF(and)27 b FB(H)1240 3536 y Fw(0)1293 3572 y FF(are)e(isomorphic)e (with)i(a)g(color)f(preserving)h(isomorphism.)257 3864 y Fp(Step)h(2:)31 b(Obtaining)25 b(the)h(canonical)f(code)257 3956 y SDict begin H.S end 257 3956 a 257 3956 a SDict begin 14.5 H.A end 257 3956 a 257 3956 a SDict begin [ /View [/XYZ H.V] /Dest (section*.5) cvn H.B /DEST pdfmark end 257 3956 a 104 x FF(On)31 b(input)e(an)i(edge-colored)g(graph)d FB(G)33 b FF(with)d FB(n)g FF(v)o(ertices,)i(maximum)d(de)o(gree)h(3,)i (a)f(planar)257 4180 y(embedding)19 b FC(\032)p FF(,)h(a)g(starting)f (v)o(erte)o(x)g FC(v)h FF(and)g(a)g(starting)f(edge)h FB(e)27 b FE(=)h FF(\()p FB(u)p FC(;)17 b(v)p FF(\),)k(we)f(w)o(ant)g (to)f(construct)257 4301 y(a)28 b(canon)f(for)f FB(G)r FF(.)39 b(F)o(or)27 b(this,)g(we)h(compute)e(\002rst)i(in)f (logarithmic)e(space)j(an)f(\()p FB(n)p FC(;)17 b FF(3\)-uni)n(v)o (ersal)257 4421 y(e)o(xploration)k(sequence)h Fv(U)t FF(.)31 b(Then,)22 b(starting)f(at)i FC(v)f FF(and)g FB(e)h FF(we)f(transv)o(erse)e FB(G)k FF(according)e(to)g Fv(U)257 4541 y FF(and)k FC(\032)f FF(gi)n(ving)f(the)h(list)i FB(L)f FF(of)g(the)f(visited)g(v)o(ertices)g(as)g(label.)33 b(W)-8 b(e)26 b(can)g(rename)g(the)f(v)o(ertices)257 4662 y(according)j(to)f(their)g(\002rst)h(occurrence)h(in)g FB(L)p FF(,)g(as)f(it)f(is)g(done)g(in)h(step)f(3)g(from)g(Theorem)3346 4663 y SDict begin H.S end 3346 4663 a Black -1 x FF(3.1)p Black 3470 4595 a SDict begin H.R end 3470 4595 a 3470 4662 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (theorem.3.1) cvn H.B /ANN pdfmark end 3470 4662 a FF(.)257 4782 y(Finally)-6 b(,)22 b(we)g(can)h(c)o(ycle)f(o)o(v)o(er)f(e)n(v)o(ery)h(possible)f (pair)h(\()p FB(i)p FC(;)32 b FB(j)p FF(\),)22 b(checking)g(whether)g (it)g(is)g(an)g(edge)257 4903 y(in)28 b(the)g(renamed)h(list)e(and)h (outputting)f(its)g(color)h(if)h(this)e(is)h(the)g(case.)42 b(This)27 b(output)g(can)i(be)257 5023 y(considered)c(as)g(a)g (canonical)g(colored)f(adjacenc)o(y)h(matrix)f(for)f FB(G)r FF(.)404 5147 y(The)30 b(authors)g(pro)o(v)o(e)f(that)h(this)g (method)f(is)h(correct)h(by)g(sho)n(wing)d(the)j(follo)n(wing:)39 b(F)o(or)257 5268 y(tw)o(o)30 b(graphs)f FB(G)805 5283 y FA(1)873 5268 y FF(and)f FB(G)1119 5283 y FA(2)1187 5268 y FF(with)i(their)g(respecti)n(v)o(e)g(embedding)f FC(\032)2563 5283 y FA(1)2600 5268 y FF(,)i FC(\032)2707 5283 y FA(2)2744 5268 y FF(,)h(with)e(starting)f(v)o(er)n(-)257 5388 y(tices)24 b FC(v)503 5403 y FA(1)540 5388 y FC(;)17 b(v)621 5403 y FA(2)682 5388 y FF(and)24 b(edges)f FB(e)1144 5403 y FA(1)1182 5388 y FC(;)17 b FB(e)1268 5403 y FA(2)1305 5388 y FF(,)24 b(if)g(the)f(canons)h(coincide)f(then)h(the)f(graphs)h (are)g(isomorphic,)p Black Black eop end %%Page: 11 11 TeXDict begin 11 10 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 257 266 a SDict begin H.S end 257 266 a Black Black 257 266 a SDict begin H.R end 257 266 a 257 266 a SDict begin [ /View [/XYZ H.V] /Dest (page.11) cvn H.B /DEST pdfmark end 257 266 a Black 307 x FF(and)24 b(moreo)o(v)o(er)l(,)f(if)f FB(G)1006 588 y FA(1)1067 573 y FF(is)h(isomorphic)g(to)e FB(G)1803 588 y FA(2)1840 573 y FF(,)j(then)g(there)g(is)f(some)g(choice)h(of)g (the)g(parame-)257 693 y(ters)h(that)f(mak)o(es)h(their)f(respecti)n(v) o(e)g(canons)h(equal.)257 890 y SDict begin H.S end 257 890 a 257 890 a SDict begin 14.5 H.A end 257 890 a 257 890 a SDict begin [ /View [/XYZ H.V] /Dest (section.4) cvn H.B /DEST pdfmark end 257 890 a 164 x FG(4)144 b(Planar)34 b(Graph)g(Isomor)o(phism)257 1285 y FF(In)23 b(this)f(section)g(we)h (describe)g(the)g(log-space)g(algorithm)e(for)i(planar)g(graph)g (isomorphism.)257 1405 y(A)28 b(pre)n(vious)f(step)g(to)n(w)o(ards)g (this)g(result)g(w)o(as)h(a)g(logarithmic)d(space)k(isomorphism)c(test) i(for)257 1526 y(partial)32 b(2-trees)g([)880 1526 y SDict begin H.S end 880 1526 a Black FF(1)p Black 930 1459 a SDict begin H.R end 930 1459 a 930 1526 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.ADK08) cvn H.B /ANN pdfmark end 930 1526 a FF(].)52 b(P)o(artial)32 b(2-trees)g(are)g(a)g (subclass)f(of)h(the)g(planar)g(graphs)g(coinciding)257 1646 y(with)21 b(that)g(of)h(series-parallel)f(graphs)h(and)f(contain)g (all)g(outer)n(-planar)h(graphs.)29 b(F)o(or)22 b(pro)o(ving)257 1767 y(this)32 b(result)g(Arvind,)i(Das)f(and)g(K\366bler)f(represent)h (a)g(partial)g(2-tree)g(as)g(a)g(tree)g(of)g(c)o(ycles.)257 1887 y(Similar)24 b(to)f(Lindell')-5 b(s)22 b(algorithm)h([)1499 1887 y SDict begin H.S end 1499 1887 a Black FF(14)p Black 1598 1820 a SDict begin H.R end 1598 1820 a 1598 1887 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.Lin92) cvn H.B /ANN pdfmark end 1598 1887 a FF(])i(the)o(y)d(compare)i(tw)o(o)g (such)f(tree)i(representations)e(up)257 2007 y(to)35 b(isomorphism,)f(de\002ning)h(a)g(canonical)f(ordering)h(procedure,)i (which)d(\002nally)h(gi)n(v)o(es)e(a)257 2128 y(canonization)24 b(algorithm.)404 2251 y(In)29 b(the)g(isomorphism)d(algorithm)i(for)h (general)h(planar)f(graphs)f(a)i(similar)e(representa-)257 2372 y(tion)20 b(is)f(used,)i(namely)f(a)g(tree)h(of)f(triconnected)g (components.)28 b(A)20 b FB(triconnected)f(component)257 2492 y FF(is)29 b(a)g(3-connected)g(graph,)g(a)g(c)o(ycle)g(or)g(a)g (3-bond,)g(i.e.)42 b(tw)o(o)29 b(v)o(ertices)f(connected)h(by)f(three) 257 2613 y(edges.)404 2736 y(W)-8 b(e)27 b(gi)n(v)o(e)f(a)h(log-space)g (algorithm)f(for)h(the)g FB(gr)o(aph)e(canonization)g(pr)l(oblem)h FF(for)h(planar)257 2857 y(graphs,)33 b(to)f(which)f(planar)g(graph)h (isomorphism)d(reduces.)52 b(The)31 b(canonization)g(in)l(v)n(olv)o(es) 257 2977 y(assigning)24 b(to)g(each)i(graph)e(an)h(isomorphism)d(in)l (v)n(ariant)i(string)g(of)h(polynomial)e(length.)404 3101 y(The)37 b(algorithm)f(decomposes)g(\002rst)i(the)f(planar)g (graph)h(into)e(its)h(biconnected)f(com-)257 3221 y(ponents)c(and)g (constructs)g(a)g FB(biconnected)g(component)g(tr)l(ee)h FF(in)f(log-space)g([)3014 3221 y SDict begin H.S end 3014 3221 a Black FF(1)p Black 3064 3154 a SDict begin H.R end 3064 3154 a 3064 3221 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.ADK08) cvn H.B /ANN pdfmark end 3064 3221 a FF(].)54 b(Then,)34 b(it)257 3342 y(further)40 b(decomposes)e(the)i (biconnected)f(planar)g(components)f(into)h(their)g(triconnected)257 3462 y(components)29 b(to)g(obtain)g(a)h(triconnected)f(component)f (tree)j(in)e(log-space.)45 b(Hopcroft)29 b(and)257 3582 y(T)-8 b(arjan)20 b([)562 3584 y SDict begin H.S end 562 3584 a Black -2 x FF(9)p Black 612 3515 a SDict begin H.R end 612 3515 a 612 3582 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.HT73) cvn H.B /ANN pdfmark end 612 3582 a FF(])g(presented)g(a)g(sequential)f(algorithm)f(for)j(the)e (decomposition)f(of)i(a)g(biconnected)257 3703 y(planar)36 b(graph)f(into)g(its)g(triconnected)g(components.)61 b(This)34 b(method)h(can)h(be)f(adapted)h(to)257 3823 y(w)o(ork)30 b(in)g(log-space.)45 b(The)30 b(algorithm)f(recursi)n(v)o (ely)f(remo)o(v)o(es)h(separating)g(pairs)h(from)f(the)257 3944 y(graph)k(and)g(puts)f(a)h(cop)o(y)g(of)g(the)f(separating)h(pair) g(in)f(each)i(of)f(the)f FB(split)g(components)g FF(so)257 4064 y(formed,)26 b(i.e.)35 b(the)26 b(nodes)f(in)h(the)g(separating)g (pair)g(are)h(connected)f(by)f(a)i FB(virtual)e(edg)o(e)p FF(.)34 b(The)257 4184 y(decomposition)d(stops)h(when)h(the)f(split)g (components)f(become)i(triconnected.)54 b(De\002ne)34 b(a)257 4305 y(node)27 b(for)g(each)h(separating)e(pair)h(and)g(each)g (component)f(and)h(connect)f(a)i FB(separ)o(ating)c(pair)257 4425 y(node)f FF(with)e(a)i FB(triconnected)f(component)g(node)g FF(if)g(the)h(separating)f(pair)g(is)g(contained)g(in)g(the)257 4545 y(component.)41 b(The)29 b(resulting)f(graph)g(is)g(a)h(tree,)h (the)f FB(triconnected)f(component)g(tr)l(ee)p FF(.)42 b(This)257 4666 y(decomposition)28 b(is)g(unique)h([)1303 4667 y SDict begin H.S end 1303 4667 a Black -1 x FF(15)p Black 1403 4599 a SDict begin H.R end 1403 4599 a 1403 4666 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.M37) cvn H.B /ANN pdfmark end 1403 4666 a FF(].)45 b(Datta)29 b(et.)44 b(al.)30 b([)2052 4667 y SDict begin H.S end 2052 4667 a Black -1 x FF(6)p Black 2102 4599 a SDict begin H.R end 2102 4599 a 2102 4666 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.DLNTW08) cvn H.B /ANN pdfmark end 2102 4666 a FF(])f(pro)o(v)o(e,)h(that)f(such)g(a)h(decomposition)257 4786 y(can)38 b(be)e(computed)g(in)h(log-space.)66 b(Figure)1878 4786 y SDict begin H.S end 1878 4786 a Black FF(2)p Black 1928 4719 a SDict begin H.R end 1928 4719 a 1928 4786 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (figure.2) cvn H.B /ANN pdfmark end 1928 4786 a 37 w FF(sho)n(ws)35 b(an)i(e)o(xample)f (of)h(the)f(decomposi-)257 4907 y(tion)24 b(of)h(a)g(biconnected)g (planar)g(graph)d FB(G)r FF(.)31 b(Its)25 b(triconnected)f(components)g (are)f FB(G)3141 4922 y FA(1)3178 4907 y FC(;)17 b(:)g(:)g(:)e(;)g FB(G)3458 4922 y FA(4)257 5027 y FF(and)36 b(the)f(corresponding)g (triconnected)g(component)f(tree)i(is)f FB(T)12 b FF(.)63 b(In)33 b FB(G)r FF(,)38 b(the)e(pairs)f(\()p FB(a)p FC(;)17 b FB(b)p FF(\))257 5147 y(and)29 b(\()p FB(c)p FC(;)17 b FB(d)s FF(\))27 b(are)i(the)f(separating)g(pairs.)41 b(Since)29 b(the)f(3-connected)g(separating)g(pair)g(\()p FB(c)p FC(;)17 b FB(d)s FF(\))28 b(is)257 5268 y(connected)d(by)f(an)g (edge)h(in)d FB(G)r FF(,)j(we)f(also)g(get)g Fv(f)p FB(c)p FC(;)17 b FB(d)s Fv(g)24 b FF(as)g(triple-bond)d FB(G)2711 5283 y FA(3)2749 5268 y FF(.)30 b(The)25 b(virtual)e(edges)257 5388 y(corresponding)h(to)h(the)f(separating)h(pairs)f(are)i(dra)o(wn)e (with)g(dashed)h(lines.)p Black Black eop end %%Page: 12 12 TeXDict begin 12 11 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 257 266 a SDict begin H.S end 257 266 a Black Black 257 266 a SDict begin H.R end 257 266 a 257 266 a SDict begin [ /View [/XYZ H.V] /Dest (page.12) cvn H.B /DEST pdfmark end 257 266 a Black Black Black Black 332 1303 a currentpoint currentpoint translate 0.72 0.72 scale neg exch neg exch translate 332 1303 a 0 setgray 3 setlinewidth np 739 892 9.6 0 360.001 arc fil 0 setgray np 739 892 9.6 0 360.001 arc st 3 setlinewidth np 739 892 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 973 657 9.6 0 360.001 arc fil 0 setgray np 973 657 9.6 0 360.001 arc st 3 setlinewidth np 973 657 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 504 657 9.6 0 360.001 arc fil 0 setgray np 504 657 9.6 0 360.001 arc st 3 setlinewidth np 504 657 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 739 657 9.6 0 360.001 arc fil 0 setgray np 739 657 9.6 0 360.001 arc st 3 setlinewidth np 739 657 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 739 422 9.6 0 360.001 arc fil 0 setgray np 739 422 9.6 0 360.001 arc st 3 setlinewidth np 739 422 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 4355 869 9.6 0 360.001 arc fil 0 setgray np 4355 869 9.6 0 360.001 arc st 3 setlinewidth np 4355 869 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 4214 869 9.6 0 360.001 arc fil 0 setgray np 4214 869 9.6 0 360.001 arc st 3 setlinewidth np 4214 869 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 4026 235 9.6 0 360.001 arc fil 0 setgray np 4026 235 9.6 0 360.001 arc st 3 setlinewidth np 4026 235 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 3885 235 9.6 0 360.001 arc fil 0 setgray np 3885 235 9.6 0 360.001 arc st 3 setlinewidth np 3885 235 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 1091 657 9.6 0 360.001 arc fil 0 setgray np 1091 657 9.6 0 360.001 arc st 3 setlinewidth np 1091 657 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 1443 892 9.6 0 360.001 arc fil 0 setgray np 1443 892 9.6 0 360.001 arc st 3 setlinewidth np 1443 892 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 1913 892 9.6 0 360.001 arc fil 0 setgray np 1913 892 9.6 0 360.001 arc st 3 setlinewidth np 1913 892 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 1913 516 9.6 0 360.001 arc fil 0 setgray np 1913 516 9.6 0 360.001 arc st 3 setlinewidth np 1913 516 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 1443 516 9.6 0 360.001 arc fil 0 setgray np 1443 516 9.6 0 360.001 arc st 3 setlinewidth np 1443 516 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 1678 516 9.6 0 360.001 arc fil 0 setgray np 1678 516 9.6 0 360.001 arc st 3 setlinewidth np 1678 516 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 2382 422 9.6 0 360.001 arc fil 0 setgray np 2382 422 9.6 0 360.001 arc st 3 setlinewidth np 2382 422 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 2382 892 9.6 0 360.001 arc fil 0 setgray np 2382 892 9.6 0 360.001 arc st 3 setlinewidth np 2382 892 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 2617 657 9.6 0 360.001 arc fil 0 setgray np 2617 657 9.6 0 360.001 arc st 3 setlinewidth np 2617 657 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 2147 657 9.6 0 360.001 arc fil 0 setgray np 2147 657 9.6 0 360.001 arc st 3 setlinewidth np 2147 657 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 2922 422 9.6 0 360.001 arc fil 0 setgray np 2922 422 9.6 0 360.001 arc st 3 setlinewidth np 2922 422 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 3274 657 9.6 0 360.001 arc fil 0 setgray np 3274 657 9.6 0 360.001 arc st 3 setlinewidth np 3274 657 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 2922 892 9.6 0 360.001 arc fil 0 setgray np 2922 892 9.6 0 360.001 arc st 3 setlinewidth np 2922 892 9.6 0 360.001 arc st 3 setlinewidth np 1443 516 a 1449 486 li st 3 setlinewidth np 1455 457 a 1461 427 li st 3 setlinewidth np 1462 427 a 1471 409 li st 3 setlinewidth np 1480 392 a 1489 374 li st 3 setlinewidth np 1488 374 a 1502.4 360.2 li st 3 setlinewidth np 1518 346 a 1532.4 332.2 li st 3 setlinewidth np 1532 332 a 1548.8 321.8 li st 3 setlinewidth np 1566 311 a 1582.8 300.8 li st 3 setlinewidth np 1582 300 a 1602.4 294 li st 3 setlinewidth np 1623 289 a 1643.4 283 li st 3 setlinewidth np 1643 283 a 1662.2 282.4 li st 3 setlinewidth np 1682 282 a 1701.2 281.4 li st 3 setlinewidth np 1701 282 a 1720.8 286.2 li st 3 setlinewidth np 1741 291 a 1760.8 295.2 li st 3 setlinewidth np 1761 296 a 1779.6 305.6 li st 3 setlinewidth np 1798 316 a 1816.6 325.6 li st 3 setlinewidth np 1817 325 a 1830.8 338.2 li st 3 setlinewidth np 1844 353 a 1857.8 366.2 li st 3 setlinewidth np 1858 366 a 1869.4 381.6 li st 3 setlinewidth np 1881 398 a 1892.4 413.6 li st 3 setlinewidth np 1892 415 a 1898 433 li st 3 setlinewidth np 1905 452 a 1911 470 li st 3 setlinewidth np 1911 471 a 1911.6 485.4 li st 3 setlinewidth np 1912 501 a 1912.6 515.4 li st 3 setlinewidth np 1443 892 a 1449 874 li st 3 setlinewidth np 1455 856 a 1461 838 li st 3 setlinewidth np 1462 838 a 1471 827.2 li st 3 setlinewidth np 1480 817 a 1489 806.2 li st 3 setlinewidth np 1488 807 a 1502.4 798.6 li st 3 setlinewidth np 1518 790 a 1532.4 781.6 li st 3 setlinewidth np 1532 781 a 1548.8 775 li st 3 setlinewidth np 1566 769 a 1582.8 763 li st 3 setlinewidth np 1582 763 a 1602.4 759.4 li st 3 setlinewidth np 1623 756 a 1643.4 752.4 li st 3 setlinewidth np 1643 752 a 1662.2 751.4 li st 3 setlinewidth np 1682 752 a 1701.2 751.4 li st 3 setlinewidth np 1701 751 a 1720.8 753.4 li st 3 setlinewidth np 1741 757 a 1760.8 759.4 li st 3 setlinewidth np 1761 760 a 1779.6 765.4 li st 3 setlinewidth np 1798 771 a 1816.6 776.4 li st 3 setlinewidth np 1817 777 a 1830.8 784.8 li st 3 setlinewidth np 1844 794 a 1857.8 801.8 li st 3 setlinewidth np 1858 802 a 1869.4 811 li st 3 setlinewidth np 1881 821 a 1892.4 830 li st 3 setlinewidth np 1892 831 a 1898 841.8 li st 3 setlinewidth np 1905 854 a 1911 864.8 li st 3 setlinewidth np 1911 865 a 1911.6 873.4 li st 3 setlinewidth np 1912 883 a 1912.6 891.4 li st 3 setlinewidth np 1443 892 a 1449 909.4 li st 3 setlinewidth np 1455 928 a 1461 945.4 li st 3 setlinewidth np 1462 946 a 1471 956.2 li st 3 setlinewidth np 1480 967 a 1489 977.2 li st 3 setlinewidth np 1488 978 a 1502.4 985.8 li st 3 setlinewidth np 1518 994 a 1532.4 1001.8 li st 3 setlinewidth np 1532 1003 a 1548.8 1008.4 li st 3 setlinewidth np 1566 1015 a 1582.8 1020.4 li st 3 setlinewidth np 1582 1022 a 1602.4 1025 li st 3 setlinewidth np 1623 1028 a 1643.4 1031 li st 3 setlinewidth np 1643 1032 a 1662.2 1032 li st 3 setlinewidth np 1682 1033 a 1701.2 1033 li st 3 setlinewidth np 1701 1033 a 1720.8 1030 li st 3 setlinewidth np 1741 1027 a 1760.8 1024 li st 3 setlinewidth np 1761 1025 a 1779.6 1019 li st 3 setlinewidth np 1798 1013 a 1816.6 1007 li st 3 setlinewidth np 1817 1007 a 1830.8 998.6 li st 3 setlinewidth np 1844 991 a 1857.8 982.6 li st 3 setlinewidth np 1858 982 a 1869.4 972.4 li st 3 setlinewidth np 1881 963 a 1892.4 953.4 li st 3 setlinewidth np 1892 953 a 1898 941.6 li st 3 setlinewidth np 1905 931 a 1911 919.6 li st 3 setlinewidth np 1911 919 a 1911.6 910 li st 3 setlinewidth np 1912 901 a 1912.6 892 li st 3 setlinewidth np 836 657 254.4 247.38 472.621 arc st 3 setlinewidth np 641 657 254.4 67.3798 292.621 arc st 3 setlinewidth np 2480 657 254.4 247.38 472.621 arc st 3 setlinewidth np 3020 657 254.4 247.38 472.621 arc st 3 setlinewidth np 504.4 657.8 a 739 422.6 li 973.6 657.8 li 739 892.4 li 504.4 657.8 li 973.6 657.8 li st 3 setlinewidth np 4057 603 54.6 90.0002 180 arc st 3 setlinewidth np 4057 524 54.6 180 270.001 arc st 3 setlinewidth np 4370 524 54.6 270.001 360.001 arc st 3 setlinewidth np 4370 603 54.6 0 90.0002 arc st 3 setlinewidth np 4003 603.2 a 4003 524.6 li st 3 setlinewidth np 4057.6 470 a 4370.8 470 li st 3 setlinewidth np 4425.4 524.6 a 4425.4 603.2 li st 3 setlinewidth np 4370.8 657.8 a 4057.6 657.8 li st 3 setlinewidth np 3885.4 305.6 a 3815.2 470 li st 3 setlinewidth np 4026.4 305.6 a 4097.2 470 li st 3 setlinewidth np 4214.2 869 a 4355.2 869 li st 3 setlinewidth np 4128 884 54.6 90.0002 180 arc st 3 setlinewidth np 4128 876 54.6 180 270.001 arc st 3 setlinewidth np 4441 876 54.6 270.001 360.001 arc st 3 setlinewidth np 4441 884 54.6 0 90.0002 arc st 3 setlinewidth np 4073.2 884.6 a 4073.2 876.8 li st 3 setlinewidth np 4128.4 822.2 a 4441.6 822.2 li st 3 setlinewidth np 4496.2 876.8 a 4496.2 884.6 li st 3 setlinewidth np 4441.6 939.8 a 4128.4 939.8 li st 3 setlinewidth np 4157.8 651.2 a 4228.6 815.6 li st 3 setlinewidth np 4157.8 651.2 a 4228.6 815.6 li st 3 setlinewidth np 4198 1236 54.6 90.0002 180 arc st 3 setlinewidth np 4198 1158 54.6 180 270.001 arc st 3 setlinewidth np 4511 1158 54.6 270.001 360.001 arc st 3 setlinewidth np 4511 1236 54.6 0 90.0002 arc st 3 setlinewidth np 4144 1236.8 a 4144 1158.8 li st 3 setlinewidth np 4198.6 1103.6 a 4511.8 1103.6 li st 3 setlinewidth np 4566.4 1158.8 a 4566.4 1236.8 li st 3 setlinewidth np 4511.8 1292 a 4198.6 1292 li st 3 setlinewidth np 4267 936.8 a 4337.2 1101.2 li st 3 setlinewidth np 3885.4 234.8 a 4026.4 234.8 li st 3 setlinewidth np 3799 250 54.6 90.0002 180 arc st 3 setlinewidth np 3799 242 54.6 180 270.001 arc st 3 setlinewidth np 4112 242 54.6 270.001 360.001 arc st 3 setlinewidth np 4112 250 54.6 0 90.0002 arc st 3 setlinewidth np 3744.4 250.4 a 3744.4 242.6 li st 3 setlinewidth np 3799.6 188 a 4112.8 188 li st 3 setlinewidth np 4167.4 242.6 a 4167.4 250.4 li st 3 setlinewidth np 4112.8 305.6 a 3799.6 305.6 li st 3 setlinewidth np 3541 603 54.6 90.0002 180 arc st 3 setlinewidth np 3541 524 54.6 180 270.001 arc st 3 setlinewidth np 3854 524 54.6 270.001 360.001 arc st 3 setlinewidth np 3854 603 54.6 0 90.0002 arc st 3 setlinewidth np 3486.4 603.2 a 3486.4 524.6 li st 3 setlinewidth np 3541 470 a 3854.2 470 li st 3 setlinewidth np 3908.8 524.6 a 3908.8 603.2 li st 3 setlinewidth np 3854.2 657.8 a 3541 657.8 li st 3 setlinewidth np 3682 1236 54.6 90.0002 180 arc st 3 setlinewidth np 3682 1158 54.6 180 270.001 arc st 3 setlinewidth np 3995 1158 54.6 270.001 360.001 arc st 3 setlinewidth np 3995 1236 54.6 0 90.0002 arc st 3 setlinewidth np 3627.4 1236.8 a 3627.4 1158.8 li st 3 setlinewidth np 3682 1103.6 a 3995.2 1103.6 li st 3 setlinewidth np 4049.8 1158.8 a 4049.8 1236.8 li st 3 setlinewidth np 3995.2 1292 a 3682 1292 li st 3 setlinewidth np 4144 939.8 a 3956.2 1103.6 li st 3 setlinewidth np 2383 892.4 a 2617.6 657.8 li st 3 setlinewidth np 2617.6 657.8 a 2383 422.6 li st 3 setlinewidth np 2383 422.6 a 2147.8 657.8 li st 3 setlinewidth np 2147.8 657.8 a 2383 892.4 li st 3 setlinewidth np 2147 657 a 2174.6 657 li st 3 setlinewidth np 2203 657 a 2230.6 657 li st 3 setlinewidth np 2258 657 a 2285.6 657 li st 3 setlinewidth np 2313 657 a 2340.6 657 li st 3 setlinewidth np 2368 657 a 2395.6 657 li st 3 setlinewidth np 2424 657 a 2451.6 657 li st 3 setlinewidth np 2479 657 a 2506.6 657 li st 3 setlinewidth np 2534 657 a 2561.6 657 li st 3 setlinewidth np 2589 657 a 2616.6 657 li st 3 setlinewidth np 2922 422 a 2922 449 li st 3 setlinewidth np 2922 478 a 2922 505 li st 3 setlinewidth np 2922 533 a 2922 560 li st 3 setlinewidth np 2922 588 a 2922 615 li st 3 setlinewidth np 2922 643 a 2922 670 li st 3 setlinewidth np 2922 699 a 2922 726 li st 3 setlinewidth np 2922 754 a 2922 781 li st 3 setlinewidth np 2922 809 a 2922 836 li st 3 setlinewidth np 2922 864 a 2922 891 li st 3 setlinewidth np 1443.4 892.4 a 1913.2 892.4 li st 3 setlinewidth np 1913.2 516.8 a 1443.4 516.8 li st 683 613 a FB(f)973 633 y(b)91 b(e)668 398 y(c)668 984 y(d)4378 914 y(d)4049 281 y(b)433 633 y(a)3791 281 y(a)4120 915 y(c)3648 595 y(G)3722 610 y FA(1)1913 492 y FB(b)1622 472 y(f)1349 492 y(a)1913 867 y(d)4165 595 y(G)4239 610 y FA(2)4306 1229 y FB(G)4380 1244 y FA(4)2617 633 y FB(b)2312 398 y(c)2312 984 y(d)2077 633 y(a)1171 b(e)2852 398 y(c)2852 984 y(d)2826 1182 y(G)2900 1197 y FA(4)2122 1182 y FB(G)2196 1197 y FA(2)1300 1182 y FB(G)1374 1197 y FA(3)398 1175 y Fi(b)385 1196 y FB(G)1300 220 y(G)1374 235 y FA(1)2032 b FB(T)3789 1228 y(G)3863 1243 y FA(3)1349 868 y FB(c)332 1303 y currentpoint currentpoint translate 1 0.72 div 1 0.72 div scale neg exch neg exch translate 332 1303 a 257 1614 a FF(Figure)28 b(2:)658 1614 y SDict begin H.S end 658 1614 a Black Black 658 1614 a SDict begin H.R end 658 1614 a 658 1614 a SDict begin [ /View [/XYZ H.V] /Dest (figure.2) cvn H.B /DEST pdfmark end 658 1614 a FF(Decomposition)e(of)h(a)h(biconnected)f(planar)g(graph)g(into) g(a)h(triconnected)e(com-)257 1734 y(ponent)e(tree.)p Black 404 2027 a(The)31 b(triconnected)f(components)g(can)h(be)g (canonized)g(in)g(log-space)g([)2910 2028 y SDict begin H.S end 2910 2028 a Black -1 x FF(5)p Black 2960 1960 a SDict begin H.R end 2960 1960 a 2960 2027 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.DLN08) cvn H.B /ANN pdfmark end 2960 2027 a FF(].)49 b(Hence,)33 b(for)257 2147 y(triconnected)25 b(component)f(trees,)i(compute)e(their)h(canonical)g (in)l(v)n(ariant)f(in)h(log-space,)g(i.e.)257 2268 y(tw)o(o)g (biconnected)f(graphs)h(are)g(isomorphic)e(if)i(their)g(trees)g(are)g (found)f(to)h(be)g(equal.)404 2391 y(In)i(section)824 2392 y SDict begin H.S end 824 2392 a Black -1 x FF(4.1)p Black 948 2323 a SDict begin H.R end 948 2323 a 948 2391 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (subsection.4.1) cvn H.B /ANN pdfmark end 948 2391 a FF(,)h(we)g(summarize,)f(ho)n(w)f (to)h(canonize)h(biconnected)e(planar)i(graphs)f(by)257 2511 y(applying)e(tree)h(canonization)f(ideas)g(from)h([)1825 2511 y SDict begin H.S end 1825 2511 a Black FF(14)p Black 1925 2444 a SDict begin H.R end 1925 2444 a 1925 2511 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.Lin92) cvn H.B /ANN pdfmark end 1925 2511 a FF(])g(to)f(their)g(triconnected)h (component)e(trees.)257 2631 y(Note,)39 b(that)c(pairwise)h (isomorphism)d(of)k(tw)o(o)e(trees)h(labelled)g(with)f(the)h(canons)g (of)g(their)257 2752 y(components)26 b(does)h(not)g(imply)f (isomorphism)f(of)i(the)g(corresponding)f(graphs.)38 b(Lindell)16 b(')-5 b(s)257 2872 y(algorithm)22 b(and)h(comple)o(xity)f (analysis)g(had)h(to)g(be)h(modi\002ed)e(in)h(a)h(non-tri)n(vial)d(w)o (ay)j(for)f(this)257 2992 y(step)i(to)f(w)o(ork)h(in)f(log-space.)404 3115 y(In)k(section)825 3116 y SDict begin H.S end 825 3116 a Black -1 x FF(4.2)p Black 949 3048 a SDict begin H.R end 949 3048 a 949 3115 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (subsection.4.2) cvn H.B /ANN pdfmark end 949 3115 a FF(,)h(we)f(describe,)g(ho)n(w)f(to)g(canonize)h(planar)g(graphs)f (using)g(their)h(bicon-)257 3236 y(nected)35 b(component)f(trees,)j (again)d(using)f(the)i(basic)f(structure)h(of)f(Lindell)16 b(')-5 b(s)34 b(algorithm.)257 3356 y(The)26 b(comparison)e(algorithm)f (refers)j(to)f(the)g(biconnected)g(component)f(tree)h(of)g(the)g (planar)257 3477 y(graph)31 b(and)g(when)g(comparing)f(biconnected)g (components,)h(to)g(their)f(triconnected)g(com-)257 3597 y(ponent)35 b(trees.)61 b(This)34 b(requires)h(a)g(detailed)f(analysis) h(of)g(the)f(interferences)i(of)f(both)f(tree)257 3717 y(structures.)257 3878 y SDict begin H.S end 257 3878 a 257 3878 a SDict begin 14.5 H.A end 257 3878 a 257 3878 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.4.1) cvn H.B /DEST pdfmark end 257 3878 a 151 x Ft(4.1)119 b(Canonization)31 b(of)f(biconnected)i(planar)f(graphs)257 4222 y FF(Let)41 b FB(S)57 b FF(and)41 b FB(T)54 b FF(be)41 b(tw)o(o)f(triconnected)h (component)f(trees)h(for)h(the)f(biconnected)f(planar)257 4342 y(graphs)h FB(G)46 b FF(and)f FB(H)t FF(,)j(respecti)n(v)o(ely)-6 b(.)84 b FB(S)59 b FF(and)43 b FB(T)56 b FF(are)44 b(rooted)f(at)g (separating)g(pair)g(nodes,)257 4463 y(say)38 b FB(s)j FE(=)g FF(\()p FB(a)p FC(;)17 b FB(b)p FF(\))32 b(and)h FB(t)42 b FE(=)g FF(\()p FB(a)1287 4427 y Fw(0)1314 4463 y FC(;)17 b FB(b)1423 4427 y Fw(0)1450 4463 y FF(\).)53 b(Therefore)33 b(we)f(also)g(write)g FB(S)2629 4478 y FA(\()p Fr(a)p Fs(;)p Fr(b)p FA(\))2793 4463 y FF(and)g FB(T)3028 4478 y FA(\()p Fr(a)3083 4458 y Fh(0)3105 4478 y Fs(;)p Fr(b)3166 4458 y Fh(0)3188 4478 y FA(\))3214 4463 y FF(.)53 b(The)o(y)257 4583 y(ha)n(v)o(e)20 b(separating)f(pair)h (nodes)f(at)h(odd)g(le)n(v)o(els)e(and)i(triconnected)f(component)g (nodes)g(at)h(e)n(v)o(en)257 4703 y(le)n(v)o(els.)30 b(Figure)827 4704 y SDict begin H.S end 827 4704 a Black -1 x FF(3)p Black 876 4636 a SDict begin H.R end 876 4636 a 876 4703 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (figure.3) cvn H.B /ANN pdfmark end 876 4703 a 25 w FF(sho)n(ws)24 b(tw)o(o)g(trees)h(to)g(be)g(compared.)404 4826 y(Similar)38 b(as)g(in)g(Lindell')-5 b(s)37 b(algorithm,)j(we)f(de\002ne)g(the)g (isomorphism)c(order)k(of)g(tw)o(o)257 4947 y(triconnected)23 b(component)e(trees)i FB(S)39 b FF(and)22 b FB(T)35 b FF(rooted)23 b(at)f(separating)h(pairs)k FB(s)h FE(=)g FF(\()p FB(a)p FC(;)17 b FB(b)p FF(\))22 b(and)h FB(t)29 b FE(=)257 5067 y FF(\()p FB(a)340 5031 y Fw(0)368 5067 y FC(;)17 b FB(b)477 5031 y Fw(0)503 5067 y FF(\).)31 b FB(S)657 5082 y FA(\()p Fr(a)p Fs(;)p Fr(b)p FA(\))815 5067 y FC(<)878 5082 y Fg(T)946 5067 y FB(T)1005 5082 y FA(\()p Fr(a)1060 5063 y Fh(0)1082 5082 y Fs(;)p Fr(b)1143 5063 y Fh(0)1164 5082 y FA(\))1215 5067 y FF(if:)257 5172 y SDict begin H.S end 257 5172 a 257 5172 a SDict begin 14.5 H.A end 257 5172 a 257 5172 a SDict begin [ /View [/XYZ H.V] /Dest (Item.1) cvn H.B /DEST pdfmark end 257 5172 a Black 378 5313 a FF(1.)p Black 73 w Fv(j)p FB(S)611 5328 y FA(\()p Fr(a)p Fs(;)p Fr(b)p FA(\))742 5313 y Fv(j)d FC(<)g Fv(j)p FB(T)960 5328 y FA(\()p Fr(a)1015 5309 y Fh(0)1037 5328 y Fs(;)p Fr(b)1098 5309 y Fh(0)1119 5328 y FA(\))1146 5313 y Fv(j)c FF(or)257 5388 y SDict begin H.S end 257 5388 a 257 5388 a SDict begin 14.5 H.A end 257 5388 a 257 5388 a SDict begin [ /View [/XYZ H.V] /Dest (Item.2) cvn H.B /DEST pdfmark end 257 5388 a Black Black eop end %%Page: 13 13 TeXDict begin 13 12 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 257 266 a SDict begin H.S end 257 266 a Black Black 257 266 a SDict begin H.R end 257 266 a 257 266 a SDict begin [ /View [/XYZ H.V] /Dest (page.13) cvn H.B /DEST pdfmark end 257 266 a Black Black Black Black 257 1734 a currentpoint currentpoint translate 0.72 0.72 scale neg exch neg exch translate 257 1734 a 0 setgray 3 setlinewidth np 1840 172 9.6 0 360.001 arc fil 0 setgray np 1840 172 9.6 0 360.001 arc st 3 setlinewidth np 1840 172 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 1699 172 9.6 0 360.001 arc fil 0 setgray np 1699 172 9.6 0 360.001 arc st 3 setlinewidth np 1699 172 9.6 0 360.001 arc st 3 setlinewidth np 1699 172.6 a 1839.4 172.6 li st 3 setlinewidth np 1613 188 54.6 90.0002 180 arc st 3 setlinewidth np 1613 180 54.6 180 270.001 arc st 3 setlinewidth np 1926 180 54.6 270.001 360.001 arc st 3 setlinewidth np 1926 188 54.6 0 90.0002 arc st 3 setlinewidth np 1558 188.8 a 1558 181 li st 3 setlinewidth np 1612.6 125.8 a 1925.8 125.8 li st 3 setlinewidth np 1980.4 181 a 1980.4 188.8 li st 3 setlinewidth np 1925.8 243.4 a 1612.6 243.4 li st 1863 219 a FB(b)-308 b(a)p 0 setgray 3 setlinewidth np 971 947 9.6 0 360.001 arc fil 0 setgray np 971 947 9.6 0 360.001 arc st 3 setlinewidth np 971 947 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 830 947 9.6 0 360.001 arc fil 0 setgray np 830 947 9.6 0 360.001 arc st 3 setlinewidth np 830 947 9.6 0 360.001 arc st 3 setlinewidth np 595 1370.8 a 595 1464.4 li st 3 setlinewidth np 641.8 1370.8 a 736 1464.4 li st 3 setlinewidth np 595 1464.4 a 548.2 1605.4 li 641.8 1605.4 li closepath st 3 setlinewidth np 736 1464.4 a 689.2 1605.4 li 782.8 1605.4 li closepath st 3 setlinewidth np 548.2 1370.8 a 454 1464.4 li st 3 setlinewidth np 454 1464.4 a 407.2 1605.4 li 501.4 1605.4 li closepath st 0 setgray 3 setlinewidth np 1558 947 9.6 0 360.001 arc fil 0 setgray np 1558 947 9.6 0 360.001 arc st 3 setlinewidth np 1558 947 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 1417 947 9.6 0 360.001 arc fil 0 setgray np 1417 947 9.6 0 360.001 arc st 3 setlinewidth np 1417 947 9.6 0 360.001 arc st 3 setlinewidth np 1417 947.8 a 1558 947.8 li st 3 setlinewidth np 1401 963 54.6 90.0002 180 arc st 3 setlinewidth np 1401 955 54.6 180 270.001 arc st 3 setlinewidth np 1574 955 54.6 270.001 360.001 arc st 3 setlinewidth np 1574 963 54.6 0 90.0002 arc st 3 setlinewidth np 1346.2 963.4 a 1346.2 955.6 li st 3 setlinewidth np 1401.4 901 a 1573.6 901 li st 3 setlinewidth np 1628.2 955.6 a 1628.2 963.4 li st 3 setlinewidth np 1573.6 1018 a 1401.4 1018 li st 3 setlinewidth np 2144.8 1370.8 a 2144.8 1464.4 li st 3 setlinewidth np 2191.6 1370.8 a 2285.8 1464.4 li st 3 setlinewidth np 2144.8 1464.4 a 2098 1605.4 li 2191.6 1605.4 li closepath st 3 setlinewidth np 2285.8 1464.4 a 2239 1605.4 li 2332.6 1605.4 li closepath st 3 setlinewidth np 2098 1370.8 a 2003.8 1464.4 li st 3 setlinewidth np 2003.8 1464.4 a 1957 1605.4 li 2051.2 1605.4 li closepath st 3 setlinewidth np 1064.8 1464.4 a 1018 1605.4 li 1111.6 1605.4 li closepath st 3 setlinewidth np 1205.8 1464.4 a 1158.4 1605.4 li 1252.6 1605.4 li closepath st 3 setlinewidth np 1064.8 1464.4 a 1111.6 1370.8 li st 3 setlinewidth np 1205.8 1464.4 a 1158.4 1370.8 li st 3 setlinewidth np 1417 1464.4 a 1370.2 1605.4 li 1463.8 1605.4 li closepath st 3 setlinewidth np 1558 1464.4 a 1510.6 1605.4 li 1604.8 1605.4 li closepath st 3 setlinewidth np 1417 1464.4 a 1463.8 1370.8 li st 3 setlinewidth np 1558 1464.4 a 1510.6 1370.8 li st 0 setgray 3 setlinewidth np 2215 947 9.6 0 360.001 arc fil 0 setgray np 2215 947 9.6 0 360.001 arc st 3 setlinewidth np 2215 947 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 2075 947 9.6 0 360.001 arc fil 0 setgray np 2075 947 9.6 0 360.001 arc st 3 setlinewidth np 2075 947 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 3202 947 9.6 0 360.001 arc fil 0 setgray np 3202 947 9.6 0 360.001 arc st 3 setlinewidth np 3202 947 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 3061 947 9.6 0 360.001 arc fil 0 setgray np 3061 947 9.6 0 360.001 arc st 3 setlinewidth np 3061 947 9.6 0 360.001 arc st 3 setlinewidth np 2825.8 1370.8 a 2825.8 1464.4 li st 3 setlinewidth np 2872.6 1370.8 a 2966.8 1464.4 li st 3 setlinewidth np 2825.8 1464.4 a 2779 1605.4 li 2872.6 1605.4 li closepath st 3 setlinewidth np 2966.8 1464.4 a 2920 1605.4 li 3013.6 1605.4 li closepath st 3 setlinewidth np 2779 1370.8 a 2684.8 1464.4 li st 3 setlinewidth np 2684.8 1464.4 a 2638 1605.4 li 2732.2 1605.4 li closepath st 0 setgray 3 setlinewidth np 3789 947 9.6 0 360.001 arc fil 0 setgray np 3789 947 9.6 0 360.001 arc st 3 setlinewidth np 3789 947 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 3648 947 9.6 0 360.001 arc fil 0 setgray np 3648 947 9.6 0 360.001 arc st 3 setlinewidth np 3648 947 9.6 0 360.001 arc st 3 setlinewidth np 3647.8 947.8 a 3788.8 947.8 li st 3 setlinewidth np 3632 963 54.6 90.0002 180 arc st 3 setlinewidth np 3632 955 54.6 180 270.001 arc st 3 setlinewidth np 3804 955 54.6 270.001 360.001 arc st 3 setlinewidth np 3804 963 54.6 0 90.0002 arc st 3 setlinewidth np 3577.6 963.4 a 3577.6 955.6 li st 3 setlinewidth np 3632.2 901 a 3804.4 901 li st 3 setlinewidth np 3859 955.6 a 3859 963.4 li st 3 setlinewidth np 3804.4 1018 a 3632.2 1018 li st 3 setlinewidth np 4375.6 1370.8 a 4375.6 1464.4 li st 3 setlinewidth np 4422.4 1370.8 a 4516.6 1464.4 li st 3 setlinewidth np 4375.6 1464.4 a 4328.8 1605.4 li 4422.4 1605.4 li closepath st 3 setlinewidth np 4516.6 1464.4 a 4469.8 1605.4 li 4563.4 1605.4 li closepath st 3 setlinewidth np 4328.8 1370.8 a 4234.6 1464.4 li st 3 setlinewidth np 4234.6 1464.4 a 4187.8 1605.4 li 4282 1605.4 li closepath st 3 setlinewidth np 3295.6 1464.4 a 3248.8 1605.4 li 3342.4 1605.4 li closepath st 3 setlinewidth np 3436.6 1464.4 a 3389.2 1605.4 li 3483.4 1605.4 li closepath st 3 setlinewidth np 3295.6 1464.4 a 3342.4 1370.8 li st 3 setlinewidth np 3436.6 1464.4 a 3389.2 1370.8 li st 3 setlinewidth np 3647.8 1464.4 a 3601 1605.4 li 3694.6 1605.4 li closepath st 3 setlinewidth np 3788.8 1464.4 a 3742 1605.4 li 3835.6 1605.4 li closepath st 3 setlinewidth np 3647.8 1464.4 a 3694.6 1370.8 li st 3 setlinewidth np 3788.8 1464.4 a 3742 1370.8 li st 0 setgray 3 setlinewidth np 4446 947 9.6 0 360.001 arc fil 0 setgray np 4446 947 9.6 0 360.001 arc st 3 setlinewidth np 4446 947 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 4305 947 9.6 0 360.001 arc fil 0 setgray np 4305 947 9.6 0 360.001 arc st 3 setlinewidth np 4305 947 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 4070 172 9.6 0 360.001 arc fil 0 setgray np 4070 172 9.6 0 360.001 arc st 3 setlinewidth np 4070 172 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 3930 172 9.6 0 360.001 arc fil 0 setgray np 3930 172 9.6 0 360.001 arc st 3 setlinewidth np 3930 172 9.6 0 360.001 arc st 3 setlinewidth np 1651.6 243.4 a 1276 478 li st 3 setlinewidth np 1722.4 243.4 a 1675 361 li st 3 setlinewidth np 1886.8 243.4 a 2144.8 478 li st 3 setlinewidth np 1816 243.4 a 1863.4 361 li st 3 setlinewidth np 1769.2 243.4 a 1769.2 361 li st 3 setlinewidth np 1182.4 665.8 a 900.4 901 li st 3 setlinewidth np 829.6 947.8 a 970.6 947.8 li st 3 setlinewidth np 853.6 1018 a 829.6 1088.8 li st 3 setlinewidth np 900.4 1018 a 900.4 1088.8 li st 3 setlinewidth np 947.2 1018 a 970.6 1088.8 li st 3 setlinewidth np 533 1315 54.6 90.0002 180 arc st 3 setlinewidth np 533 1237 54.6 180 270.001 arc st 3 setlinewidth np 658 1237 54.6 270.001 360.001 arc st 3 setlinewidth np 658 1315 54.6 0 90.0002 arc st 3 setlinewidth np 477.4 1315.6 a 477.4 1237.6 li st 3 setlinewidth np 532.6 1182.4 a 657.4 1182.4 li st 3 setlinewidth np 712.6 1237.6 a 712.6 1315.6 li st 3 setlinewidth np 657.4 1370.8 a 532.6 1370.8 li st 3 setlinewidth np 814 963 54.6 90.0002 180 arc st 3 setlinewidth np 814 955 54.6 180 270.001 arc st 3 setlinewidth np 987 955 54.6 270.001 360.001 arc st 3 setlinewidth np 987 963 54.6 0 90.0002 arc st 3 setlinewidth np 759.4 963.4 a 759.4 955.6 li st 3 setlinewidth np 814 901 a 986.2 901 li st 3 setlinewidth np 1041.4 955.6 a 1041.4 963.4 li st 3 setlinewidth np 986.2 1018 a 814 1018 li st 3 setlinewidth np 1276 665.8 a 1299.4 736.6 li st 3 setlinewidth np 1229.2 665.8 a 1205.8 736.6 li st 3 setlinewidth np 1322.8 665.8 a 1463.8 901 li st 3 setlinewidth np 1487.2 1018 a 1487.2 1182.4 li st 3 setlinewidth np 1190 611 54.6 90.0002 180 arc st 3 setlinewidth np 1190 533 54.6 180 270.001 arc st 3 setlinewidth np 1315 533 54.6 270.001 360.001 arc st 3 setlinewidth np 1315 611 54.6 0 90.0002 arc st 3 setlinewidth np 1135 611.2 a 1135 533.2 li st 3 setlinewidth np 1190.2 478 a 1315 478 li st 3 setlinewidth np 1370.2 533.2 a 1370.2 611.2 li st 3 setlinewidth np 1315 665.8 a 1190.2 665.8 li st 3 setlinewidth np 806.2 1018 a 595 1182.4 li st 3 setlinewidth np 1073 1315 54.6 90.0002 180 arc st 3 setlinewidth np 1073 1237 54.6 180 270.001 arc st 3 setlinewidth np 1198 1237 54.6 270.001 360.001 arc st 3 setlinewidth np 1198 1315 54.6 0 90.0002 arc st 3 setlinewidth np 1018 1315.6 a 1018 1237.6 li st 3 setlinewidth np 1072.6 1182.4 a 1198 1182.4 li st 3 setlinewidth np 1252.6 1237.6 a 1252.6 1315.6 li st 3 setlinewidth np 1198 1370.8 a 1072.6 1370.8 li st 3 setlinewidth np 994 1018 a 1135 1182.4 li st 3 setlinewidth np 1425 1315 54.6 90.0002 180 arc st 3 setlinewidth np 1425 1237 54.6 180 270.001 arc st 3 setlinewidth np 1550 1237 54.6 270.001 360.001 arc st 3 setlinewidth np 1550 1315 54.6 0 90.0002 arc st 3 setlinewidth np 1370.2 1315.6 a 1370.2 1237.6 li st 3 setlinewidth np 1424.8 1182.4 a 1550.2 1182.4 li st 3 setlinewidth np 1604.8 1237.6 a 1604.8 1315.6 li st 3 setlinewidth np 1550.2 1370.8 a 1424.8 1370.8 li st 3 setlinewidth np 2074.6 947.8 a 2215.6 947.8 li st 3 setlinewidth np 2144.8 665.8 a 2144.8 901 li st 3 setlinewidth np 2144.8 1018 a 2144.8 1182.4 li st 3 setlinewidth np 2082 611 54.6 90.0002 180 arc st 3 setlinewidth np 2082 533 54.6 180 270.001 arc st 3 setlinewidth np 2208 533 54.6 270.001 360.001 arc st 3 setlinewidth np 2208 611 54.6 0 90.0002 arc st 3 setlinewidth np 2027.2 611.2 a 2027.2 533.2 li st 3 setlinewidth np 2082.4 478 a 2207.8 478 li st 3 setlinewidth np 2262.4 533.2 a 2262.4 611.2 li st 3 setlinewidth np 2207.8 665.8 a 2082.4 665.8 li st 3 setlinewidth np 2082 1315 54.6 90.0002 180 arc st 3 setlinewidth np 2082 1237 54.6 180 270.001 arc st 3 setlinewidth np 2208 1237 54.6 270.001 360.001 arc st 3 setlinewidth np 2208 1315 54.6 0 90.0002 arc st 3 setlinewidth np 2027.2 1315.6 a 2027.2 1237.6 li st 3 setlinewidth np 2082.4 1182.4 a 2207.8 1182.4 li st 3 setlinewidth np 2262.4 1237.6 a 2262.4 1315.6 li st 3 setlinewidth np 2207.8 1370.8 a 2082.4 1370.8 li st 3 setlinewidth np 2059 963 54.6 90.0002 180 arc st 3 setlinewidth np 2059 955 54.6 180 270.001 arc st 3 setlinewidth np 2231 955 54.6 270.001 360.001 arc st 3 setlinewidth np 2231 963 54.6 0 90.0002 arc st 3 setlinewidth np 2003.8 963.4 a 2003.8 955.6 li st 3 setlinewidth np 2059 901 a 2231.2 901 li st 3 setlinewidth np 2285.8 955.6 a 2285.8 963.4 li st 3 setlinewidth np 2231.2 1018 a 2059 1018 li st 3 setlinewidth np 1910 783 a 1922.6 783 li st 3 setlinewidth np 1943 783 a 1955.6 783 li st 3 setlinewidth np 1975 783 a 1987.6 783 li st 3 setlinewidth np 2008 783 a 2020.6 783 li st 3 setlinewidth np 2041 783 a 2053.6 783 li st 3 setlinewidth np 2073 783 a 2085.6 783 li st 3 setlinewidth np 2106 783 a 2118.6 783 li st 3 setlinewidth np 2139 783 a 2151.6 783 li st 3 setlinewidth np 2171 783 a 2183.6 783 li st 3 setlinewidth np 2204 783 a 2216.6 783 li st 3 setlinewidth np 2237 783 a 2249.6 783 li st 3 setlinewidth np 2269 783 a 2281.6 783 li st 3 setlinewidth np 2302 783 a 2314.6 783 li st 3 setlinewidth np 2335 783 a 2347.6 783 li st 3 setlinewidth np 2367 783 a 2379.6 783 li st 3 setlinewidth np 2380 783 a 2380 795 li st 3 setlinewidth np 2380 816 a 2380 828 li st 3 setlinewidth np 2380 849 a 2380 861 li st 3 setlinewidth np 2380 882 a 2380 894 li st 3 setlinewidth np 2380 915 a 2380 927 li st 3 setlinewidth np 2380 948 a 2380 960 li st 3 setlinewidth np 2380 981 a 2380 993 li st 3 setlinewidth np 2380 1014 a 2380 1026 li st 3 setlinewidth np 2380 1047 a 2380 1059 li st 3 setlinewidth np 2380 1079 a 2380 1091 li st 3 setlinewidth np 2380 1112 a 2380 1124 li st 3 setlinewidth np 2380 1145 a 2380 1157 li st 3 setlinewidth np 2380 1178 a 2380 1190 li st 3 setlinewidth np 2380 1211 a 2380 1223 li st 3 setlinewidth np 2380 1244 a 2380 1256 li st 3 setlinewidth np 2380 1277 a 2380 1289 li st 3 setlinewidth np 2380 1310 a 2380 1322 li st 3 setlinewidth np 2380 1343 a 2380 1355 li st 3 setlinewidth np 2380 1376 a 2380 1388 li st 3 setlinewidth np 2380 1409 a 2380 1421 li st 3 setlinewidth np 2380 1442 a 2380 1454 li st 3 setlinewidth np 2380 1475 a 2380 1487 li st 3 setlinewidth np 2380 1508 a 2380 1520 li st 3 setlinewidth np 2380 1541 a 2380 1553 li st 3 setlinewidth np 2380 1574 a 2380 1586 li st 3 setlinewidth np 2380 1606 a 2380 1618 li st 3 setlinewidth np 2380 1639 a 2380 1651 li st 3 setlinewidth np 2380 1652 a 2368 1652 li st 3 setlinewidth np 2347 1652 a 2335 1652 li st 3 setlinewidth np 2314 1652 a 2302 1652 li st 3 setlinewidth np 2282 1652 a 2270 1652 li st 3 setlinewidth np 2249 1652 a 2237 1652 li st 3 setlinewidth np 2216 1652 a 2204 1652 li st 3 setlinewidth np 2184 1652 a 2172 1652 li st 3 setlinewidth np 2151 1652 a 2139 1652 li st 3 setlinewidth np 2118 1652 a 2106 1652 li st 3 setlinewidth np 2086 1652 a 2074 1652 li st 3 setlinewidth np 2053 1652 a 2041 1652 li st 3 setlinewidth np 2020 1652 a 2008 1652 li st 3 setlinewidth np 1988 1652 a 1976 1652 li st 3 setlinewidth np 1955 1652 a 1943 1652 li st 3 setlinewidth np 1923 1652 a 1911 1652 li st 3 setlinewidth np 1910 1652 a 1910 1639.4 li st 3 setlinewidth np 1910 1619 a 1910 1606.4 li st 3 setlinewidth np 1910 1586 a 1910 1573.4 li st 3 setlinewidth np 1910 1553 a 1910 1540.4 li st 3 setlinewidth np 1910 1520 a 1910 1507.4 li st 3 setlinewidth np 1910 1487 a 1910 1474.4 li st 3 setlinewidth np 1910 1454 a 1910 1441.4 li st 3 setlinewidth np 1910 1421 a 1910 1408.4 li st 3 setlinewidth np 1910 1388 a 1910 1375.4 li st 3 setlinewidth np 1910 1355 a 1910 1342.4 li st 3 setlinewidth np 1910 1322 a 1910 1309.4 li st 3 setlinewidth np 1910 1289 a 1910 1276.4 li st 3 setlinewidth np 1910 1257 a 1910 1244.4 li st 3 setlinewidth np 1910 1224 a 1910 1211.4 li st 3 setlinewidth np 1910 1191 a 1910 1178.4 li st 3 setlinewidth np 1910 1158 a 1910 1145.4 li st 3 setlinewidth np 1910 1125 a 1910 1112.4 li st 3 setlinewidth np 1910 1092 a 1910 1079.4 li st 3 setlinewidth np 1910 1059 a 1910 1046.4 li st 3 setlinewidth np 1910 1026 a 1910 1013.4 li st 3 setlinewidth np 1910 993 a 1910 980.4 li st 3 setlinewidth np 1910 960 a 1910 947.4 li st 3 setlinewidth np 1910 927 a 1910 914.4 li st 3 setlinewidth np 1910 894 a 1910 881.4 li st 3 setlinewidth np 1910 861 a 1910 848.4 li st 3 setlinewidth np 1910 828 a 1910 815.4 li st 3 setlinewidth np 1910 795 a 1910 782.4 li st 3 setlinewidth np 1112 783 a 1118.6 793.2 li st 3 setlinewidth np 1128 810 a 1134.6 820.2 li st 3 setlinewidth np 1145 837 a 1151.6 847.2 li st 3 setlinewidth np 1161 863 a 1167.6 873.2 li st 3 setlinewidth np 1178 890 a 1184.6 900.2 li st 3 setlinewidth np 1194 917 a 1200.6 927.2 li st 3 setlinewidth np 1211 944 a 1217.6 954.2 li st 3 setlinewidth np 1227 971 a 1233.6 981.2 li st 3 setlinewidth np 1244 997 a 1250.6 1007.2 li st 3 setlinewidth np 1260 1024 a 1266.6 1034.2 li st 3 setlinewidth np 1277 1051 a 1283.6 1061.2 li st 3 setlinewidth np 1293 1078 a 1299.6 1088.2 li st 3 setlinewidth np 1300 1088 a 1300 1099.4 li st 3 setlinewidth np 1300 1123 a 1300 1134.4 li st 3 setlinewidth np 1300 1157 a 1300 1168.4 li st 3 setlinewidth np 1300 1192 a 1300 1203.4 li st 3 setlinewidth np 1300 1226 a 1300 1237.4 li st 3 setlinewidth np 1300 1261 a 1300 1272.4 li st 3 setlinewidth np 1300 1295 a 1300 1306.4 li st 3 setlinewidth np 1300 1329 a 1300 1340.4 li st 3 setlinewidth np 1300 1364 a 1300 1375.4 li st 3 setlinewidth np 1300 1398 a 1300 1409.4 li st 3 setlinewidth np 1300 1433 a 1300 1444.4 li st 3 setlinewidth np 1300 1467 a 1300 1478.4 li st 3 setlinewidth np 1300 1502 a 1300 1513.4 li st 3 setlinewidth np 1300 1536 a 1300 1547.4 li st 3 setlinewidth np 1300 1571 a 1300 1582.4 li st 3 setlinewidth np 1300 1605 a 1300 1616.4 li st 3 setlinewidth np 1300 1640 a 1300 1651.4 li st 3 setlinewidth np 1300 1652 a 1288 1652 li st 3 setlinewidth np 1267 1652 a 1255 1652 li st 3 setlinewidth np 1233 1652 a 1221 1652 li st 3 setlinewidth np 1200 1652 a 1188 1652 li st 3 setlinewidth np 1167 1652 a 1155 1652 li st 3 setlinewidth np 1134 1652 a 1122 1652 li st 3 setlinewidth np 1101 1652 a 1089 1652 li st 3 setlinewidth np 1068 1652 a 1056 1652 li st 3 setlinewidth np 1035 1652 a 1023 1652 li st 3 setlinewidth np 1002 1652 a 990 1652 li st 3 setlinewidth np 969 1652 a 957 1652 li st 3 setlinewidth np 936 1652 a 924 1652 li st 3 setlinewidth np 902 1652 a 890 1652 li st 3 setlinewidth np 869 1652 a 857 1652 li st 3 setlinewidth np 836 1652 a 824 1652 li st 3 setlinewidth np 803 1652 a 791 1652 li st 3 setlinewidth np 770 1652 a 758 1652 li st 3 setlinewidth np 737 1652 a 725 1652 li st 3 setlinewidth np 704 1652 a 692 1652 li st 3 setlinewidth np 671 1652 a 659 1652 li st 3 setlinewidth np 638 1652 a 626 1652 li st 3 setlinewidth np 604 1652 a 592 1652 li st 3 setlinewidth np 571 1652 a 559 1652 li st 3 setlinewidth np 538 1652 a 526 1652 li st 3 setlinewidth np 505 1652 a 493 1652 li st 3 setlinewidth np 472 1652 a 460 1652 li st 3 setlinewidth np 439 1652 a 427 1652 li st 3 setlinewidth np 406 1652 a 394 1652 li st 3 setlinewidth np 373 1652 a 361 1652 li st 3 setlinewidth np 360 1652 a 360 1640 li st 3 setlinewidth np 360 1617 a 360 1605 li st 3 setlinewidth np 360 1583 a 360 1571 li st 3 setlinewidth np 360 1548 a 360 1536 li st 3 setlinewidth np 360 1514 a 360 1502 li st 3 setlinewidth np 360 1480 a 360 1468 li st 3 setlinewidth np 360 1445 a 360 1433 li st 3 setlinewidth np 360 1411 a 360 1399 li st 3 setlinewidth np 360 1376 a 360 1364 li st 3 setlinewidth np 360 1342 a 360 1330 li st 3 setlinewidth np 360 1307 a 360 1295 li st 3 setlinewidth np 360 1273 a 360 1261 li st 3 setlinewidth np 360 1238 a 360 1226 li st 3 setlinewidth np 360 1204 a 360 1192 li st 3 setlinewidth np 360 1169 a 360 1157 li st 3 setlinewidth np 360 1135 a 360 1123 li st 3 setlinewidth np 360 1101 a 360 1089 li st 3 setlinewidth np 360 1088 a 370.2 1080.8 li st 3 setlinewidth np 388 1070 a 398.2 1062.8 li st 3 setlinewidth np 415 1051 a 425.2 1043.8 li st 3 setlinewidth np 442 1032 a 452.2 1024.8 li st 3 setlinewidth np 469 1014 a 479.2 1006.8 li st 3 setlinewidth np 497 995 a 507.2 987.8 li st 3 setlinewidth np 524 976 a 534.2 968.8 li st 3 setlinewidth np 551 958 a 561.2 950.8 li st 3 setlinewidth np 578 939 a 588.2 931.8 li st 3 setlinewidth np 606 920 a 616.2 912.8 li st 3 setlinewidth np 633 902 a 643.2 894.8 li st 3 setlinewidth np 660 883 a 670.2 875.8 li st 3 setlinewidth np 687 865 a 697.2 857.8 li st 3 setlinewidth np 715 846 a 725.2 838.8 li st 3 setlinewidth np 742 827 a 752.2 819.8 li st 3 setlinewidth np 769 809 a 779.2 801.8 li st 3 setlinewidth np 796 790 a 806.2 782.8 li st 3 setlinewidth np 807 783 a 819 783 li st 3 setlinewidth np 839 783 a 851 783 li st 3 setlinewidth np 872 783 a 884 783 li st 3 setlinewidth np 904 783 a 916 783 li st 3 setlinewidth np 937 783 a 949 783 li st 3 setlinewidth np 969 783 a 981 783 li st 3 setlinewidth np 1002 783 a 1014 783 li st 3 setlinewidth np 1034 783 a 1046 783 li st 3 setlinewidth np 1067 783 a 1079 783 li st 3 setlinewidth np 1100 783 a 1112 783 li st 3 setlinewidth np 3342 783 a 3348.6 793.2 li st 3 setlinewidth np 3359 810 a 3365.6 820.2 li st 3 setlinewidth np 3375 837 a 3381.6 847.2 li st 3 setlinewidth np 3392 863 a 3398.6 873.2 li st 3 setlinewidth np 3408 890 a 3414.6 900.2 li st 3 setlinewidth np 3425 917 a 3431.6 927.2 li st 3 setlinewidth np 3441 944 a 3447.6 954.2 li st 3 setlinewidth np 3458 971 a 3464.6 981.2 li st 3 setlinewidth np 3474 997 a 3480.6 1007.2 li st 3 setlinewidth np 3491 1024 a 3497.6 1034.2 li st 3 setlinewidth np 3507 1051 a 3513.6 1061.2 li st 3 setlinewidth np 3524 1078 a 3530.6 1088.2 li st 3 setlinewidth np 3530 1088 a 3530 1099.4 li st 3 setlinewidth np 3530 1123 a 3530 1134.4 li st 3 setlinewidth np 3530 1157 a 3530 1168.4 li st 3 setlinewidth np 3530 1192 a 3530 1203.4 li st 3 setlinewidth np 3530 1226 a 3530 1237.4 li st 3 setlinewidth np 3530 1261 a 3530 1272.4 li st 3 setlinewidth np 3530 1295 a 3530 1306.4 li st 3 setlinewidth np 3530 1329 a 3530 1340.4 li st 3 setlinewidth np 3530 1364 a 3530 1375.4 li st 3 setlinewidth np 3530 1398 a 3530 1409.4 li st 3 setlinewidth np 3530 1433 a 3530 1444.4 li st 3 setlinewidth np 3530 1467 a 3530 1478.4 li st 3 setlinewidth np 3530 1502 a 3530 1513.4 li st 3 setlinewidth np 3530 1536 a 3530 1547.4 li st 3 setlinewidth np 3530 1571 a 3530 1582.4 li st 3 setlinewidth np 3530 1605 a 3530 1616.4 li st 3 setlinewidth np 3530 1640 a 3530 1651.4 li st 3 setlinewidth np 3530 1652 a 3518 1652 li st 3 setlinewidth np 3497 1652 a 3485 1652 li st 3 setlinewidth np 3464 1652 a 3452 1652 li st 3 setlinewidth np 3431 1652 a 3419 1652 li st 3 setlinewidth np 3398 1652 a 3386 1652 li st 3 setlinewidth np 3365 1652 a 3353 1652 li st 3 setlinewidth np 3332 1652 a 3320 1652 li st 3 setlinewidth np 3299 1652 a 3287 1652 li st 3 setlinewidth np 3266 1652 a 3254 1652 li st 3 setlinewidth np 3232 1652 a 3220 1652 li st 3 setlinewidth np 3199 1652 a 3187 1652 li st 3 setlinewidth np 3166 1652 a 3154 1652 li st 3 setlinewidth np 3133 1652 a 3121 1652 li st 3 setlinewidth np 3100 1652 a 3088 1652 li st 3 setlinewidth np 3067 1652 a 3055 1652 li st 3 setlinewidth np 3034 1652 a 3022 1652 li st 3 setlinewidth np 3001 1652 a 2989 1652 li st 3 setlinewidth np 2968 1652 a 2956 1652 li st 3 setlinewidth np 2934 1652 a 2922 1652 li st 3 setlinewidth np 2901 1652 a 2889 1652 li st 3 setlinewidth np 2868 1652 a 2856 1652 li st 3 setlinewidth np 2835 1652 a 2823 1652 li st 3 setlinewidth np 2802 1652 a 2790 1652 li st 3 setlinewidth np 2769 1652 a 2757 1652 li st 3 setlinewidth np 2736 1652 a 2724 1652 li st 3 setlinewidth np 2703 1652 a 2691 1652 li st 3 setlinewidth np 2670 1652 a 2658 1652 li st 3 setlinewidth np 2637 1652 a 2625 1652 li st 3 setlinewidth np 2603 1652 a 2591 1652 li st 3 setlinewidth np 2591 1652 a 2591 1640 li st 3 setlinewidth np 2591 1617 a 2591 1605 li st 3 setlinewidth np 2591 1583 a 2591 1571 li st 3 setlinewidth np 2591 1548 a 2591 1536 li st 3 setlinewidth np 2591 1514 a 2591 1502 li st 3 setlinewidth np 2591 1480 a 2591 1468 li st 3 setlinewidth np 2591 1445 a 2591 1433 li st 3 setlinewidth np 2591 1411 a 2591 1399 li st 3 setlinewidth np 2591 1376 a 2591 1364 li st 3 setlinewidth np 2591 1342 a 2591 1330 li st 3 setlinewidth np 2591 1307 a 2591 1295 li st 3 setlinewidth np 2591 1273 a 2591 1261 li st 3 setlinewidth np 2591 1238 a 2591 1226 li st 3 setlinewidth np 2591 1204 a 2591 1192 li st 3 setlinewidth np 2591 1169 a 2591 1157 li st 3 setlinewidth np 2591 1135 a 2591 1123 li st 3 setlinewidth np 2591 1101 a 2591 1089 li st 3 setlinewidth np 2591 1088 a 2601.2 1080.8 li st 3 setlinewidth np 2618 1070 a 2628.2 1062.8 li st 3 setlinewidth np 2646 1051 a 2656.2 1043.8 li st 3 setlinewidth np 2673 1032 a 2683.2 1024.8 li st 3 setlinewidth np 2700 1014 a 2710.2 1006.8 li st 3 setlinewidth np 2727 995 a 2737.2 987.8 li st 3 setlinewidth np 2755 976 a 2765.2 968.8 li st 3 setlinewidth np 2782 958 a 2792.2 950.8 li st 3 setlinewidth np 2809 939 a 2819.2 931.8 li st 3 setlinewidth np 2836 920 a 2846.2 912.8 li st 3 setlinewidth np 2864 902 a 2874.2 894.8 li st 3 setlinewidth np 2891 883 a 2901.2 875.8 li st 3 setlinewidth np 2918 865 a 2928.2 857.8 li st 3 setlinewidth np 2945 846 a 2955.2 838.8 li st 3 setlinewidth np 2973 827 a 2983.2 819.8 li st 3 setlinewidth np 3000 809 a 3010.2 801.8 li st 3 setlinewidth np 3027 790 a 3037.2 782.8 li st 3 setlinewidth np 3037 783 a 3049 783 li st 3 setlinewidth np 3070 783 a 3082 783 li st 3 setlinewidth np 3102 783 a 3114 783 li st 3 setlinewidth np 3135 783 a 3147 783 li st 3 setlinewidth np 3167 783 a 3179 783 li st 3 setlinewidth np 3200 783 a 3212 783 li st 3 setlinewidth np 3233 783 a 3245 783 li st 3 setlinewidth np 3265 783 a 3277 783 li st 3 setlinewidth np 3298 783 a 3310 783 li st 3 setlinewidth np 3330 783 a 3342 783 li st 3 setlinewidth np 3882.4 243.4 a 3506.8 478 li st 3 setlinewidth np 3953.2 243.4 a 3905.8 361 li st 3 setlinewidth np 4117.6 243.4 a 4375.6 478 li st 3 setlinewidth np 4046.8 243.4 a 4094.2 361 li st 3 setlinewidth np 4000 243.4 a 4000 361 li st 3 setlinewidth np 3413.2 665.8 a 3131.2 901 li st 3 setlinewidth np 3061 947.8 a 3201.4 947.8 li st 3 setlinewidth np 3084.4 1018 a 3061 1088.8 li st 3 setlinewidth np 3131.2 1018 a 3131.2 1088.8 li st 3 setlinewidth np 3178 1018 a 3201.4 1088.8 li st 3 setlinewidth np 2763 1315 54.6 90.0002 180 arc st 3 setlinewidth np 2763 1237 54.6 180 270.001 arc st 3 setlinewidth np 2889 1237 54.6 270.001 360.001 arc st 3 setlinewidth np 2889 1315 54.6 0 90.0002 arc st 3 setlinewidth np 2708.2 1315.6 a 2708.2 1237.6 li st 3 setlinewidth np 2763.4 1182.4 a 2888.8 1182.4 li st 3 setlinewidth np 2943.4 1237.6 a 2943.4 1315.6 li st 3 setlinewidth np 2888.8 1370.8 a 2763.4 1370.8 li st 3 setlinewidth np 3045 963 54.6 90.0002 180 arc st 3 setlinewidth np 3045 955 54.6 180 270.001 arc st 3 setlinewidth np 3217 955 54.6 270.001 360.001 arc st 3 setlinewidth np 3217 963 54.6 0 90.0002 arc st 3 setlinewidth np 2990.2 963.4 a 2990.2 955.6 li st 3 setlinewidth np 3044.8 901 a 3217 901 li st 3 setlinewidth np 3272.2 955.6 a 3272.2 963.4 li st 3 setlinewidth np 3217 1018 a 3044.8 1018 li st 3 setlinewidth np 3506.8 665.8 a 3530.2 736.6 li st 3 setlinewidth np 3460 665.8 a 3436.6 736.6 li st 3 setlinewidth np 3553.6 665.8 a 3694.6 901 li st 3 setlinewidth np 3718 1018 a 3718 1182.4 li st 3 setlinewidth np 3421 611 54.6 90.0002 180 arc st 3 setlinewidth np 3421 533 54.6 180 270.001 arc st 3 setlinewidth np 3546 533 54.6 270.001 360.001 arc st 3 setlinewidth np 3546 611 54.6 0 90.0002 arc st 3 setlinewidth np 3365.8 611.2 a 3365.8 533.2 li st 3 setlinewidth np 3421 478 a 3545.8 478 li st 3 setlinewidth np 3601 533.2 a 3601 611.2 li st 3 setlinewidth np 3545.8 665.8 a 3421 665.8 li st 3 setlinewidth np 3037 1018 a 2825.8 1182.4 li st 3 setlinewidth np 3303 1315 54.6 90.0002 180 arc st 3 setlinewidth np 3303 1237 54.6 180 270.001 arc st 3 setlinewidth np 3429 1237 54.6 270.001 360.001 arc st 3 setlinewidth np 3429 1315 54.6 0 90.0002 arc st 3 setlinewidth np 3248.8 1315.6 a 3248.8 1237.6 li st 3 setlinewidth np 3303.4 1182.4 a 3428.8 1182.4 li st 3 setlinewidth np 3483.4 1237.6 a 3483.4 1315.6 li st 3 setlinewidth np 3428.8 1370.8 a 3303.4 1370.8 li st 3 setlinewidth np 3224.8 1018 a 3365.8 1182.4 li st 3 setlinewidth np 3656 1315 54.6 90.0002 180 arc st 3 setlinewidth np 3656 1237 54.6 180 270.001 arc st 3 setlinewidth np 3781 1237 54.6 270.001 360.001 arc st 3 setlinewidth np 3781 1315 54.6 0 90.0002 arc st 3 setlinewidth np 3601 1315.6 a 3601 1237.6 li st 3 setlinewidth np 3655.6 1182.4 a 3781 1182.4 li st 3 setlinewidth np 3835.6 1237.6 a 3835.6 1315.6 li st 3 setlinewidth np 3781 1370.8 a 3655.6 1370.8 li st 3 setlinewidth np 4305.4 947.8 a 4446.4 947.8 li st 3 setlinewidth np 4375.6 665.8 a 4375.6 901 li st 3 setlinewidth np 4375.6 1018 a 4375.6 1182.4 li st 3 setlinewidth np 4313 611 54.6 90.0002 180 arc st 3 setlinewidth np 4313 533 54.6 180 270.001 arc st 3 setlinewidth np 4438 533 54.6 270.001 360.001 arc st 3 setlinewidth np 4438 611 54.6 0 90.0002 arc st 3 setlinewidth np 4258.6 611.2 a 4258.6 533.2 li st 3 setlinewidth np 4313.2 478 a 4438.6 478 li st 3 setlinewidth np 4493.2 533.2 a 4493.2 611.2 li st 3 setlinewidth np 4438.6 665.8 a 4313.2 665.8 li st 3 setlinewidth np 4313 1315 54.6 90.0002 180 arc st 3 setlinewidth np 4313 1237 54.6 180 270.001 arc st 3 setlinewidth np 4438 1237 54.6 270.001 360.001 arc st 3 setlinewidth np 4438 1315 54.6 0 90.0002 arc st 3 setlinewidth np 4258.6 1315.6 a 4258.6 1237.6 li st 3 setlinewidth np 4313.2 1182.4 a 4438.6 1182.4 li st 3 setlinewidth np 4493.2 1237.6 a 4493.2 1315.6 li st 3 setlinewidth np 4438.6 1370.8 a 4313.2 1370.8 li st 3 setlinewidth np 4290 963 54.6 90.0002 180 arc st 3 setlinewidth np 4290 955 54.6 180 270.001 arc st 3 setlinewidth np 4462 955 54.6 270.001 360.001 arc st 3 setlinewidth np 4462 963 54.6 0 90.0002 arc st 3 setlinewidth np 4234.6 963.4 a 4234.6 955.6 li st 3 setlinewidth np 4289.8 901 a 4462 901 li st 3 setlinewidth np 4516.6 955.6 a 4516.6 963.4 li st 3 setlinewidth np 4462 1018 a 4289.8 1018 li st 3 setlinewidth np 3929.8 172.6 a 4070.2 172.6 li st 3 setlinewidth np 3843 188 54.6 90.0002 180 arc st 3 setlinewidth np 3843 180 54.6 180 270.001 arc st 3 setlinewidth np 4157 180 54.6 270.001 360.001 arc st 3 setlinewidth np 4157 188 54.6 0 90.0002 arc st 3 setlinewidth np 3788.8 188.8 a 3788.8 181 li st 3 setlinewidth np 3843.4 125.8 a 4156.6 125.8 li st 3 setlinewidth np 4211.2 181 a 4211.2 188.8 li st 3 setlinewidth np 4156.6 243.4 a 3843.4 243.4 li st 4 setlinewidth np [ 4 27.125 ] 0 setdash 313 1723 a 313 971.2 li st [] 0 setdash [] 0 setdash 4 setlinewidth np [ 4 26.6478 ] 0 setdash 313 971.2 a 1158.4 407.8 li st [] 0 setdash [] 0 setdash 4 setlinewidth np [ 4 26.1 ] 0 setdash 1158.4 407.8 a 1463.8 407.8 li st [] 0 setdash [] 0 setdash 4 setlinewidth np [ 4 27.1302 ] 0 setdash 1463.8 407.8 a 1651.6 736.6 li st [] 0 setdash [] 0 setdash 4 setlinewidth np [ 4 26.6875 ] 0 setdash 1651.6 736.6 a 1651.6 1723 li st [] 0 setdash [] 0 setdash 4 setlinewidth np [ 4 27.0233 ] 0 setdash 1651.6 1723 a 313 1723 li st [] 0 setdash [] 0 setdash 4 setlinewidth np [ 4 35.3158 ] 0 setdash 2543.8 1723 a 2543.8 971.2 li st [] 0 setdash [] 0 setdash 4 setlinewidth np [ 4 34.8992 ] 0 setdash 2543.8 971.2 a 3389.2 407.8 li st [] 0 setdash [] 0 setdash 4 setlinewidth np [ 4 33.625 ] 0 setdash 3389.2 407.8 a 3694.6 407.8 li st [] 0 setdash [] 0 setdash 4 setlinewidth np [ 4 33.3562 ] 0 setdash 3694.6 407.8 a 3882.4 736.6 li st [] 0 setdash [] 0 setdash 4 setlinewidth np [ 4 35.28 ] 0 setdash 3882.4 736.6 a 3882.4 1723 li st [] 0 setdash [] 0 setdash 4 setlinewidth np [ 4 35.2353 ] 0 setdash 3882.4 1723 a 2543.8 1723 li st [] 0 setdash [] 0 setdash 4 setlinewidth np [ 4 35.3846 ] 0 setdash 4117.6 1723 a 4634.2 1723 li st [] 0 setdash [] 0 setdash 4 setlinewidth np [ 4 34.5588 ] 0 setdash 4634.2 1723 a 4634.2 407.8 li st [] 0 setdash [] 0 setdash 4 setlinewidth np [ 4 35.3846 ] 0 setdash 4634.2 407.8 a 4117.6 407.8 li st [] 0 setdash [] 0 setdash 4 setlinewidth np [ 4 34.5588 ] 0 setdash 4117.6 407.8 a 4117.6 1723 li st [] 0 setdash [] 0 setdash 3 setlinewidth np 4141 783 a 4153.6 783 li st 3 setlinewidth np 4174 783 a 4186.6 783 li st 3 setlinewidth np 4206 783 a 4218.6 783 li st 3 setlinewidth np 4239 783 a 4251.6 783 li st 3 setlinewidth np 4272 783 a 4284.6 783 li st 3 setlinewidth np 4304 783 a 4316.6 783 li st 3 setlinewidth np 4337 783 a 4349.6 783 li st 3 setlinewidth np 4369 783 a 4381.6 783 li st 3 setlinewidth np 4402 783 a 4414.6 783 li st 3 setlinewidth np 4435 783 a 4447.6 783 li st 3 setlinewidth np 4467 783 a 4479.6 783 li st 3 setlinewidth np 4500 783 a 4512.6 783 li st 3 setlinewidth np 4533 783 a 4545.6 783 li st 3 setlinewidth np 4565 783 a 4577.6 783 li st 3 setlinewidth np 4598 783 a 4610.6 783 li st 3 setlinewidth np 4610 783 a 4610 795 li st 3 setlinewidth np 4610 816 a 4610 828 li st 3 setlinewidth np 4610 849 a 4610 861 li st 3 setlinewidth np 4610 882 a 4610 894 li st 3 setlinewidth np 4610 915 a 4610 927 li st 3 setlinewidth np 4610 948 a 4610 960 li st 3 setlinewidth np 4610 981 a 4610 993 li st 3 setlinewidth np 4610 1014 a 4610 1026 li st 3 setlinewidth np 4610 1047 a 4610 1059 li st 3 setlinewidth np 4610 1079 a 4610 1091 li st 3 setlinewidth np 4610 1112 a 4610 1124 li st 3 setlinewidth np 4610 1145 a 4610 1157 li st 3 setlinewidth np 4610 1178 a 4610 1190 li st 3 setlinewidth np 4610 1211 a 4610 1223 li st 3 setlinewidth np 4610 1244 a 4610 1256 li st 3 setlinewidth np 4610 1277 a 4610 1289 li st 3 setlinewidth np 4610 1310 a 4610 1322 li st 3 setlinewidth np 4610 1343 a 4610 1355 li st 3 setlinewidth np 4610 1376 a 4610 1388 li st 3 setlinewidth np 4610 1409 a 4610 1421 li st 3 setlinewidth np 4610 1442 a 4610 1454 li st 3 setlinewidth np 4610 1475 a 4610 1487 li st 3 setlinewidth np 4610 1508 a 4610 1520 li st 3 setlinewidth np 4610 1541 a 4610 1553 li st 3 setlinewidth np 4610 1574 a 4610 1586 li st 3 setlinewidth np 4610 1606 a 4610 1618 li st 3 setlinewidth np 4610 1639 a 4610 1651 li st 3 setlinewidth np 4610 1652 a 4598 1652 li st 3 setlinewidth np 4578 1652 a 4566 1652 li st 3 setlinewidth np 4545 1652 a 4533 1652 li st 3 setlinewidth np 4512 1652 a 4500 1652 li st 3 setlinewidth np 4480 1652 a 4468 1652 li st 3 setlinewidth np 4447 1652 a 4435 1652 li st 3 setlinewidth np 4415 1652 a 4403 1652 li st 3 setlinewidth np 4382 1652 a 4370 1652 li st 3 setlinewidth np 4349 1652 a 4337 1652 li st 3 setlinewidth np 4317 1652 a 4305 1652 li st 3 setlinewidth np 4284 1652 a 4272 1652 li st 3 setlinewidth np 4251 1652 a 4239 1652 li st 3 setlinewidth np 4219 1652 a 4207 1652 li st 3 setlinewidth np 4186 1652 a 4174 1652 li st 3 setlinewidth np 4153 1652 a 4141 1652 li st 3 setlinewidth np 4141 1652 a 4141 1639.4 li st 3 setlinewidth np 4141 1619 a 4141 1606.4 li st 3 setlinewidth np 4141 1586 a 4141 1573.4 li st 3 setlinewidth np 4141 1553 a 4141 1540.4 li st 3 setlinewidth np 4141 1520 a 4141 1507.4 li st 3 setlinewidth np 4141 1487 a 4141 1474.4 li st 3 setlinewidth np 4141 1454 a 4141 1441.4 li st 3 setlinewidth np 4141 1421 a 4141 1408.4 li st 3 setlinewidth np 4141 1388 a 4141 1375.4 li st 3 setlinewidth np 4141 1355 a 4141 1342.4 li st 3 setlinewidth np 4141 1322 a 4141 1309.4 li st 3 setlinewidth np 4141 1289 a 4141 1276.4 li st 3 setlinewidth np 4141 1257 a 4141 1244.4 li st 3 setlinewidth np 4141 1224 a 4141 1211.4 li st 3 setlinewidth np 4141 1191 a 4141 1178.4 li st 3 setlinewidth np 4141 1158 a 4141 1145.4 li st 3 setlinewidth np 4141 1125 a 4141 1112.4 li st 3 setlinewidth np 4141 1092 a 4141 1079.4 li st 3 setlinewidth np 4141 1059 a 4141 1046.4 li st 3 setlinewidth np 4141 1026 a 4141 1013.4 li st 3 setlinewidth np 4141 993 a 4141 980.4 li st 3 setlinewidth np 4141 960 a 4141 947.4 li st 3 setlinewidth np 4141 927 a 4141 914.4 li st 3 setlinewidth np 4141 894 a 4141 881.4 li st 3 setlinewidth np 4141 861 a 4141 848.4 li st 3 setlinewidth np 4141 828 a 4141 815.4 li st 3 setlinewidth np 4141 795 a 4141 782.4 li st 4 setlinewidth np [ 4 26.1176 ] 0 setdash 1886.8 1723 a 2403.4 1723 li st [] 0 setdash [] 0 setdash 4 setlinewidth np [ 4 27.2143 ] 0 setdash 2403.4 1723 a 2403.4 407.8 li st [] 0 setdash [] 0 setdash 4 setlinewidth np [ 4 26.1176 ] 0 setdash 2403.4 407.8 a 1886.8 407.8 li st [] 0 setdash [] 0 setdash 4 setlinewidth np [ 4 27.2143 ] 0 setdash 1886.8 407.8 a 1886.8 1723 li st [] 0 setdash [] 0 setdash 3 setlinewidth np 2115.4 664.6 a 2098 736.6 li st 3 setlinewidth np 2098 665.8 a 2051.2 736.6 li st 3 setlinewidth np 4346.8 664.6 a 4328.8 736.6 li st 3 setlinewidth np 4328.8 665.8 a 4282 736.6 li st 1727 100 a(s)1204 604 y(G)1278 619 y FA(1)1675 1274 y Ff(.)31 b(.)g(.)1675 945 y(.)g(.)g(.)-805 b(.)31 b(.)g(.)1581 569 y(.)g(.)g(.)2096 603 y FB(G)2170 618 y Fr(k)835 862 y FB(s)874 877 y FA(1)783 1274 y Ff(.)g(.)g(.)2905 b(.)31 b(.)g(.)3906 945 y(.)g(.)g(.)-805 b(.)31 b(.)g(.)3812 569 y(.)g(.)g(.)3014 1274 y(.)g(.)g(.)3953 101 y FB(t)3836 219 y(a)3886 183 y Fw(0)4094 219 y FB(b)4144 183 y Fw(0)4331 603 y FB(H)4403 618 y Fr(k)3438 604 y FB(H)3510 619 y FA(1)3061 862 y FB(t)3089 877 y FA(1)2173 851 y FB(s)2212 866 y Fr(l)2230 876 y Fd(k)4399 851 y FB(t)4427 866 y Fr(l)4445 876 y Fd(k)1469 852 y FB(s)1508 867 y Fr(l)1526 877 y Fl(1)3695 852 y FB(t)3723 867 y Fr(l)3741 877 y Fl(1)1300 52 y FB(S)1365 67 y FA(\()p Fr(a)p Fs(;)p Fr(b)p FA(\))3530 52 y FB(T)3589 67 y FA(\()p Fr(a)3644 48 y Fh(0)3667 67 y Fs(;)p Fr(b)3717 48 y Fh(0)3738 67 y FA(\))830 745 y FB(S)895 760 y FA(1)2262 734 y FB(S)2327 749 y Fr(l)2345 759 y Fd(k)3061 745 y FB(T)3120 760 y FA(1)4493 734 y FB(T)4552 749 y Fr(l)4570 759 y Fd(k)1182 359 y FB(S)1245 374 y Fr(G)1295 384 y Fl(1)2168 358 y FB(S)2231 373 y Fr(G)2281 383 y Fd(k)4399 358 y FB(T)4460 373 y Fr(H)4508 383 y Fd(k)3413 359 y FB(T)3474 374 y Fr(H)3522 384 y Fl(1)257 1734 y currentpoint currentpoint translate 1 0.72 div 1 0.72 div scale neg exch neg exch translate 257 1734 a Black 1066 2045 a FF(Figure)25 b(3:)1459 2045 y SDict begin H.S end 1459 2045 a Black Black 1459 2045 a SDict begin H.R end 1459 2045 a 1459 2045 a SDict begin [ /View [/XYZ H.V] /Dest (figure.3) cvn H.B /DEST pdfmark end 1459 2045 a FF(T)m(riconnected)f(component)g (trees.)p Black Black Black 378 2333 a(2.)p Black 48 w Fv(j)p FB(S)586 2348 y FA(\()p Fr(a)p Fs(;)p Fr(b)p FA(\))717 2333 y Fv(j)k FE(=)g Fv(j)p FB(T)935 2348 y FA(\()p Fr(a)990 2329 y Fh(0)1012 2348 y Fs(;)p Fr(b)1073 2329 y Fh(0)1094 2348 y FA(\))1121 2333 y Fv(j)d FF(b)n(ut)f(#)5 b FB(s)27 b FC(<)h FF(#)p FB(t)e FF(or)257 2403 y SDict begin H.S end 257 2403 a 257 2403 a SDict begin 14.5 H.A end 257 2403 a 257 2403 a SDict begin [ /View [/XYZ H.V] /Dest (Item.3) cvn H.B /DEST pdfmark end 257 2403 a Black 378 2540 a FF(3.)p Black 75 w Fv(j)p FB(S)613 2555 y FA(\()p Fr(a)p Fs(;)p Fr(b)p FA(\))745 2540 y Fv(j)31 b FE(=)i Fv(j)p FB(T)971 2555 y FA(\()p Fr(a)1026 2536 y Fh(0)1048 2555 y Fs(;)p Fr(b)1109 2536 y Fh(0)1130 2555 y FA(\))1156 2540 y Fv(j)p FF(,)27 b(#)5 b FB(s)32 b FE(=)g FF(#)p FB(t)h FE(=)f FB(k)r FF(,)c(b)n(ut)e(\()p FB(S)2002 2555 y Fr(G)2052 2565 y Fl(1)2086 2540 y FC(;)17 b(:)g(:)g(:)e(;)i FB(S)2357 2555 y Fr(G)2407 2565 y Fd(k)2437 2540 y FF(\))32 b FC(<)2565 2555 y Fg(T)2636 2540 y FF(\()p FB(T)2730 2555 y Fr(H)2778 2565 y Fl(1)2812 2540 y FC(;)17 b(:)g(:)g(:)e(;)i FB(T)3081 2555 y Fr(H)3129 2565 y Fd(k)3160 2540 y FF(\))27 b(le)o(xico-)501 2660 y(graphically)-6 b(,)22 b(where)i(we)f(assume)f(that)h FB(S)1949 2675 y Fr(G)1999 2685 y Fl(1)2059 2660 y Fq(6)2122 2675 y Fg(T)2190 2660 y FC(:)17 b(:)g(:)26 b Fq(6)2388 2675 y Fg(T)2455 2660 y FB(S)2518 2675 y Fr(G)2568 2685 y Fd(k)2622 2660 y FF(and)d FB(T)2850 2675 y Fr(H)2898 2685 y Fl(1)2959 2660 y Fq(6)3022 2675 y Fg(T)3089 2660 y FC(:)17 b(:)g(:)27 b Fq(6)3288 2675 y Fg(T)3355 2660 y FB(T)3416 2675 y Fr(H)3464 2685 y Fd(k)501 2781 y FF(are)32 b(the)f(ordered)h(subtrees)e(of)h FB(S)1679 2796 y FA(\()p Fr(a)p Fs(;)p Fr(b)p FA(\))1842 2781 y FF(and)g FB(T)2076 2796 y FA(\()p Fr(a)2131 2776 y Fh(0)2153 2796 y Fs(;)p Fr(b)2214 2776 y Fh(0)2236 2796 y FA(\))2262 2781 y FF(,)i(respecti)n (v)o(ely)-6 b(.)47 b(T)-8 b(o)31 b(compute)f(the)501 2901 y(order)j(between)f(the)g(subtrees)g FB(S)1684 2916 y Fr(G)1734 2926 y Fd(i)1788 2901 y FF(and)g FB(T)2025 2916 y Fr(H)2073 2926 y Fd(i)2127 2901 y FF(we)h(compare)f(le)o (xicographically)e(the)501 3021 y(canons)23 b(of)d FB(G)978 3036 y Fr(i)1023 3021 y FF(and)25 b FB(H)1264 3036 y Fr(i)1309 3021 y FF(and)e FB(r)l(ecur)o(sively)f FF(the)h(subtrees)f (rooted)g(at)h(the)f(children)g(of)f FB(G)3473 3036 y Fr(i)501 3142 y FF(and)27 b FB(H)744 3157 y Fr(i)767 3142 y FF(.)k(Note)24 b(that)h(these)f(children)h(are)g(again)g (separating)f(pair)h(nodes.)257 3206 y SDict begin H.S end 257 3206 a 257 3206 a SDict begin 14.5 H.A end 257 3206 a 257 3206 a SDict begin [ /View [/XYZ H.V] /Dest (Item.4) cvn H.B /DEST pdfmark end 257 3206 a Black 378 3348 a FF(4.)p Black 74 w Fv(j)p FB(S)612 3363 y FA(\()p Fr(a)p Fs(;)p Fr(b)p FA(\))743 3348 y Fv(j)30 b FE(=)g Fv(j)p FB(T)965 3363 y FA(\()p Fr(a)1020 3344 y Fh(0)1043 3363 y Fs(;)p Fr(b)1104 3344 y Fh(0)1125 3363 y FA(\))1151 3348 y Fv(j)p FF(,)c(#)5 b FB(s)30 b FE(=)g FF(#)p FB(t)h FE(=)f FB(k)r FF(,)c(\()p FB(S)1834 3363 y Fr(G)1884 3373 y Fl(1)1947 3348 y Fq(6)2010 3363 y Fg(T)2080 3348 y FC(:)17 b(:)g(:)29 b Fq(6)2281 3363 y Fg(T)2350 3348 y FB(S)2413 3363 y Fr(G)2463 3373 y Fd(k)2494 3348 y FF(\))h FE(=)2620 3363 y Fg(T)2689 3348 y FF(\()p FB(T)2783 3363 y Fr(H)2831 3373 y Fl(1)2894 3348 y Fq(6)2957 3363 y Fg(T)3027 3348 y FC(:)17 b(:)g(:)29 b Fq(6)3228 3363 y Fg(T)3297 3348 y FB(T)3358 3363 y Fr(H)3406 3373 y Fd(k)3437 3348 y FF(\),)501 3469 y(b)n(ut)41 b(\()p FB(O)773 3484 y FA(1)811 3469 y FC(;)17 b(:)g(:)g(:)e(;)i FB(O)1096 3484 y Fr(p)1133 3469 y FF(\))59 b FC(<)h FF(\()p FB(O)1453 3484 y FA(1)1507 3432 y Fw(0)1534 3469 y FC(;)17 b(:)g(:)g(:)e(;)i FB(O)1819 3484 y Fr(p)1872 3432 y Fw(0)1900 3469 y FF(\))42 b(le)o(xicographically)-6 b(,)43 b(where)g FB(O)3097 3484 y Fr(j)3161 3469 y FF(and)f FB(O)3429 3484 y Fr(j)3468 3432 y Fw(0)501 3589 y FF(are)33 b(the)f(orientation)f(counters)g(of)h (the)47 b FB(j)1954 3553 y Fr(t)q(h)2043 3589 y FF(isomorphism)29 b(classes)34 b FB(I)2949 3604 y Fr(j)3004 3589 y FF(and)g FB(I)3220 3553 y Fw(0)3225 3618 y Fr(j)3281 3589 y FF(of)e(all)501 3709 y(the)k FB(S)722 3724 y Fr(G)772 3734 y Fd(i)793 3709 y FF(')-5 b(s)36 b(and)f(the)g FB(T)1293 3724 y Fr(H)1341 3734 y Fd(i)1364 3709 y FF(')-5 b(s.)62 b(\(The)36 b(concept)f(of)h(orientation)e(counter)h(is)g(e)o(xplained)501 3830 y(later\).)404 4062 y(W)-8 b(e)26 b(say)f(that)g(tw)o(o)g (triconnected)g(component)f(trees)i FB(S)2332 4077 y Fr(e)2391 4062 y FF(and)g FB(T)2620 4077 y Fr(e)2649 4058 y Fh(0)2701 4062 y FF(are)g FB(equal)f(accor)l(ding)257 4183 y(to)e(the)g(isomorphism)e(or)l(der)r FF(,)j(denoted)e(by)h FB(S)1836 4198 y Fr(e)1897 4183 y FE(=)1960 4198 y Fg(T)2027 4183 y FB(T)2086 4198 y Fr(e)2115 4178 y Fh(0)2142 4183 y FF(,)g(if)h(neither)f FB(S)2640 4198 y Fr(e)2701 4183 y FC(<)2764 4198 y Fg(T)2831 4183 y FB(T)2890 4198 y Fr(e)2919 4178 y Fh(0)2968 4183 y FF(nor)h FB(T)3184 4198 y Fr(e)3213 4178 y Fh(0)3267 4183 y FC(<)3330 4198 y Fg(T)3397 4183 y FB(S)3462 4198 y Fr(e)257 4303 y FF(holds.)38 b(T)-8 b(w)o(o)27 b(trees)g(are)h FE(=)1165 4318 y Fg(T)1205 4303 y FF(-equal,)g(precisely)f(when)g(the)g(underlying)f(graphs)h(are) h(isomor)n(-)257 4423 y(phic.)404 4544 y(W)-8 b(e)35 b(summarize)f(no)n(w)-6 b(,)36 b(ho)n(w)e(we)h(can)g(compute)f(the)g (isomorphism)e(order)j(when)g(we)257 4665 y(compare)d(subtrees)f (rooted)h(at)f(separating)h(pairs,)h(e.g.)e FB(S)2314 4680 y FA(\()p Fr(a)p Fs(;)p Fr(b)p FA(\))2477 4665 y FF(and)h FB(T)2712 4680 y FA(\()p Fr(a)2767 4660 y Fh(0)2789 4680 y Fs(;)p Fr(b)2850 4660 y Fh(0)2871 4680 y FA(\))2898 4665 y FF(,)h(and)f(when)f(we)257 4785 y(compare)25 b(subtrees)g (rooted)f(at)h(triconnected)f(components,)g(e.g.)g FB(S)2633 4800 y Fr(G)2683 4810 y Fd(i)2730 4785 y FF(and)h FB(T)2960 4800 y Fr(H)3015 4810 y Fd(j)3038 4785 y FF(.)404 4906 y(Comparing)33 b FB(S)951 4921 y FA(\()p Fr(a)p Fs(;)p Fr(b)p FA(\))1115 4906 y FF(and)h FB(T)1352 4921 y FA(\()p Fr(a)1407 4902 y Fh(0)1429 4921 y Fs(;)p Fr(b)1479 4902 y Fh(0)1500 4921 y FA(\))1560 4906 y FF(is)f(similar)g(to)g(the)g (comparison)g(of)g(subtrees)h(in)f(Lin-)257 5026 y(dell')-5 b(s)24 b(algorithm.)29 b(W)-8 b(e)25 b(mak)o(e)f(a)h(cross-comparison)e (of)h(the)h(children)f(and)g(store)g(the)g(coun-)257 5147 y(ters)h FB(c)470 5162 y Fs(<)517 5147 y FC(;)17 b FB(c)603 5162 y Fz(=)649 5147 y FC(;)g FB(c)735 5162 y Fs(>)806 5147 y FF(for)25 b(their)g(order)g(pro\002le.)404 5268 y(Assume,)i(both)g(subtrees)g(are)h(of)g(equal)f(size,)h(i.e.)f Fv(j)p FB(S)2280 5283 y Fr(G)2330 5293 y Fd(i)2352 5268 y Fv(j)33 b FE(=)g Fv(j)p FB(T)2582 5283 y Fr(H)2637 5293 y Fd(j)2660 5268 y Fv(j)f FE(=)j FB(N)6 b FF(,)29 b(both)e(rooted)g(at)257 5388 y(triconnected)e(component)e(nodes)g FB(G)1572 5403 y Fr(i)1619 5388 y FF(and)k FB(H)1872 5403 y Fr(j)1895 5388 y FF(,)e(respecti)n(v)o(ely)-6 b(.)p Black Black eop end %%Page: 14 14 TeXDict begin 14 13 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 257 266 a SDict begin H.S end 257 266 a Black Black 257 266 a SDict begin H.R end 257 266 a 257 266 a SDict begin [ /View [/XYZ H.V] /Dest (page.14) cvn H.B /DEST pdfmark end 257 266 a Black 404 573 a FF(First,)33 b(we)e(compare)h(the)f(types)g(of)f FB(G)1749 588 y Fr(i)1803 573 y FF(and)j FB(H)2062 588 y Fr(j)2085 573 y FF(.)51 b(W)-8 b(e)32 b(say)g(that)f(bonds)f Fq(6)3001 588 y Fg(T)3072 573 y FF(c)o(ycles)h(and)257 693 y(c)o(ycles)k Fq(6)603 708 y Fg(T)678 693 y FF(3-connected)h (components.)61 b(3-bonds)35 b(are)h(al)o(w)o(ays)f(equal.)63 b(If)36 b(both)e(are)i(c)o(y-)257 814 y(cles)c(or)f(3-connected)g (components)f(then)h(we)h(construct)e(the)h(canons)g(of)f FB(G)2984 829 y Fr(i)3038 814 y FF(and)j FB(H)3297 829 y Fr(j)3351 814 y FF(and)257 934 y(compare)25 b(all)g(of)g(them)f (bit-by-bit.)404 1063 y(T)-8 b(o)31 b(canonize)g(a)h(c)o(ycle,)g(we)g (tra)n(v)o(erse)f(it)g(starting)f(from)h(the)g(virtual)g(edge)g(which)g (cor)n(-)257 1183 y(responds)f(to)g(its)g(parent)h(\(i.e.)48 b(the)31 b(parent)f(node)h(of)e FB(G)2201 1198 y Fr(i)2223 1183 y FF(\),)j(and)f(then)f(tra)n(v)o(ersing)g(the)g(entire)257 1303 y(c)o(ycle)h(along)f(the)g(edges)h(encountered.)48 b(There)31 b(are)h(tw)o(o)e(possible)f(tra)n(v)o(ersals)h(depending)257 1424 y(on)d(which)g(direction)g(of)g(the)h(starting)e(edge)h(is)g (chosen.)38 b(Thus,)28 b(a)f(c)o(ycle)h(has)f(tw)o(o)g(possible)257 1544 y(canons.)404 1673 y(T)-8 b(o)39 b(canonize)h(a)g(3-connected)g (component)c FB(G)2105 1688 y Fr(i)2128 1673 y FF(,)43 b(we)d(use)g(the)f(log-space)h(algorithm)257 1793 y(from)26 b(Datta,)h(Limaye,)e(and)i(Nimbhorkar)e([)1817 1794 y SDict begin H.S end 1817 1794 a Black -1 x FF(5)p Black 1867 1726 a SDict begin H.R end 1867 1726 a 1867 1793 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.DLN08) cvn H.B /ANN pdfmark end 1867 1793 a FF(].)35 b(The)26 b(canon)g(depends)g (on)g(the)g(direction)f(of)257 1914 y(the)h(starting)e(edge)i(and)f (additionally)-6 b(,)23 b(on)j(the)f(embedding)f(of)i(the)f(component)d FB(G)3169 1929 y Fr(i)3192 1914 y FF(.)33 b(F)o(or)25 b(3-)257 2034 y(connected)f(components,)e(there)i(are)h(tw)o(o)e (possible)f(embeddings.)29 b(Hence,)24 b(we)g(ha)n(v)o(e)f(up)h(to)257 2154 y(four)h(possible)f(canons.)404 2283 y(In)29 b(the)g(bit-by-bit)f (comparison,)h(we)g(ha)n(v)o(e)g(to)g(distinguish)d(se)n(v)o(eral)j (cases.)44 b(When)29 b(we)257 2404 y(reach)i(virtual)d(edges)i(in)f (the)g(comparison)f(steps,)i(we)g(go)f(into)g(recursion)g(at)g(the)h (subtrees)257 2524 y(rooted)21 b(at)h(the)f(corresponding)f(separating) h(pairs.)29 b(If)22 b(we)g(\002nd)f(in)g(the)g(recursion)g(that)g(one)g (of)257 2644 y(the)29 b(subtrees)g(is)g(smaller)f(than)h(the)g(other)l (,)h(then)f(we)h(ha)n(v)o(e)e(found)h(an)g(inequality)f(between)257 2765 y(the)g(current)g(canons)f(we)h(compare.)39 b(W)-8 b(e)28 b(eliminate)e(the)i(canons)f(which)g(are)i(not)e(found)g(to)257 2885 y(be)35 b(minimal.)59 b(At)34 b(the)h(end,)i(if)e(there)g(remains) f(a)h(canon)g(for)e FB(G)2556 2900 y Fr(i)2613 2885 y FF(and)i(for)i FB(H)3027 2900 y Fr(j)3050 2885 y FF(,)g(then)d(both)257 3006 y(subtrees)25 b FB(S)672 3021 y Fr(G)722 3031 y Fd(i)769 3006 y FF(and)f FB(T)998 3021 y Fr(H)1053 3031 y Fd(j)1101 3006 y FF(are)h(equal)g(up)g(to)f(step)h(3.)257 3231 y SDict begin H.S end 257 3231 a 257 3231 a SDict begin 14.5 H.A end 257 3231 a 257 3231 a SDict begin [ /View [/XYZ H.V] /Dest (section*.6) cvn H.B /DEST pdfmark end 257 3231 a 94 x Fp(Orientation)31 b(counters.)100 b FF(Here)31 b(it)f(does)f(not)h(su)p FE(\016)p FF(ce)g(to)g(stop)f(after)i(step)e (3.)46 b(W)-8 b(e)31 b(need)f(a)257 3445 y(further)g(comparison)f(step) g(to)g(ensure)h(that)d FB(G)k FF(and)h FB(H)j FF(are)30 b(indeed)f(isomorphic.)44 b(W)-8 b(e)30 b(gi)n(v)o(e)257 3566 y(an)j(e)o(xample)e(illustrating)f(this)h(in)h(Figure)1795 3566 y SDict begin H.S end 1795 3566 a Black FF(4)p Black 1844 3498 a SDict begin H.R end 1844 3498 a 1844 3566 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (figure.4) cvn H.B /ANN pdfmark end 1844 3566 a FF(.)53 b(Assume)32 b(that)k FB(s)c FF(and)g FB(t)i FF(ha)n(v)o(e)e(tw)o(o)g(children)257 3686 y(each,)d FB(G)567 3701 y FA(1)605 3686 y FF(,)f FB(G)732 3701 y FA(2)800 3686 y FF(and)33 b FB(H)1049 3701 y FA(1)1087 3686 y FF(,)f FB(H)1216 3701 y FA(2)1285 3686 y FF(such)e(that)e FB(G)1750 3701 y FA(1)1826 3686 y Fj(\033)40 b FB(H)2001 3701 y FA(1)2070 3686 y FF(and)28 b FB(G)2316 3701 y FA(2)2392 3686 y Fj(\033)40 b FB(H)2567 3701 y FA(2)2605 3686 y FF(.)48 b(Still)30 b(we)h(cannot)f(con-)257 3806 y(clude)35 b(that)d FB(G)37 b FF(and)g FB(H)i FF(are)d(isomorphic) d(because)i(it)f(is)h(possible)e(that)h(the)h(isomorphism)257 3927 y(between)c FB(G)694 3942 y FA(1)764 3927 y FF(and)k FB(H)1015 3942 y FA(1)1086 3927 y FF(maps)d FB(a)h FF(to)f FB(a)1572 3891 y Fw(0)1632 3927 y FF(and)h FB(b)f FF(to)h FB(b)2069 3891 y Fw(0)2095 3927 y FF(,)i(b)n(ut)d(the)h(isomorphism)d (between)h FB(G)3458 3942 y FA(2)257 4047 y FF(and)37 b FB(H)510 4062 y FA(2)582 4047 y FF(maps)d FB(a)g FF(to)g FB(b)1090 4011 y Fw(0)1151 4047 y FF(and)g FB(b)g FF(to)g FB(a)1575 4011 y Fw(0)1602 4047 y FF(.)59 b(Then)35 b(these)f(tw)o(o)g (isomorphisms)d(cannot)j(be)g(e)o(x-)257 4168 y(tended)25 b(to)f(an)h(isomorphism)e(between)f FB(G)27 b FF(and)g FB(H)t FF(.)404 4296 y(T)-8 b(o)33 b(handle)g(this)g(problem,)i(we)f (introduce)f(the)g(notion)g(of)g(an)h FB(orientation)e(of)h(a)g(sep-) 257 4417 y(ar)o(ating)j(pair)r FF(.)70 b(A)38 b(separating)g(pair)g (gets)f(an)h(orientation)f(from)h(subtrees)f(rooted)h(at)g(its)257 4537 y(children.)51 b(Also,)32 b(e)n(v)o(ery)f(subtree)g(rooted)g(at)h (a)g(triconnected)f(component)f(node)h(gi)n(v)o(es)f(an)257 4657 y(orientation)f(to)g(the)h(parent)g(separating)g(pair)-5 b(.)45 b(If)30 b(the)g(orientation)f(is)g(consistent,)h(then)f(we)257 4778 y(de\002ne)c FB(S)590 4793 y FA(\()p Fr(a)p Fs(;)p Fr(b)p FA(\))749 4778 y FE(=)812 4793 y Fg(T)879 4778 y FB(T)938 4793 y FA(\()p Fr(a)993 4774 y Fh(0)1015 4793 y Fs(;)p Fr(b)1076 4774 y Fh(0)1098 4793 y FA(\))1148 4778 y FF(and)f(we)h(will)e(sho)n(w)h(that)d FB(G)27 b FF(and)f FB(H)j FF(are)c(isomorphic)e(in)h(this)f(case.)404 4907 y(W)-8 b(e)26 b(de\002ne)h(the)g FB(orientation)d(given)i(to)g (the)g(par)l(ent)g(separ)o(ating)e(pair)i(of)38 b(G)3059 4922 y Fr(i)3108 4907 y FF(and)29 b FB(H)3363 4922 y Fr(j)3412 4907 y FF(as)257 5027 y(the)k(direction)e(in)h(which)g(the)g (minimum)e(canon)j(tra)n(v)o(erses)f(this)f(edge.)54 b(If)32 b(the)h(minimum)257 5147 y(canons)22 b(are)h(obtained)e(for)h (both)f(choices)h(of)g(directions)f(of)h(the)g(edge)g(then)g(we)g(say)g (that)f FB(S)3423 5162 y Fr(G)3473 5172 y Fd(i)257 5268 y FF(and)j FB(T)486 5283 y Fr(H)541 5293 y Fd(j)588 5268 y FF(are)h FB(symmetric)e(about)g(their)g(par)l(ent)h(separ)o(ating)d (pair)r FF(,)j(and)g(thus)f(do)g(not)h(gi)n(v)o(e)e(an)257 5388 y(orientation.)p Black Black eop end %%Page: 15 15 TeXDict begin 15 14 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 257 266 a SDict begin H.S end 257 266 a Black Black 257 266 a SDict begin H.R end 257 266 a 257 266 a SDict begin [ /View [/XYZ H.V] /Dest (page.15) cvn H.B /DEST pdfmark end 257 266 a Black Black Black Black 410 1489 a currentpoint currentpoint translate 0.72 0.72 scale neg exch neg exch translate 410 1489 a 3 setlinewidth np 1069.8 867.2 a 1304.4 1101.8 li 1069.8 1337 li 600 1337 li 600 867.2 li 1069.8 1337 li 1069.8 867.2 li 600 1337 li st 3 setlinewidth np 1680 867.2 a 1445.4 1101.8 li 1680 1337 li 2149.8 1337 li 2149.8 867.2 li 1680 1337 li 1680 867.2 li 2149.8 1337 li st 0 setgray 3 setlinewidth np 600 397 9.6 0 360.001 arc fil 0 setgray np 600 397 9.6 0 360.001 arc st 3 setlinewidth np 600 397 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 1070 397 9.6 0 360.001 arc fil 0 setgray np 1070 397 9.6 0 360.001 arc st 3 setlinewidth np 1070 397 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 1305 632 9.6 0 360.001 arc fil 0 setgray np 1305 632 9.6 0 360.001 arc st 3 setlinewidth np 1305 632 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 835 632 9.6 0 360.001 arc fil 0 setgray np 835 632 9.6 0 360.001 arc st 3 setlinewidth np 835 632 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 600 1336 9.6 0 360.001 arc fil 0 setgray np 600 1336 9.6 0 360.001 arc st 3 setlinewidth np 600 1336 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 1070 1336 9.6 0 360.001 arc fil 0 setgray np 1070 1336 9.6 0 360.001 arc st 3 setlinewidth np 1070 1336 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 835 1102 9.6 0 360.001 arc fil 0 setgray np 835 1102 9.6 0 360.001 arc st 3 setlinewidth np 835 1102 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 1070 632 9.6 0 360.001 arc fil 0 setgray np 1070 632 9.6 0 360.001 arc st 3 setlinewidth np 1070 632 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 1070 1102 9.6 0 360.001 arc fil 0 setgray np 1070 1102 9.6 0 360.001 arc st 3 setlinewidth np 1070 1102 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 1305 1102 9.6 0 360.001 arc fil 0 setgray np 1305 1102 9.6 0 360.001 arc st 3 setlinewidth np 1305 1102 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 1680 397 9.6 0 360.001 arc fil 0 setgray np 1680 397 9.6 0 360.001 arc st 3 setlinewidth np 1680 397 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 2150 397 9.6 0 360.001 arc fil 0 setgray np 2150 397 9.6 0 360.001 arc st 3 setlinewidth np 2150 397 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 2385 632 9.6 0 360.001 arc fil 0 setgray np 2385 632 9.6 0 360.001 arc st 3 setlinewidth np 2385 632 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 1915 632 9.6 0 360.001 arc fil 0 setgray np 1915 632 9.6 0 360.001 arc st 3 setlinewidth np 1915 632 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 2150 867 9.6 0 360.001 arc fil 0 setgray np 2150 867 9.6 0 360.001 arc st 3 setlinewidth np 2150 867 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 1680 867 9.6 0 360.001 arc fil 0 setgray np 1680 867 9.6 0 360.001 arc st 3 setlinewidth np 1680 867 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 1680 867 9.6 0 360.001 arc fil 0 setgray np 1680 867 9.6 0 360.001 arc st 3 setlinewidth np 1680 867 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 1680 1336 9.6 0 360.001 arc fil 0 setgray np 1680 1336 9.6 0 360.001 arc st 3 setlinewidth np 1680 1336 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 2150 1336 9.6 0 360.001 arc fil 0 setgray np 2150 1336 9.6 0 360.001 arc st 3 setlinewidth np 2150 1336 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 2150 867 9.6 0 360.001 arc fil 0 setgray np 2150 867 9.6 0 360.001 arc st 3 setlinewidth np 2150 867 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 1915 1102 9.6 0 360.001 arc fil 0 setgray np 1915 1102 9.6 0 360.001 arc st 3 setlinewidth np 1915 1102 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 1680 1102 9.6 0 360.001 arc fil 0 setgray np 1680 1102 9.6 0 360.001 arc st 3 setlinewidth np 1680 1102 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 2150 632 9.6 0 360.001 arc fil 0 setgray np 2150 632 9.6 0 360.001 arc st 3 setlinewidth np 2150 632 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 1446 1102 9.6 0 360.001 arc fil 0 setgray np 1446 1102 9.6 0 360.001 arc st 3 setlinewidth np 1446 1102 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 3348 162 9.6 0 360.001 arc fil 0 setgray np 3348 162 9.6 0 360.001 arc st 3 setlinewidth np 3348 162 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 3629 162 9.6 0 360.001 arc fil 0 setgray np 3629 162 9.6 0 360.001 arc st 3 setlinewidth np 3629 162 9.6 0 360.001 arc st 3 setlinewidth np 3555 515 a 3657 617 li 3555 718.4 li 3555 515 li 3351.6 718.4 li 3555 718.4 li 3351.6 515 li 3351.6 718.4 li st 3 setlinewidth np 3348 515 a 3374.4 515 li st 3 setlinewidth np 3409 515 a 3435.4 515 li st 3 setlinewidth np 3471 515 a 3497.4 515 li st 3 setlinewidth np 3532 515 a 3558.4 515 li st 3 setlinewidth np 3966 1247 a 3864 1349 li 3966 1450.4 li 3966 1247 li 4169.4 1450.4 li 3966 1450.4 li 4169.4 1247 li 4169.4 1450.4 li st 3 setlinewidth np 4174 1247 a 4148.2 1247 li st 3 setlinewidth np 4112 1247 a 4086.2 1247 li st 3 setlinewidth np 4050 1247 a 4024.2 1247 li st 3 setlinewidth np 3989 1247 a 3963.2 1247 li st 3 setlinewidth np 4118.4 515 a 4220.4 617 li 4118.4 718.4 li 4118.4 515 li 3915 718.4 li 4118.4 718.4 li 3915 515 li 3915 718.4 li st 3 setlinewidth np 3911 515 a 3937.4 515 li st 3 setlinewidth np 3973 515 a 3999.4 515 li st 3 setlinewidth np 4034 515 a 4060.4 515 li st 3 setlinewidth np 4096 515 a 4122.4 515 li st 0 setgray 3 setlinewidth np 600 867 9.6 0 360.001 arc fil 0 setgray np 600 867 9.6 0 360.001 arc st 3 setlinewidth np 600 867 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 1070 867 9.6 0 360.001 arc fil 0 setgray np 1070 867 9.6 0 360.001 arc st 3 setlinewidth np 1070 867 9.6 0 360.001 arc st 3 setlinewidth np 3308 953 54.6 90.0002 180 arc st 3 setlinewidth np 3308 875 54.6 180 270.001 arc st 3 setlinewidth np 3668 875 54.6 270.001 360.001 arc st 3 setlinewidth np 3668 953 54.6 0 90.0002 arc st 3 setlinewidth np 3253.2 953 a 3253.2 875 li st 3 setlinewidth np 3308.4 820.4 a 3668.4 820.4 li st 3 setlinewidth np 3723 875 a 3723 953 li st 3 setlinewidth np 3668.4 1008.2 a 3308.4 1008.2 li st 3 setlinewidth np 3347.4 890.6 a 3629.4 890.6 li st 3 setlinewidth np 2745 1422 54.6 90.0002 180 arc st 3 setlinewidth np 2745 1250 54.6 180 270.001 arc st 3 setlinewidth np 3105 1250 54.6 270.001 360.001 arc st 3 setlinewidth np 3105 1422 54.6 0 90.0002 arc st 3 setlinewidth np 2689.8 1422.8 a 2689.8 1250.6 li st 3 setlinewidth np 2744.4 1196 a 3104.4 1196 li st 3 setlinewidth np 3159.6 1250.6 a 3159.6 1422.8 li st 3 setlinewidth np 3104.4 1478 a 2744.4 1478 li st 3 setlinewidth np 3308 1422 54.6 90.0002 180 arc st 3 setlinewidth np 3308 1250 54.6 180 270.001 arc st 3 setlinewidth np 3668 1250 54.6 270.001 360.001 arc st 3 setlinewidth np 3668 1422 54.6 0 90.0002 arc st 3 setlinewidth np 3253.2 1422.8 a 3253.2 1250.6 li st 3 setlinewidth np 3308.4 1196 a 3668.4 1196 li st 3 setlinewidth np 3723 1250.6 a 3723 1422.8 li st 3 setlinewidth np 3668.4 1478 a 3308.4 1478 li st 3 setlinewidth np 3347.4 1008.2 a 3065.4 1196 li st 3 setlinewidth np 3488.4 1008.2 a 3488.4 1196 li st 3 setlinewidth np 600 397.4 a 1069.8 397.4 li 1304.4 632.6 li 1069.8 867.2 li 1069.8 397.4 li 600 867.2 li 1069.8 867.2 li 600 397.4 li 600 867.2 li st 3 setlinewidth np 1069.8 632.6 a 1304.4 632.6 li st 3 setlinewidth np 1680 397.4 a 2149.8 397.4 li 2384.4 632.6 li 2149.8 867.2 li 2149.8 397.4 li 1680 867.2 li 2149.8 867.2 li 1680 397.4 li 1680 867.2 li st 3 setlinewidth np 1069.8 1101.8 a 1304.4 1101.8 li st 3 setlinewidth np 1445.4 1101.8 a 1680 1101.8 li st 3 setlinewidth np 2149.8 632.6 a 2384.4 632.6 li st 3 setlinewidth np 3347.4 162.8 a 3629.4 162.8 li st 3 setlinewidth np 2745 695 54.6 90.0002 180 arc st 3 setlinewidth np 2745 522 54.6 180 270.001 arc st 3 setlinewidth np 3105 522 54.6 270.001 360.001 arc st 3 setlinewidth np 3105 695 54.6 0 90.0002 arc st 3 setlinewidth np 2689.8 695 a 2689.8 522.8 li st 3 setlinewidth np 2744.4 468.2 a 3104.4 468.2 li st 3 setlinewidth np 3159.6 522.8 a 3159.6 695 li st 3 setlinewidth np 3104.4 749.6 a 2744.4 749.6 li st 3 setlinewidth np 3308 225 54.6 90.0002 180 arc st 3 setlinewidth np 3308 147 54.6 180 270.001 arc st 3 setlinewidth np 3668 147 54.6 270.001 360.001 arc st 3 setlinewidth np 3668 225 54.6 0 90.0002 arc st 3 setlinewidth np 3253.2 225.2 a 3253.2 147.2 li st 3 setlinewidth np 3308.4 92 a 3668.4 92 li st 3 setlinewidth np 3723 147.2 a 3723 225.2 li st 3 setlinewidth np 3668.4 280.4 a 3308.4 280.4 li st 3 setlinewidth np 3347.4 280.4 a 3065.4 468.2 li st 3 setlinewidth np 3488.4 280.4 a 3488.4 468.2 li st 3 setlinewidth np 3308 695 54.6 90.0002 180 arc st 3 setlinewidth np 3308 522 54.6 180 270.001 arc st 3 setlinewidth np 3668 522 54.6 270.001 360.001 arc st 3 setlinewidth np 3668 695 54.6 0 90.0002 arc st 3 setlinewidth np 3253.2 695 a 3253.2 522.8 li st 3 setlinewidth np 3308.4 468.2 a 3668.4 468.2 li st 3 setlinewidth np 3723 522.8 a 3723 695 li st 3 setlinewidth np 3668.4 749.6 a 3308.4 749.6 li st 3 setlinewidth np 3872 1422 54.6 90.0002 180 arc st 3 setlinewidth np 3872 1250 54.6 180 270.001 arc st 3 setlinewidth np 4232 1250 54.6 270.001 360.001 arc st 3 setlinewidth np 4232 1422 54.6 0 90.0002 arc st 3 setlinewidth np 3817.2 1422.8 a 3817.2 1250.6 li st 3 setlinewidth np 3871.8 1196 a 4231.8 1196 li st 3 setlinewidth np 4286.4 1250.6 a 4286.4 1422.8 li st 3 setlinewidth np 4231.8 1478 a 3871.8 1478 li st 3 setlinewidth np 3872 695 54.6 90.0002 180 arc st 3 setlinewidth np 3872 522 54.6 180 270.001 arc st 3 setlinewidth np 4232 522 54.6 270.001 360.001 arc st 3 setlinewidth np 4232 695 54.6 0 90.0002 arc st 3 setlinewidth np 3817.2 695 a 3817.2 522.8 li st 3 setlinewidth np 3871.8 468.2 a 4231.8 468.2 li st 3 setlinewidth np 4286.4 522.8 a 4286.4 695 li st 3 setlinewidth np 4231.8 749.6 a 3871.8 749.6 li st 3 setlinewidth np 3629.4 280.4 a 3910.8 468.2 li st 3 setlinewidth np 3348 1242 a 3374.4 1242 li st 3 setlinewidth np 3409 1242 a 3435.4 1242 li st 3 setlinewidth np 3471 1242 a 3497.4 1242 li st 3 setlinewidth np 3532 1242 a 3558.4 1242 li st 3 setlinewidth np 2784 1337 a 3018.6 1337 li st 3 setlinewidth np 3555 1242.8 a 3657 1344.8 li 3555 1446.2 li 3555 1242.8 li 3351.6 1446.2 li 3555 1446.2 li 3351.6 1242.8 li 3351.6 1446.2 li st 3 setlinewidth np 2784 1336 a 2793 1342 li st 3 setlinewidth np 2811 1356 a 2820 1362 li st 3 setlinewidth np 2820 1363 a 2831.4 1366.6 li st 3 setlinewidth np 2854 1376 a 2865.4 1379.6 li st 3 setlinewidth np 2865 1380 a 2877 1380 li st 3 setlinewidth np 2914 1380 a 2926 1380 li st 3 setlinewidth np 2926 1380 a 2938 1377 li st 3 setlinewidth np 2963 1371 a 2975 1368 li st 3 setlinewidth np 2975 1368 a 2983.4 1362 li st 3 setlinewidth np 3001 1350 a 3009.4 1344 li st 3 setlinewidth np 3009 1344 a 3012 1341 li st 3 setlinewidth np 3016 1338 a 3019 1335 li st 3 setlinewidth np 2784 608.6 a 3018.6 608.6 li st 3 setlinewidth np 2784 1336 a 2793 1329.4 li st 3 setlinewidth np 2811 1316 a 2820 1309.4 li st 3 setlinewidth np 2820 1310 a 2831.4 1305.8 li st 3 setlinewidth np 2854 1297 a 2865.4 1292.8 li st 3 setlinewidth np 2865 1293 a 2877 1293 li st 3 setlinewidth np 2914 1293 a 2926 1293 li st 3 setlinewidth np 2926 1293 a 2938 1295.4 li st 3 setlinewidth np 2963 1302 a 2975 1304.4 li st 3 setlinewidth np 2975 1305 a 2983.4 1310.4 li st 3 setlinewidth np 3001 1323 a 3009.4 1328.4 li st 3 setlinewidth np 3009 1329 a 3012 1331.4 li st 3 setlinewidth np 3016 1335 a 3019 1337.4 li st 3 setlinewidth np 2782 605 a 2791 598.4 li st 3 setlinewidth np 2809 585 a 2818 578.4 li st 3 setlinewidth np 2818 579 a 2829.4 574.8 li st 3 setlinewidth np 2853 566 a 2864.4 561.8 li st 3 setlinewidth np 2864 561 a 2876 561 li st 3 setlinewidth np 2913 561 a 2925 561 li st 3 setlinewidth np 2925 561 a 2937 563.4 li st 3 setlinewidth np 2961 571 a 2973 573.4 li st 3 setlinewidth np 2973 574 a 2981.4 579.4 li st 3 setlinewidth np 2999 592 a 3007.4 597.4 li st 3 setlinewidth np 3008 597 a 3011 599.4 li st 3 setlinewidth np 3014 604 a 3017 606.4 li st 3 setlinewidth np 4214.4 617 a 4122 617 li st 3 setlinewidth np 2784 608 a 2793 614 li st 3 setlinewidth np 2811 628 a 2820 634 li st 3 setlinewidth np 2820 635 a 2831.4 638.6 li st 3 setlinewidth np 2854 648 a 2865.4 651.6 li st 3 setlinewidth np 2865 652 a 2877 652 li st 3 setlinewidth np 2914 652 a 2926 652 li st 3 setlinewidth np 2926 652 a 2938 649 li st 3 setlinewidth np 2963 643 a 2975 640 li st 3 setlinewidth np 2975 640 a 2983.4 634 li st 3 setlinewidth np 3001 622 a 3009.4 616 li st 3 setlinewidth np 3009 616 a 3012 613 li st 3 setlinewidth np 3016 610 a 3019 607 li st 3 setlinewidth np 3652.8 1344.8 a 3558.6 1344.8 li st 3 setlinewidth np 3651 617 a 3561.6 617 li st 3 setlinewidth np 3964.2 1349.6 a 3873.6 1349.6 li st 3629 983 a FB(b)3679 947 y Fw(0)506 889 y FB(a)p 3 setlinewidth np 3629.4 1008.2 a 3910.8 1196 li st 457 372 a(G)2220 889 y(b)2270 853 y Fw(0)1140 889 y FB(b)1542 374 y(H)3301 255 y(a)1586 889 y(a)1636 853 y Fw(0)3019 584 y FI(b)3019 1312 y(b)3064 1279 y Fw(0)3629 255 y FB(b)2737 1312 y FI(a)2782 1279 y Fw(0)3277 548 y FI(a)3301 983 y FB(a)3351 947 y Fw(0)4193 1276 y FI(b)4238 1243 y Fw(0)3841 548 y Fa(a)3606 1276 y FI(b)3651 1243 y Fw(0)3277 1276 y FI(a)3322 1243 y Fw(0)3841 1276 y FI(a)3886 1243 y Fw(0)4169 548 y FI(b)3580 430 y(G)3648 444 y FA(1)4144 430 y FI(G)4212 444 y FA(2)2737 584 y FI(a)4171 1158 y(H)4237 1172 y FA(2)3606 548 y FI(b)3584 1158 y(H)3650 1172 y FA(1)2712 430 y FI(G)2780 444 y FA(0)2620 136 y FB(S)2685 151 y FA(\()p Fr(a)p Fs(;)p Fr(b)p FA(\))2716 1157 y FI(H)2782 1171 y FA(0)p 0 setgray 3 setlinewidth np 3348 890 9.6 0 360.001 arc fil 0 setgray np 3348 890 9.6 0 360.001 arc st 3 setlinewidth np 3348 890 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 3629 890 9.6 0 360.001 arc fil 0 setgray np 3629 890 9.6 0 360.001 arc st 3 setlinewidth np 3629 890 9.6 0 360.001 arc st 2620 863 a FB(T)2679 878 y FA(\()p Fr(a)2734 859 y Fh(0)2756 878 y Fs(;)p Fr(b)2806 859 y Fh(0)2827 878 y FA(\))410 1489 y currentpoint currentpoint translate 1 0.72 div 1 0.72 div scale neg exch neg exch translate 410 1489 a Black 1680 1800 a FF(Figure)25 b(4:)2073 1800 y SDict begin H.S end 2073 1800 a Black Black 2073 1800 a SDict begin H.R end 2073 1800 a 2073 1800 a SDict begin [ /View [/XYZ H.V] /Dest (figure.4) cvn H.B /DEST pdfmark end 2073 1800 a Black Black 404 2092 a FF(W)-8 b(e)28 b(de\002ne)g(the)f FB(orientation)f(given)h(to)g(the)h (virtual)e(edg)o(e)i(in)f(the)g(par)l(ent)g(triconnected)257 2212 y(component)g(of)41 b FF(the)26 b(corresponding)g(separating)h (pair)g(node)f(\()p FB(a)p FC(;)17 b FB(b)p FF(\))27 b(or)g(\()p FB(a)2853 2176 y Fw(0)2880 2212 y FC(;)17 b FB(b)2989 2176 y Fw(0)3015 2212 y FF(\))27 b(by)g(consid-)257 2333 y(ering)g(all)f(the)g(orientations)f(gi)n(v)o(en)h(to)g(the)g (separating)g(pair)h(of)f(their)h(children)d FB(G)3119 2348 y FA(1)3156 2333 y FC(;)17 b(:)g(:)g(:)e(;)g FB(G)3436 2348 y Fr(k)3470 2333 y FF(,)257 2453 y(respecti)n(v)o(ely)-6 b(.)31 b(W)-8 b(e)26 b(\002rst)f(order)h(the)f(subtrees,)g(say)h FB(S)2104 2468 y Fr(G)2154 2478 y Fl(1)2216 2453 y Fq(6)2279 2468 y Fg(T)2347 2453 y Fv(\001)17 b(\001)g(\001)28 b Fq(6)2547 2468 y Fg(T)2615 2453 y FB(S)2678 2468 y Fr(G)2728 2478 y Fd(k)2784 2453 y FF(and)e FB(T)3015 2468 y Fr(H)3063 2478 y Fl(1)3125 2453 y Fq(6)3188 2468 y Fg(T)3256 2453 y Fv(\001)17 b(\001)g(\001)28 b Fq(6)3456 2468 y Fg(T)257 2573 y FB(T)318 2588 y Fr(H)366 2598 y Fd(k)398 2573 y FF(,)42 b(and)d(partition)e(them)i(into)e(isomorphism)g(classes,)42 b(say)e FB(I)2566 2588 y FA(1)2604 2573 y FC(;)17 b(:)g(:)g(:)f(;)j FB(I)2853 2588 y Fr(p)2929 2573 y FF(and)41 b FB(I)3152 2537 y Fw(0)3147 2604 y FA(1)3185 2573 y FC(;)17 b(:)g(:)g(:)e(;)k FB(I)3433 2537 y Fw(0)3433 2598 y Fr(p)3470 2573 y FF(.)257 2694 y(Let)25 b FB(I)458 2709 y Fr(j)503 2694 y FF(be)e(the)f(smallest) f(isomorphism)f(class)i(such)g(that)g(there)h(are)g(more)f(components)f (that)257 2814 y(gi)n(v)o(e)31 b(the)i(orientation)e FB(a)41 b Fv(!)h FB(b)32 b FF(to)g(the)g(parent)h(than)f FB(b)41 b Fv(!)h FB(a)32 b FF(\(or)g(vice)h(v)o(ersa\).)53 b(Then,)34 b(we)257 2935 y(de\002ne)26 b FB(a)h Fv(!)h FB(b)d FF(to)g(be)g(the)f FB(r)l(efer)l(ence)i(orientation)d FF(\()p FB(b)28 b Fv(!)g FB(a)c FF(otherwise\).)31 b(F)o(or)25 b(each)g(isomor)n(-)257 3055 y(phism)h(class)i FB(I)793 3070 y Fr(j)816 3055 y FF(,)f(we)g(compute)f(no)n(w)g(the)g (orientation)f(counters)h FB(O)2629 3070 y Fr(j)2683 3055 y FE(=)31 b FF(\()p FB(c)2854 3019 y Fw(!)2864 3085 y Fr(j)2923 3055 y FC(;)17 b FB(c)3009 3019 y Fw( )3019 3085 y Fr(j)3077 3055 y FF(\))27 b(such)f(that)257 3175 y FB(c)301 3139 y Fw(!)311 3205 y Fr(j)398 3175 y FF(is)h(the)h(number) f(of)h(children)g(in)h FB(I)1590 3190 y Fr(j)1642 3175 y FF(which)e(gi)n(v)o(e)g(the)g(reference)j(orientation)d(and)g FB(c)3332 3139 y Fw( )3342 3205 y Fr(j)3429 3175 y FF(is)257 3296 y(the)e(number)f(of)h(children)g(in)h FB(I)1340 3311 y Fr(j)1388 3296 y FF(which)f(gi)n(v)o(e)e(the)i(re)n(v)o(erse)f (orientation.)404 3418 y(Recall)32 b(the)g(e)o(xample)f(of)h(Figure) 1620 3418 y SDict begin H.S end 1620 3418 a Black FF(4)p Black 1669 3351 a SDict begin H.R end 1669 3351 a 1669 3418 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (figure.4) cvn H.B /ANN pdfmark end 1669 3418 a FF(.)53 b(The)32 b(graphs)d FB(G)34 b FF(and)g FB(H)j FF(ha)n(v)o(e)32 b(the)g(same)g(tricon-)257 3539 y(nected)h(component)e(trees)i(b)n(ut)f (are)h(not)f(isomorphic.)53 b(In)32 b FB(S)2426 3554 y FA(\()p Fr(a)p Fs(;)p Fr(b)p FA(\))2558 3539 y FF(,)i(the)e(3-bonds)g (form)g(one)257 3659 y(isomorphism)19 b(class)24 b FB(I)1050 3674 y FA(1)1109 3659 y FF(and)e(the)f(other)h(tw)o(o)f(components)f (form)h(the)h(second)f(isomorphism)257 3780 y(class)27 b FB(I)511 3795 y FA(2)549 3780 y FF(,)e(as)g(the)o(y)f(all)g(are)i (pairwise)e(isomorphic.)30 b(The)24 b(non-isomorphism)e(is)j(detected)g (by)257 3900 y(comparing)i(the)h(directions)f(gi)n(v)o(en)f(to)h(the)h (parent)g(separating)f(pair)-5 b(.)39 b(W)-8 b(e)29 b(ha)n(v)o(e)34 b FB(p)g FE(=)f FF(2)28 b(iso-)257 4020 y(morphism)k(classes)i(and)g (for)g(the)g(orientation)e(counters)i(we)g(ha)n(v)o(e)g FB(O)2768 4035 y FA(1)2850 4020 y FE(=)44 b FB(O)3029 4035 y FA(1)3083 3984 y Fw(0)3155 4020 y FE(=)h FF(\(0)p FC(;)17 b FF(0\),)257 4141 y(whereas)29 b FB(O)684 4156 y FA(2)754 4141 y FE(=)34 b FF(\(2)p FC(;)17 b FF(0\))27 b(and)h FB(O)1330 4156 y FA(2)1384 4105 y Fw(0)1444 4141 y FE(=)34 b FF(\(1)p FC(;)17 b FF(1\))27 b(and)h(hence)g FB(O)2280 4156 y FA(2)2334 4105 y Fw(0)2389 4141 y FF(is)g(le)o (xicographically)e(smaller)257 4261 y(than)f FB(O)526 4276 y FA(2)563 4261 y FF(.)31 b(Therefore)26 b(we)f(ha)n(v)o(e)f FB(T)1446 4276 y FA(\()p Fr(a)1501 4257 y Fh(0)1523 4276 y Fs(;)p Fr(b)1584 4257 y Fh(0)1606 4276 y FA(\))1660 4261 y FC(<)1723 4276 y Fg(T)1790 4261 y FB(S)1855 4276 y FA(\()p Fr(a)p Fs(;)p Fr(b)p FA(\))1986 4261 y FF(.)257 4446 y SDict begin H.S end 257 4446 a 257 4446 a SDict begin 14.5 H.A end 257 4446 a 257 4446 a SDict begin [ /View [/XYZ H.V] /Dest (section*.7) cvn H.B /DEST pdfmark end 257 4446 a 96 x Fp(Complexity)-7 b(.)99 b FF(W)-8 b(e)34 b(ar)n(gue)f(no)n(w)-6 b(,)34 b(that)e(we)h(can)g(do)g(the)f(four)h (comparison)f(steps)g(in)h(log-)257 4662 y(space.)j(The)27 b(\002rst)f(and)g(the)h(second)f(step)g(are)h(similar)e(to)h(Lindell') -5 b(s)25 b(algorithm.)34 b(W)-8 b(e)27 b(de\002ne)257 4783 y(the)j FB(size)f(of)g(a)g(separ)o(ating)f(pair)g(node)i FF(as)f(2)h(and)f(the)h FB(size)f(of)g(a)g(triconnected)g(component)257 4903 y FF(as)23 b(the)g(number)f(of)h(v)o(ertices)f(in)g(the)h (component.)29 b(F)o(or)22 b(the)h(third)f(and)h(fourth)f(step,)h(we)g (ha)n(v)o(e)257 5023 y(the)i(follo)n(wing)e(cases:)p Black 403 5268 a Fv(\017)p Black 48 w FF(When)c(we)h(compare)f(tw)o(o)g (triconnected)g(components)c FB(G)2508 5283 y Fr(i)2550 5268 y FF(and)21 b FB(H)2797 5283 y Fr(j)2820 5268 y FF(,)f(then)f(we)h(ha)n(v)o(e)f(up)501 5388 y(to)27 b(four)h(canons.)38 b(Suppose,)28 b(we)f(construct)g(and)g(compare)h(tw)o(o)f(canons)e FB(C)3155 5403 y Fs(g)3220 5388 y FF(and)g FB(C)3457 5403 y Fr(h)p Black Black eop end %%Page: 16 16 TeXDict begin 16 15 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 257 266 a SDict begin H.S end 257 266 a Black Black 257 266 a SDict begin H.R end 257 266 a 257 266 a SDict begin [ /View [/XYZ H.V] /Dest (page.16) cvn H.B /DEST pdfmark end 257 266 a Black 501 573 a FF(and)24 b(reach)g(separating)f(pairs)g(\()p FB(a)p FC(;)17 b FB(b)p FF(\))23 b(and)g(\()p FB(a)2039 537 y Fw(0)2067 573 y FC(;)17 b FB(b)2176 537 y Fw(0)2202 573 y FF(\).)30 b(W)-8 b(e)24 b(store)f(the)g(canons)h(which)f(are)501 693 y(not)h(eliminated,)e(which)i(of)g(them)e FB(C)1787 708 y Fs(g)1848 693 y FF(and)g FB(C)2082 708 y Fr(h)2144 693 y FF(are)j(and)f(the)g(direction)f(of)h(the)g(virtual)501 814 y(edges)h(\()p FB(a)p FC(;)17 b FB(b)p FF(\))24 b(and)h(\()p FB(a)1237 777 y Fw(0)1264 814 y FC(;)17 b FB(b)1373 777 y Fw(0)1399 814 y FF(\).)31 b(Hence,)26 b(we)f(need)g FB(O)p FF(\(1\))g(bits.)p Black 403 1035 a Fv(\017)p Black 48 w FF(When)35 b(we)g(compare)g(tw)o(o)g(separating)f(pairs)h (\()p FB(a)p FC(;)17 b FB(b)p FF(\))35 b(and)g(\()p FB(a)2665 999 y Fw(0)2692 1035 y FC(;)17 b FB(b)2801 999 y Fw(0)2827 1035 y FF(\),)38 b(then)c(we)h(mak)o(e)501 1155 y(a)50 b(cross-comparison)f(as)g(in)h(Lindell')-5 b(s)48 b(algorithm.)103 b(Hence,)57 b(we)50 b(need)f(coun-)501 1276 y(ters)29 b FB(c)718 1291 y Fs(<)765 1276 y FC(;)17 b FB(c)851 1291 y Fz(=)897 1276 y FC(;)g FB(c)983 1291 y Fs(>)1058 1276 y FF(to)29 b(store)g(the)g(order)h(pro\002le.)43 b(This)29 b(w)o(ay)-6 b(,)30 b(we)f(get)g(the)g(isomorphism)501 1396 y(classes.)42 b(W)-8 b(e)30 b(further)e(store)h(the)f(orientation) g(counters)g FB(O)2586 1411 y Fr(j)2638 1396 y FF(and)g FB(O)2892 1411 y Fr(j)2931 1360 y Fw(0)2987 1396 y FF(for)j FB(I)3177 1411 y Fr(j)3229 1396 y FF(and)g FB(I)3442 1360 y Fw(0)3447 1426 y Fr(j)3470 1396 y FF(.)501 1517 y(W)-8 b(e)25 b(need)g FB(O)p FF(\(log)16 b Fv(j)r FB(I)1183 1532 y Fr(j)1207 1517 y Fv(j)p FF(\))25 b(bits)f(on)g(the)h(w)o (ork-tape)g(for)g(all)f(the)h(counters.)404 1770 y(Ho)n(we)n(v)o(er)l (,)42 b(we)f(cannot)f(guarantee)g(yet,)k(that)c(the)g(algorithm)f(w)o (orks)h(in)g(log-space.)257 1890 y(Let)35 b FB(S)488 1905 y Fr(C)574 1890 y FF(be)g(the)g(subtree)f(rooted)g(at)h(node)d FB(C)40 b FF(in)34 b(a)h(triconnected)f(component)g(tree.)60 b(The)257 2011 y(problem)30 b(is,)h(that)f(the)g(subtrees)g(\(i.e.)47 b(the)30 b(children)g(of)f FB(C)t FF(\))i(where)g(we)f(go)g(into)g (recursion)257 2131 y(might)24 b(be)h(of)g(size)f FC(>)29 b Fv(j)p FB(S)1096 2146 y Fr(C)1147 2131 y Fv(j)p FC(=)p FF(2,)c(we)g(call)f(this)g(a)h FB(lar)l(g)o(e)g(c)o(hild)p FF(.)404 2255 y(T)-8 b(o)28 b(get)g(around)h(this)e(problem,)i(we)f (\002rst)h(check)g(whether)f(the)h(nodes)d FB(C)33 b FF(and)27 b FB(C)3255 2219 y Fw(0)3311 2255 y FF(ha)n(v)o(e)257 2375 y(a)35 b(lar)n(ge)g(child.)58 b(If)35 b(so)f(then)g(we)h(compare)f (them)g(a)g(priori)g(and)g(store)g(the)g(result)g(of)h(their)257 2496 y(comparison)f(and)g(the)g(orientation)f(gi)n(v)o(en)g(to)h(the)g (parent.)59 b(Because)33 b FB(C)39 b FF(and)32 b FB(C)3143 2460 y Fw(0)3204 2496 y FF(ha)n(v)o(e)i(at)257 2616 y(most)24 b(one)h(lar)n(ge)g(child,)f(this)g(needs)h(only)f FB(O)p FF(\(1\))g(additional)g(bits.)30 b(In)24 b(the)h(comparison)f(of)f FB(C)257 2737 y FF(and)e FB(C)509 2700 y Fw(0)536 2737 y FF(,)i(whene)n(v)o(er)f(we)g(w)o(ould)g(go)g(into)f(recursion)i(at)f (those)g(lar)n(ge)h(children,)f(we)h(just)e(look)257 2857 y(at)k(the)g(w)o(ork-tape)g(for)g(the)f(result)h(instead.)404 2981 y(As)33 b(seen)h(abo)o(v)o(e,)g(while)f(comparing)g(tw)o(o)g (trees)h(of)g(size)h FB(N)6 b FF(,)37 b(the)c(algorithm)f(uses)i(no)257 3101 y(space)39 b(for)f(making)f(a)h(recursi)n(v)o(e)f(call)h(for)g(a)h (subtree)f(of)g(size)g(lar)n(ger)g(than)i FB(N)s FC(=)p FF(2,)i(and)c(it)257 3222 y(uses)25 b FB(O)p FF(\(log)16 b FB(k)757 3237 y Fr(j)780 3222 y FF(\))26 b(space)g(if)f(the)g (subtrees)g(are)h(of)g(size)f(at)g(most)i FB(N)s FC(=)p FB(k)2593 3237 y Fr(j)2617 3222 y FF(,)e(where)h FB(k)2990 3237 y Fr(j)3041 3222 y Fq(>)k FF(2.)i(Hence)257 3342 y(we)25 b(get)g(the)g(same)f(recurrence)j(for)e(the)f(space)i Fv(S)p FF(\()r FB(N)6 b FF(\))25 b(as)g(Lindell:)1240 3629 y Fv(S)p FF(\()r FB(N)6 b FF(\))29 b Fq(6)f FF(max)1649 3690 y Fr(j)1781 3629 y Fv(S)1849 3488 y Fi( )1899 3561 y FB(N)p 1896 3606 77 6 v 1896 3697 a(k)1950 3712 y Fr(j)1983 3488 y Fi(!)2043 3629 y FE(+)23 b FB(O)p FF(\(log)16 b FB(k)2432 3644 y Fr(j)2455 3629 y FF(\))p FC(;)257 3915 y FF(where)29 b FB(k)583 3930 y Fr(j)640 3915 y Fq(>)35 b FF(2)28 b(for)h(all)43 b FB(j)p FF(.)e(Thus)28 b Fv(S)p FF(\()r FB(N)6 b FF(\))35 b FE(=)f FB(O)p FF(\(log)18 b FB(N)6 b FF(\).)43 b(Note)28 b(that)g(the)g(number)g FB(n)g FF(of)h(nodes)257 4035 y(of)f FB(G)k FF(is)d(in)h(general)g (smaller)f(than)i FB(N)6 b FF(,)32 b(because)e(the)g(separating)f(pair) h(nodes)f(occur)h(in)g(all)257 4155 y(components)c(split)f(o)p FE(\013)i FF(by)f(this)g(pair)-5 b(.)37 b(But)26 b(we)i(certainly)e(ha) n(v)o(e)h FB(n)k FC(<)j FB(N)k Fq(6)32 b FB(O)p FF(\()p FB(n)3043 4119 y FA(2)3080 4155 y FF(\))27 b([)3173 4157 y SDict begin H.S end 3173 4157 a Black -2 x FF(9)p Black 3223 4088 a SDict begin H.R end 3223 4088 a 3223 4155 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.HT73) cvn H.B /ANN pdfmark end 3223 4155 a FF(].)37 b(This)257 4276 y(leads)25 b(to)f(the)h(follo)n(wing)e(theorem.)257 4297 y SDict begin H.S end 257 4297 a 257 4297 a SDict begin 14.5 H.A end 257 4297 a 257 4297 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.4.1) cvn H.B /DEST pdfmark end 257 4297 a Black 200 x Fp(Theor)n(em)32 b(4.1.)p Black 44 w FB([)906 4498 y SDict begin H.S end 906 4498 a Black -1 x FB(6)p Black 956 4427 a SDict begin H.R end 956 4427 a 956 4497 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.DLNTW08) cvn H.B /ANN pdfmark end 956 4497 a FB(])e(The)h(isomorphism)d(or)l(der)i (between)h(two)g(triconnected)e(component)257 4618 y(tr)l(ees)c(of)g (biconnected)f(planar)g(gr)o(aphs)f(can)i(be)f(computed)h(in)f(lo)o (g-space)o(.)257 4803 y SDict begin H.S end 257 4803 a 257 4803 a SDict begin 14.5 H.A end 257 4803 a 257 4803 a SDict begin [ /View [/XYZ H.V] /Dest (section*.8) cvn H.B /DEST pdfmark end 257 4803 a 104 x Fp(The)35 b(canon.)101 b FF(Once)34 b(we)g(kno)n(w)f(the)g(ordering)h(among)f(the)h(subtrees,) h(it)f(is)f(straightfor)n(-)257 5027 y(w)o(ard)27 b(to)f(output)f(the)h (canon)h(of)f(the)g(triconnected)g(component)g(tree)g FB(T)12 b FF(.)36 b(W)-8 b(e)27 b(tra)n(v)o(erse)f FB(T)39 b FF(in)257 5147 y(the)23 b(tree)g(isomorphism)d(order)j(as)g(in)f (Lindell')-5 b(s)22 b(algorithm,)f(outputting)g(the)h(canon)h(of)g (each)257 5268 y(of)f(the)g(nodes)f(along)h(with)f(virtual)g(edges)h (and)g(delimiters.)28 b(That)22 b(is,)f(we)i(output)d(a)i(`[')h(while) 257 5388 y(going)h(do)n(wn)g(a)h(subtree,)g(and)f(`]')i(while)e(going)g (up)h(a)g(subtree.)p Black Black eop end %%Page: 17 17 TeXDict begin 17 16 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 257 266 a SDict begin H.S end 257 266 a Black Black 257 266 a SDict begin H.R end 257 266 a 257 266 a SDict begin [ /View [/XYZ H.V] /Dest (page.17) cvn H.B /DEST pdfmark end 257 266 a Black 404 573 a FF(W)-8 b(e)36 b(need)g(to)g(choose)f(a)h(separating) g(pair)g(as)g(root)f(for)h(the)g(tree.)65 b(Since)36 b(there)g(is)g(no)257 693 y(distinguished)26 b(separating)i(pair)l(,)h (we)f(simply)f(c)o(ycle)h(through)f(all)h(of)h(them)e(and)h(select)g (the)257 814 y(one,)34 b(which)d(leads)g(to)h(the)f(minimum)f(canon.)51 b(Let)31 b(\()p FB(a)p FC(;)17 b FB(b)p FF(\))32 b(be)f(this)g (separating)g(pair)-5 b(.)51 b(The)257 934 y(canonization)29 b(procedure)g(has)g(tw)o(o)f(steps.)43 b(In)29 b(the)f(\002rst)i(step)e (we)h(compute)f(what)h(we)g(call)257 1054 y(a)j FB(canonical)e(list)i FF(for)g FB(S)1125 1069 y FA(\()p Fr(a)p Fs(;)p Fr(b)p FA(\))1256 1054 y FF(.)50 b(This)30 b(is)h(a)h(list)e(of)h(the)g(edges) g(of)f FB(G)r FF(,)j(also)d(including)g(virtual)257 1175 y(edges.)h(In)25 b(the)g(second)f(step)h(we)g(compute)f(the)g(\002nal)h (canon)g(from)g(the)f(canonical)h(list.)257 1303 y Fp(Canon)k(of)g (separating)g(pair)g(nodes.)72 b FF(Consider)29 b(a)g(subtree)f FB(S)2546 1318 y FA(\()p Fr(a)p Fs(;)p Fr(b)p FA(\))2706 1303 y FF(rooted)g(at)h(\()p FB(a)p FC(;)17 b FB(b)p FF(\).)42 b(W)-8 b(e)257 1423 y(start)37 b(with)f(computing)g(the)h (reference)h(orientation)e(of)h(\()p FB(a)p FC(;)17 b FB(b)p FF(\))37 b(with)f(oracle)i(calls)f(to)f(the)257 1544 y(canonical)31 b(ordering)g(algorithm)e(and)i(output)f(the)h(edge) g(in)f(this)g(direction.)49 b(Then)30 b(we)i(re-)257 1664 y(cursi)n(v)o(ely)26 b(output)f(the)i(canonical)g(lists)e(of)i (the)g(subtrees)f(of)h(node)g(\()p FB(a)p FC(;)17 b FB(b)p FF(\))26 b(according)h(to)f(the)257 1785 y(increasing)h(isomorphism)d (order)-5 b(.)37 b(Among)26 b(isomorphic)f(siblings,)g(those)i(which)f (gi)n(v)o(e)f(the)257 1905 y(reference)38 b(orientation)d(to)h(the)g (parent)h(come)f(\002rst.)65 b(W)-8 b(e)37 b(denote)f(this)f(canonical) h(list)f(of)257 2025 y(edges)24 b FB(l)p FF(\()p FB(S)10 b FC(;)17 b FB(a)p FC(;)g FB(b)p FF(\).)29 b(If)24 b(there)f(is)g(no)g (reference)i(orientation)e(for)g(a)h(child,)f(tak)o(e)g(the)h (orientation)257 2146 y(of)h(the)g(parent)g(\()p FB(a)p FC(;)17 b FB(b)p FF(\).)257 2274 y Fp(Canon)k(of)f(triconnected)j (component)e(nodes.)51 b FF(Consider)20 b(the)g(subtree)g FB(S)2917 2289 y Fr(G)2967 2299 y Fd(i)3009 2274 y FF(rooted)g(at)e FB(G)3448 2289 y Fr(i)3470 2274 y FF(.)257 2394 y(Let)37 b(\()p FB(a)p FC(;)17 b FB(b)p FF(\))36 b(be)g(the)h(parent)f (separating)g(pair)h(of)f FB(S)2065 2409 y Fr(G)2115 2419 y Fd(i)2174 2394 y FF(with)f(reference)k(orientation)c(\()p FB(a)p FC(;)17 b FB(b)p FF(\).)257 2515 y(If)42 b FB(G)439 2530 y Fr(i)505 2515 y FF(is)i(a)f(3-bond)g(then)g(output)g FB(l)p FF(\()n FB(G)1676 2530 y Fr(i)1698 2515 y FC(;)17 b FB(a)p FC(;)g FB(b)p FF(\))61 b FE(=)i FF(\()p FB(a)p FC(;)17 b FB(b)p FF(\).)86 b(If)42 b FB(G)2603 2530 y Fr(i)2669 2515 y FF(is)h(a)h(c)o(ycle)g(then)f(out-)257 2635 y(put)35 b FB(l)p FF(\()n FB(G)553 2650 y Fr(i)576 2635 y FC(;)17 b FB(a)p FC(;)g FB(b)p FF(\))46 b FE(=)i FF(\()p FB(a)p FC(;)17 b FB(b)p FF(\)\()p FB(b)p FC(;)g(v)1322 2650 y FA(1)1359 2635 y FF(\)\()p FC(v)1464 2650 y FA(1)1501 2635 y FC(;)g(v)1582 2650 y FA(2)1619 2635 y FF(\))g FC(:)g(:)g(:)f FF(\()p FC(v)1866 2650 y Fr(n)1903 2635 y FC(;)h FB(a)p FF(\).)63 b(If)34 b FB(G)2290 2650 y Fr(i)2348 2635 y FF(is)h(a)h(3-connected)f(component)257 2756 y(then)e(compute)f(the)h(minimum)e(of)i(tw)o(o)f(canons)h(with)f (an)h(oracle)h(call.)55 b(That)33 b(is)f(with)h(re-)257 2876 y(spect)24 b(to)f(the)h(gi)n(v)o(en)e(reference)j(orientation)e (\()p FB(a)p FC(;)17 b FB(b)p FF(\))23 b(and)g(both)g(embeddings)g(for) f FB(G)3141 2891 y Fr(i)3163 2876 y FF(.)30 b(Output)257 2996 y(this)d(canon)g(as)h FB(l)p FF(\()n FB(G)938 3011 y Fr(i)960 2996 y FC(;)17 b FB(a)p FC(;)g FB(b)p FF(\).)38 b(V)-6 b(irtual)26 b(edges)i(are)g(output)e(in)h(the)g(direction)g(of)g (the)h(reference)257 3117 y(orientation)g(gi)n(v)o(en)f(to)i(them,)g (if)f(an)o(y)-6 b(.)42 b(Finally)-6 b(,)28 b(we)i(output)d(the)i (subtrees)f(in)g(the)h(order)g(we)257 3237 y(ha)n(v)o(e)c(virtual)f (edges)h(in)f(the)h(canon.)404 3365 y(W)-8 b(e)37 b(gi)n(v)o(e)e(an)i (e)o(xample.)64 b(Consider)37 b(the)f(canonical)h(list)e FB(l)p FF(\()p FB(S)10 b FC(;)17 b FB(a)p FC(;)g FB(b)p FF(\))36 b(of)g(edges)h(for)f(the)257 3486 y(tree)21 b FB(S)492 3501 y FA(\()p Fr(a)p Fs(;)p Fr(b)p FA(\))644 3486 y FF(of)f(Figure)1028 3487 y SDict begin H.S end 1028 3487 a Black -1 x FF(3)p Black 1078 3419 a SDict begin H.R end 1078 3419 a 1078 3486 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (figure.3) cvn H.B /ANN pdfmark end 1078 3486 a FF(.)29 b(Let)c FB(s)1329 3501 y Fr(i)1372 3486 y FF(be)c(the)f(edge)h(connecting)e(the)i(v)o(ertices)f FB(a)2817 3501 y Fr(i)2860 3486 y FF(with)f FB(b)3107 3501 y Fr(i)3130 3486 y FF(.)29 b(W)-8 b(e)21 b(also)257 3606 y(write)33 b(for)f(short)g FB(l)920 3570 y Fw(0)947 3606 y FF(\()p FB(S)1045 3621 y Fr(i)1067 3606 y FC(;)22 b FB(s)1153 3621 y Fr(i)1175 3606 y FF(\))33 b(which)f(is)g(one)g(of)h FB(l)p FF(\()p FB(S)2034 3621 y Fr(i)2056 3606 y FC(;)17 b FB(a)2148 3621 y Fr(i)2170 3606 y FC(;)g FB(b)2262 3621 y Fr(i)2284 3606 y FF(\))33 b(or)f FB(l)p FF(\()p FB(S)2591 3621 y Fr(i)2614 3606 y FC(;)17 b FB(b)2706 3621 y Fr(i)2728 3606 y FC(;)g FB(a)2820 3621 y Fr(i)2842 3606 y FF(\).)54 b(The)32 b(direction)257 3727 y(of)e FB(s)409 3742 y Fr(i)457 3727 y FF(is)24 b(as)h(described)f(abo)o(v)o (e.)30 b(Let)25 b FB(l)1539 3742 y FA(0)1604 3727 y FE(=)j FF(0.)i(Then)25 b(we)g(ha)n(v)o(e:)674 3970 y FB(l)p FF(\()p FB(S)10 b FC(;)17 b FB(a)p FC(;)g FB(b)p FF(\))82 b FE(=)h FF([)28 b(\()p FB(a)p FC(;)17 b FB(b)p FF(\))27 b FB(l)p FF(\()p FB(S)1660 3985 y Fr(G)1710 3995 y Fl(1)1743 3970 y FC(;)17 b FB(a)p FC(;)g FB(b)p FF(\))43 b FC(:)17 b(:)g(:)43 b FB(l)p FF(\()p FB(S)2279 3985 y Fr(G)2329 3995 y Fd(k)2360 3970 y FC(;)17 b FB(a)p FC(;)g FB(b)p FF(\))27 b(])p FC(;)41 b FF(where)599 4116 y FB(l)p FF(\()p FB(S)723 4131 y Fr(G)773 4141 y Fd(i)795 4116 y FC(;)17 b FB(a)p FC(;)g FB(b)p FF(\))82 b FE(=)h FF([)28 b FB(l)p FF(\()n FB(G)1434 4131 y Fr(i)1457 4116 y FC(;)17 b FB(a)p FC(;)g FB(b)p FF(\))26 b([)p FB(l)1778 4075 y Fw(0)1805 4116 y FF(\()p FB(S)1903 4131 y Fr(l)1921 4141 y Fd(i)p Fh(\000)p Fl(1)1996 4131 y Fz(+)p FA(1)2075 4116 y FC(;)c FB(s)2161 4131 y Fr(l)2179 4141 y Fd(i)p Fh(\000)p Fl(1)2254 4131 y Fz(+)p FA(1)2333 4116 y FF(\)])45 b FC(:)17 b(:)g(:)43 b FF([)p FB(l)2674 4075 y Fw(0)2701 4116 y FF(\()p FB(S)2799 4131 y Fr(l)2817 4141 y Fd(i)2840 4116 y FC(;)22 b FB(s)2926 4131 y Fr(l)2944 4141 y Fd(i)2965 4116 y FF(\)])28 b(])g(])257 4330 y SDict begin H.S end 257 4330 a 257 4330 a SDict begin 14.5 H.A end 257 4330 a 257 4330 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.4.2) cvn H.B /DEST pdfmark end 257 4330 a 132 x Ft(4.2)119 b(Canonization)31 b(of)f(planar)h(graphs)257 4666 y FF(Consider)j(the)g(decomposition)e(of)i(a)g(connected)g(planar) g(graph.)58 b(F)o(or)34 b(each)g(articulation)257 4786 y(point)d(and)i(biconnected)e(component)g(we)i(de\002ne)f(nodes)g(i.e.) 53 b FB(articulation)30 b(point)h(nodes)257 4907 y FF(and)c FB(biconnected)g(component)f(nodes)p FF(.)37 b(An)26 b(articulation)g(point)g(node)h(for)g FB(a)g FF(is)f(connected)257 5027 y(by)36 b(an)g(edge)g(to)g(the)f(nodes)h(of)g(biconnected)f (components)g(where)h FB(a)g FF(is)g(contained)f(as)h(a)257 5147 y(v)o(erte)o(x.)43 b(The)29 b(resulting)f(graph)h(is)g(a)g(tree,)i (the)e FB(biconnected)f(component)h(tr)l(ee)p FF(.)44 b(The)29 b(main)257 5268 y(di)p FE(\013)p FF(erence)d(to)g(the)f (triconnected)g(component)g(tree)h(is,)f(that)h(for)g(articulation)e (point)h(nodes,)257 5388 y(there)g(is)g(no)f(concept)h(of)g (orientation)f(as)g(for)i(separating)e(pairs.)p Black Black eop end %%Page: 18 18 TeXDict begin 18 17 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 257 266 a SDict begin H.S end 257 266 a Black Black 257 266 a SDict begin H.R end 257 266 a 257 266 a SDict begin [ /View [/XYZ H.V] /Dest (page.18) cvn H.B /DEST pdfmark end 257 266 a Black 404 573 a FF(W)-8 b(e)39 b(de\002ne)g(the)f(isomorphism)e (order)j(for)g(tw)o(o)f(biconnected)g(component)f(trees)i FB(S)3458 588 y Fr(a)257 693 y FF(and)30 b FB(T)490 708 y Fr(a)523 689 y Fh(0)579 693 y FF(rooted)g(at)f(nodes)35 b FB(s)29 b FF(and)h FB(t)h FF(corresponding)e(to)g(articulation)g (points)f FB(a)i FF(and)g FB(a)3302 657 y Fw(0)3329 693 y FF(,)h(re-)257 814 y(specti)n(v)o(ely)-6 b(.)35 b(Also)26 b(see)h(Figure)1369 815 y SDict begin H.S end 1369 815 a Black -1 x FF(5)p Black 1419 746 a SDict begin H.R end 1419 746 a 1419 814 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (figure.5) cvn H.B /ANN pdfmark end 1419 814 a FF(.)37 b(Let)26 b Fv(j)p FB(S)1725 829 y Fr(a)1762 814 y Fv(j)h FF(be)g(the)g(sum)f(of)h(the)g(sizes)f(of)h(the)g(nodes)f(in)g(the)257 934 y(tree.)41 b(The)28 b FB(size)g(of)f(an)h(articulation)d(point)i (node)h(a)g FF(is)f(de\002ned)i(as)f(1)g(and)g(the)g FB(size)f(of)h(a)g(bi-)257 1054 y(connected)c(component)f(node)j(B)d FF(is)g(the)g(size)g(of)h(its)f(triconnected)g(component)f(tree)i Fv(j)p FD(T)p FF(\()s FB(B)p FF(\))p Fv(j)p FF(.)257 1175 y(Let)h FB(S)480 1190 y Fr(a)541 1175 y FF(and)f FB(T)768 1190 y Fr(a)801 1171 y Fh(0)852 1175 y FF(be)g(biconnected)g (component)f(trees)i(rooted)f(at)g(articulation)f(points)g FB(a)h FF(and)257 1295 y FB(a)307 1259 y Fw(0)334 1295 y FF(.)31 b(De\002ne)26 b FB(S)746 1310 y Fr(a)810 1295 y FC(<)873 1310 y Fg(B)941 1295 y FB(T)1000 1310 y Fr(a)1033 1291 y Fh(0)1084 1295 y FF(if)257 1341 y SDict begin H.S end 257 1341 a 257 1341 a SDict begin 14.5 H.A end 257 1341 a 257 1341 a SDict begin [ /View [/XYZ H.V] /Dest (Item.5) cvn H.B /DEST pdfmark end 257 1341 a Black 378 1482 a FF(1.)p Black 73 w Fv(j)p FB(S)611 1497 y Fr(a)648 1482 y Fv(j)i FC(<)g Fv(j)p FB(T)866 1497 y Fr(a)899 1478 y Fh(0)925 1482 y Fv(j)d FF(or)257 1535 y SDict begin H.S end 257 1535 a 257 1535 a SDict begin 14.5 H.A end 257 1535 a 257 1535 a SDict begin [ /View [/XYZ H.V] /Dest (Item.6) cvn H.B /DEST pdfmark end 257 1535 a Black 378 1675 a FF(2.)p Black 73 w Fv(j)p FB(S)611 1690 y Fr(a)648 1675 y Fv(j)j FE(=)g Fv(j)p FB(T)866 1690 y Fr(a)899 1671 y Fh(0)925 1675 y Fv(j)d FF(b)n(ut)f(#)5 b FB(s)27 b FC(<)h FF(#)p FB(t)f FF(or)257 1728 y SDict begin H.S end 257 1728 a 257 1728 a SDict begin 14.5 H.A end 257 1728 a 257 1728 a SDict begin [ /View [/XYZ H.V] /Dest (Item.7) cvn H.B /DEST pdfmark end 257 1728 a Black 378 1868 a FF(3.)p Black 73 w Fv(j)p FB(S)611 1883 y Fr(a)648 1868 y Fv(j)h FE(=)g Fv(j)p FB(T)866 1883 y Fr(a)899 1864 y Fh(0)925 1868 y Fv(j)p FF(,)d(#)5 b FB(s)27 b FE(=)h FF(#)p FB(t)h FE(=)f FB(k)r FF(,)d(b)n(ut)f(\()p FB(S)1751 1883 y Fr(B)1792 1893 y Fl(1)1825 1868 y FC(;)17 b(:)g(:)g(:)e(;)i FB(S)2100 1883 y Fr(B)2141 1893 y Fd(k)2170 1868 y FF(\))28 b FC(<)2294 1883 y Fg(B)2361 1868 y FF(\()p FB(T)2455 1883 y Fr(B)2496 1893 y Fl(1)2536 1864 y Fh(0)2562 1868 y FC(;)17 b(:)g(:)g(:)e(;)i FB(T)2831 1883 y Fr(B)2872 1893 y Fd(k)2910 1864 y Fh(0)2935 1868 y FF(\))25 b(le)o(xicograph-)501 1989 y(ically)-6 b(,)34 b(where)f(we)g(assume)f(that)h FB(S)1777 2004 y Fr(B)1818 2014 y Fl(1)1892 1989 y Fq(6)1955 2004 y Fg(B)2037 1989 y Fv(\001)17 b(\001)g(\001)41 b Fq(6)2250 2004 y Fg(B)2332 1989 y FB(S)2399 2004 y Fr(B)2440 2014 y Fd(k)2503 1989 y FF(and)33 b FB(T)2741 2004 y Fr(B)2782 2014 y Fl(1)2822 1985 y Fh(0)2890 1989 y Fq(6)2953 2004 y Fg(B)3035 1989 y Fv(\001)17 b(\001)g(\001)41 b Fq(6)3248 2004 y Fg(B)3330 1989 y FB(T)3391 2004 y Fr(B)3432 2014 y Fd(k)3470 1985 y Fh(0)501 2109 y FF(are)31 b(the)e(ordered)h (subtrees)f(of)h FB(S)1672 2124 y Fr(a)1739 2109 y FF(and)g FB(T)1972 2124 y Fr(a)2005 2105 y Fh(0)2031 2109 y FF(,)h(respecti)n(v) o(ely)-6 b(.)43 b(T)-8 b(o)30 b(compare)f(the)h(order)501 2230 y(between)c(the)g(subtrees)g FB(S)1427 2245 y Fr(B)1468 2255 y Fd(i)1515 2230 y FF(and)g FB(T)1746 2245 y Fr(B)1794 2255 y Fd(j)1823 2225 y Fh(0)1875 2230 y FF(we)g(compare)g(the)g (triconnected)f(component)501 2350 y(trees)g FD(T)p FF(\()s FB(B)869 2365 y Fr(i)891 2350 y FF(\))h(of)i FB(B)1122 2365 y Fr(i)1169 2350 y FF(and)d FD(T)p FF(\()s FB(B)1503 2365 y Fr(j)1542 2314 y Fw(0)1569 2350 y FF(\))g(of)j FB(B)1809 2365 y Fr(j)1848 2314 y Fw(0)1900 2350 y FF(and)d(when)g(we)h (reach)f(the)g(\002rst)h(occurrences)501 2470 y(of)h(some)g (articulation)e(points)h(then)g(we)i(compare)f FB(r)l(ecur)o(sively)g FF(the)f(corresponding)501 2591 y(subtrees)j(rooted)g(at)g(the)g (children)f(of)k FB(B)1924 2606 y Fr(i)1975 2591 y FF(and)g FB(B)2222 2606 y Fr(j)2261 2555 y Fw(0)2288 2591 y FF(.)44 b(Note,)29 b(that)g(these)g(children)g(are)501 2711 y(again)24 b(articulation)g(point)g(nodes.)p Black Black Black 257 4139 a currentpoint currentpoint translate 0.72 0.72 scale neg exch neg exch translate 257 4139 a 3 setlinewidth np 595 3775.8 a 595 3869.4 li st 3 setlinewidth np 641.8 3775.8 a 736 3869.4 li st 3 setlinewidth np 595 3869.4 a 548.2 4010.4 li 641.8 4010.4 li closepath st 3 setlinewidth np 736 3869.4 a 689.2 4010.4 li 782.8 4010.4 li closepath st 3 setlinewidth np 548.2 3775.8 a 454 3869.4 li st 3 setlinewidth np 454 3869.4 a 407.2 4010.4 li 501.4 4010.4 li closepath st 3 setlinewidth np 2144.8 3775.8 a 2144.8 3869.4 li st 3 setlinewidth np 2191.6 3775.8 a 2285.8 3869.4 li st 3 setlinewidth np 2144.8 3869.4 a 2098 4010.4 li 2191.6 4010.4 li closepath st 3 setlinewidth np 2285.8 3869.4 a 2239 4010.4 li 2332.6 4010.4 li closepath st 3 setlinewidth np 2098 3775.8 a 2003.8 3869.4 li st 3 setlinewidth np 2003.8 3869.4 a 1957 4010.4 li 2051.2 4010.4 li closepath st 3 setlinewidth np 1064.8 3869.4 a 1018 4010.4 li 1111.6 4010.4 li closepath st 3 setlinewidth np 1205.8 3869.4 a 1158.4 4010.4 li 1252.6 4010.4 li closepath st 3 setlinewidth np 1064.8 3869.4 a 1111.6 3775.8 li st 3 setlinewidth np 1205.8 3869.4 a 1158.4 3775.8 li st 3 setlinewidth np 1417 3869.4 a 1370.2 4010.4 li 1463.8 4010.4 li closepath st 3 setlinewidth np 1558 3869.4 a 1510.6 4010.4 li 1604.8 4010.4 li closepath st 3 setlinewidth np 1417 3869.4 a 1463.8 3775.8 li st 3 setlinewidth np 1558 3869.4 a 1510.6 3775.8 li st 3 setlinewidth np 2825.8 3775.8 a 2825.8 3869.4 li st 3 setlinewidth np 2872.6 3775.8 a 2966.8 3869.4 li st 3 setlinewidth np 2825.8 3869.4 a 2779 4010.4 li 2872.6 4010.4 li closepath st 3 setlinewidth np 2966.8 3869.4 a 2920 4010.4 li 3013.6 4010.4 li closepath st 3 setlinewidth np 2779 3775.8 a 2684.8 3869.4 li st 3 setlinewidth np 2684.8 3869.4 a 2638 4010.4 li 2732.2 4010.4 li closepath st 3 setlinewidth np 4375.6 3775.8 a 4375.6 3869.4 li st 3 setlinewidth np 4422.4 3775.8 a 4516.6 3869.4 li st 3 setlinewidth np 4375.6 3869.4 a 4328.8 4010.4 li 4422.4 4010.4 li closepath st 3 setlinewidth np 4516.6 3869.4 a 4469.8 4010.4 li 4563.4 4010.4 li closepath st 3 setlinewidth np 4328.8 3775.8 a 4234.6 3869.4 li st 3 setlinewidth np 4234.6 3869.4 a 4187.8 4010.4 li 4282 4010.4 li closepath st 3 setlinewidth np 3295.6 3869.4 a 3248.8 4010.4 li 3342.4 4010.4 li closepath st 3 setlinewidth np 3436.6 3869.4 a 3389.2 4010.4 li 3483.4 4010.4 li closepath st 3 setlinewidth np 3295.6 3869.4 a 3342.4 3775.8 li st 3 setlinewidth np 3436.6 3869.4 a 3389.2 3775.8 li st 3 setlinewidth np 3647.8 3869.4 a 3601 4010.4 li 3694.6 4010.4 li closepath st 3 setlinewidth np 3788.8 3869.4 a 3742 4010.4 li 3835.6 4010.4 li closepath st 3 setlinewidth np 3647.8 3869.4 a 3694.6 3775.8 li st 3 setlinewidth np 3788.8 3869.4 a 3742 3775.8 li st 0 setgray 3 setlinewidth np 3996 2602 9.6 0 360.001 arc fil 0 setgray np 3996 2602 9.6 0 360.001 arc st 3 setlinewidth np 3996 2602 9.6 0 360.001 arc st 3 setlinewidth np 3997 2603 49.8 0 360.001 arc st 0 setgray 3 setlinewidth np 3127 3353 9.6 0 360.001 arc fil 0 setgray np 3127 3353 9.6 0 360.001 arc st 3 setlinewidth np 3127 3353 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 3714 3353 9.6 0 360.001 arc fil 0 setgray np 3714 3353 9.6 0 360.001 arc st 3 setlinewidth np 3714 3353 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 4371 3348 9.6 0 360.001 arc fil 0 setgray np 4371 3348 9.6 0 360.001 arc st 3 setlinewidth np 4371 3348 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 2141 3353 9.6 0 360.001 arc fil 0 setgray np 2141 3353 9.6 0 360.001 arc st 3 setlinewidth np 2141 3353 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 1483 3353 9.6 0 360.001 arc fil 0 setgray np 1483 3353 9.6 0 360.001 arc st 3 setlinewidth np 1483 3353 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 896 3348 9.6 0 360.001 arc fil 0 setgray np 896 3348 9.6 0 360.001 arc st 3 setlinewidth np 896 3348 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 1765 2602 9.6 0 360.001 arc fil 0 setgray np 1765 2602 9.6 0 360.001 arc st 3 setlinewidth np 1765 2602 9.6 0 360.001 arc st 3 setlinewidth np 3129 3355 49.8 0 360.001 arc st 3 setlinewidth np 3716 3355 49.8 0 360.001 arc st 3 setlinewidth np 4373 3350 49.8 0 360.001 arc st 3 setlinewidth np 2142 3355 49.8 0 360.001 arc st 3 setlinewidth np 1485 3355 49.8 0 360.001 arc st 3 setlinewidth np 898 3350 49.8 0 360.001 arc st 3 setlinewidth np 1767 2603 49.8 0 360.001 arc st 3 setlinewidth np 1722.4 2625 a 1276 2883 li st 3 setlinewidth np 1738 2646 a 1675 2766 li st 3 setlinewidth np 1816 2625 a 2144.8 2883 li st 3 setlinewidth np 1798.6 2644.2 a 1863.4 2766 li st 3 setlinewidth np 1769.2 2655 a 1769.2 2766 li st 3 setlinewidth np 1182.4 3070.8 a 900.4 3300 li st 3 setlinewidth np 880 3396 a 829.6 3493.8 li st 3 setlinewidth np 900.4 3399.6 a 900.4 3493.8 li st 3 setlinewidth np 920.8 3394.2 a 970.6 3493.8 li st 3 setlinewidth np 533 3720 54.6 90.0002 180 arc st 3 setlinewidth np 533 3642 54.6 180 270.001 arc st 3 setlinewidth np 658 3642 54.6 270.001 360.001 arc st 3 setlinewidth np 658 3720 54.6 0 90.0002 arc st 3 setlinewidth np 477.4 3720.6 a 477.4 3642.6 li st 3 setlinewidth np 532.6 3587.4 a 657.4 3587.4 li st 3 setlinewidth np 712.6 3642.6 a 712.6 3720.6 li st 3 setlinewidth np 657.4 3775.8 a 532.6 3775.8 li st 3 setlinewidth np 1276 3070.8 a 1299.4 3141.6 li st 3 setlinewidth np 1229.2 3070.8 a 1205.8 3141.6 li st 3 setlinewidth np 1322.8 3070.8 a 1463.8 3306 li st 3 setlinewidth np 1488.4 3405 a 1487.2 3587.4 li st 3 setlinewidth np 1190 3016 54.6 90.0002 180 arc st 3 setlinewidth np 1190 2937 54.6 180 270.001 arc st 3 setlinewidth np 1315 2937 54.6 270.001 360.001 arc st 3 setlinewidth np 1315 3016 54.6 0 90.0002 arc st 3 setlinewidth np 1135 3016.2 a 1135 2938.2 li st 3 setlinewidth np 1190.2 2883 a 1315 2883 li st 3 setlinewidth np 1370.2 2938.2 a 1370.2 3016.2 li st 3 setlinewidth np 1315 3070.8 a 1190.2 3070.8 li st 3 setlinewidth np 866.8 3389.4 a 595 3587.4 li st 3 setlinewidth np 1073 3720 54.6 90.0002 180 arc st 3 setlinewidth np 1073 3642 54.6 180 270.001 arc st 3 setlinewidth np 1198 3642 54.6 270.001 360.001 arc st 3 setlinewidth np 1198 3720 54.6 0 90.0002 arc st 3 setlinewidth np 1018 3720.6 a 1018 3642.6 li st 3 setlinewidth np 1072.6 3587.4 a 1198 3587.4 li st 3 setlinewidth np 1252.6 3642.6 a 1252.6 3720.6 li st 3 setlinewidth np 1198 3775.8 a 1072.6 3775.8 li st 3 setlinewidth np 931.6 3385.2 a 1135 3587.4 li st 3 setlinewidth np 1425 3720 54.6 90.0002 180 arc st 3 setlinewidth np 1425 3642 54.6 180 270.001 arc st 3 setlinewidth np 1550 3642 54.6 270.001 360.001 arc st 3 setlinewidth np 1550 3720 54.6 0 90.0002 arc st 3 setlinewidth np 1370.2 3720.6 a 1370.2 3642.6 li st 3 setlinewidth np 1424.8 3587.4 a 1550.2 3587.4 li st 3 setlinewidth np 1604.8 3642.6 a 1604.8 3720.6 li st 3 setlinewidth np 1550.2 3775.8 a 1424.8 3775.8 li st 3 setlinewidth np 2144.8 3070.8 a 2144.8 3306 li st 3 setlinewidth np 2144.8 3403.2 a 2144.8 3587.4 li st 3 setlinewidth np 2082 3016 54.6 90.0002 180 arc st 3 setlinewidth np 2082 2937 54.6 180 270.001 arc st 3 setlinewidth np 2208 2937 54.6 270.001 360.001 arc st 3 setlinewidth np 2208 3016 54.6 0 90.0002 arc st 3 setlinewidth np 2027.2 3016.2 a 2027.2 2938.2 li st 3 setlinewidth np 2082.4 2883 a 2207.8 2883 li st 3 setlinewidth np 2262.4 2938.2 a 2262.4 3016.2 li st 3 setlinewidth np 2207.8 3070.8 a 2082.4 3070.8 li st 3 setlinewidth np 2082 3720 54.6 90.0002 180 arc st 3 setlinewidth np 2082 3642 54.6 180 270.001 arc st 3 setlinewidth np 2208 3642 54.6 270.001 360.001 arc st 3 setlinewidth np 2208 3720 54.6 0 90.0002 arc st 3 setlinewidth np 2027.2 3720.6 a 2027.2 3642.6 li st 3 setlinewidth np 2082.4 3587.4 a 2207.8 3587.4 li st 3 setlinewidth np 2262.4 3642.6 a 2262.4 3720.6 li st 3 setlinewidth np 2207.8 3775.8 a 2082.4 3775.8 li st 3 setlinewidth np 1934 3188 a 1946.6 3188 li st 3 setlinewidth np 1967 3188 a 1979.6 3188 li st 3 setlinewidth np 2000 3188 a 2012.6 3188 li st 3 setlinewidth np 2034 3188 a 2046.6 3188 li st 3 setlinewidth np 2067 3188 a 2079.6 3188 li st 3 setlinewidth np 2100 3188 a 2112.6 3188 li st 3 setlinewidth np 2134 3188 a 2146.6 3188 li st 3 setlinewidth np 2167 3188 a 2179.6 3188 li st 3 setlinewidth np 2201 3188 a 2213.6 3188 li st 3 setlinewidth np 2234 3188 a 2246.6 3188 li st 3 setlinewidth np 2267 3188 a 2279.6 3188 li st 3 setlinewidth np 2301 3188 a 2313.6 3188 li st 3 setlinewidth np 2334 3188 a 2346.6 3188 li st 3 setlinewidth np 2367 3188 a 2379.6 3188 li st 3 setlinewidth np 2380 3188 a 2380 3200 li st 3 setlinewidth np 2380 3221 a 2380 3233 li st 3 setlinewidth np 2380 3254 a 2380 3266 li st 3 setlinewidth np 2380 3287 a 2380 3299 li st 3 setlinewidth np 2380 3320 a 2380 3332 li st 3 setlinewidth np 2380 3352 a 2380 3364 li st 3 setlinewidth np 2380 3385 a 2380 3397 li st 3 setlinewidth np 2380 3418 a 2380 3430 li st 3 setlinewidth np 2380 3451 a 2380 3463 li st 3 setlinewidth np 2380 3484 a 2380 3496 li st 3 setlinewidth np 2380 3517 a 2380 3529 li st 3 setlinewidth np 2380 3550 a 2380 3562 li st 3 setlinewidth np 2380 3583 a 2380 3595 li st 3 setlinewidth np 2380 3616 a 2380 3628 li st 3 setlinewidth np 2380 3649 a 2380 3661 li st 3 setlinewidth np 2380 3682 a 2380 3694 li st 3 setlinewidth np 2380 3715 a 2380 3727 li st 3 setlinewidth np 2380 3748 a 2380 3760 li st 3 setlinewidth np 2380 3781 a 2380 3793 li st 3 setlinewidth np 2380 3814 a 2380 3826 li st 3 setlinewidth np 2380 3847 a 2380 3859 li st 3 setlinewidth np 2380 3879 a 2380 3891 li st 3 setlinewidth np 2380 3912 a 2380 3924 li st 3 setlinewidth np 2380 3945 a 2380 3957 li st 3 setlinewidth np 2380 3978 a 2380 3990 li st 3 setlinewidth np 2380 4011 a 2380 4023 li st 3 setlinewidth np 2380 4044 a 2380 4056 li st 3 setlinewidth np 2380 4057 a 2368 4057 li st 3 setlinewidth np 2346 4057 a 2334 4057 li st 3 setlinewidth np 2313 4057 a 2301 4057 li st 3 setlinewidth np 2280 4057 a 2268 4057 li st 3 setlinewidth np 2246 4057 a 2234 4057 li st 3 setlinewidth np 2213 4057 a 2201 4057 li st 3 setlinewidth np 2180 4057 a 2168 4057 li st 3 setlinewidth np 2146 4057 a 2134 4057 li st 3 setlinewidth np 2113 4057 a 2101 4057 li st 3 setlinewidth np 2079 4057 a 2067 4057 li st 3 setlinewidth np 2046 4057 a 2034 4057 li st 3 setlinewidth np 2013 4057 a 2001 4057 li st 3 setlinewidth np 1979 4057 a 1967 4057 li st 3 setlinewidth np 1946 4057 a 1934 4057 li st 3 setlinewidth np 1934 4057 a 1934 4044.4 li st 3 setlinewidth np 1934 4024 a 1934 4011.4 li st 3 setlinewidth np 1934 3991 a 1934 3978.4 li st 3 setlinewidth np 1934 3958 a 1934 3945.4 li st 3 setlinewidth np 1934 3925 a 1934 3912.4 li st 3 setlinewidth np 1934 3892 a 1934 3879.4 li st 3 setlinewidth np 1934 3859 a 1934 3846.4 li st 3 setlinewidth np 1934 3826 a 1934 3813.4 li st 3 setlinewidth np 1934 3793 a 1934 3780.4 li st 3 setlinewidth np 1934 3760 a 1934 3747.4 li st 3 setlinewidth np 1934 3727 a 1934 3714.4 li st 3 setlinewidth np 1934 3694 a 1934 3681.4 li st 3 setlinewidth np 1934 3661 a 1934 3648.4 li st 3 setlinewidth np 1934 3628 a 1934 3615.4 li st 3 setlinewidth np 1934 3595 a 1934 3582.4 li st 3 setlinewidth np 1934 3563 a 1934 3550.4 li st 3 setlinewidth np 1934 3530 a 1934 3517.4 li st 3 setlinewidth np 1934 3497 a 1934 3484.4 li st 3 setlinewidth np 1934 3464 a 1934 3451.4 li st 3 setlinewidth np 1934 3431 a 1934 3418.4 li st 3 setlinewidth np 1934 3398 a 1934 3385.4 li st 3 setlinewidth np 1934 3365 a 1934 3352.4 li st 3 setlinewidth np 1934 3332 a 1934 3319.4 li st 3 setlinewidth np 1934 3299 a 1934 3286.4 li st 3 setlinewidth np 1934 3266 a 1934 3253.4 li st 3 setlinewidth np 1934 3233 a 1934 3220.4 li st 3 setlinewidth np 1934 3200 a 1934 3187.4 li st 3 setlinewidth np 1112 3188 a 1118.6 3198.2 li st 3 setlinewidth np 1128 3215 a 1134.6 3225.2 li st 3 setlinewidth np 1145 3241 a 1151.6 3251.2 li st 3 setlinewidth np 1161 3268 a 1167.6 3278.2 li st 3 setlinewidth np 1178 3295 a 1184.6 3305.2 li st 3 setlinewidth np 1194 3322 a 1200.6 3332.2 li st 3 setlinewidth np 1211 3349 a 1217.6 3359.2 li st 3 setlinewidth np 1227 3375 a 1233.6 3385.2 li st 3 setlinewidth np 1244 3402 a 1250.6 3412.2 li st 3 setlinewidth np 1260 3429 a 1266.6 3439.2 li st 3 setlinewidth np 1277 3456 a 1283.6 3466.2 li st 3 setlinewidth np 1293 3483 a 1299.6 3493.2 li st 3 setlinewidth np 1300 3493 a 1300 3504.4 li st 3 setlinewidth np 1300 3528 a 1300 3539.4 li st 3 setlinewidth np 1300 3562 a 1300 3573.4 li st 3 setlinewidth np 1300 3596 a 1300 3607.4 li st 3 setlinewidth np 1300 3631 a 1300 3642.4 li st 3 setlinewidth np 1300 3665 a 1300 3676.4 li st 3 setlinewidth np 1300 3700 a 1300 3711.4 li st 3 setlinewidth np 1300 3734 a 1300 3745.4 li st 3 setlinewidth np 1300 3769 a 1300 3780.4 li st 3 setlinewidth np 1300 3803 a 1300 3814.4 li st 3 setlinewidth np 1300 3838 a 1300 3849.4 li st 3 setlinewidth np 1300 3872 a 1300 3883.4 li st 3 setlinewidth np 1300 3907 a 1300 3918.4 li st 3 setlinewidth np 1300 3941 a 1300 3952.4 li st 3 setlinewidth np 1300 3975 a 1300 3986.4 li st 3 setlinewidth np 1300 4010 a 1300 4021.4 li st 3 setlinewidth np 1300 4044 a 1300 4055.4 li st 3 setlinewidth np 1300 4057 a 1288 4057 li st 3 setlinewidth np 1267 4057 a 1255 4057 li st 3 setlinewidth np 1233 4057 a 1221 4057 li st 3 setlinewidth np 1200 4057 a 1188 4057 li st 3 setlinewidth np 1167 4057 a 1155 4057 li st 3 setlinewidth np 1134 4057 a 1122 4057 li st 3 setlinewidth np 1101 4057 a 1089 4057 li st 3 setlinewidth np 1068 4057 a 1056 4057 li st 3 setlinewidth np 1035 4057 a 1023 4057 li st 3 setlinewidth np 1002 4057 a 990 4057 li st 3 setlinewidth np 969 4057 a 957 4057 li st 3 setlinewidth np 936 4057 a 924 4057 li st 3 setlinewidth np 902 4057 a 890 4057 li st 3 setlinewidth np 869 4057 a 857 4057 li st 3 setlinewidth np 836 4057 a 824 4057 li st 3 setlinewidth np 803 4057 a 791 4057 li st 3 setlinewidth np 770 4057 a 758 4057 li st 3 setlinewidth np 737 4057 a 725 4057 li st 3 setlinewidth np 704 4057 a 692 4057 li st 3 setlinewidth np 671 4057 a 659 4057 li st 3 setlinewidth np 638 4057 a 626 4057 li st 3 setlinewidth np 604 4057 a 592 4057 li st 3 setlinewidth np 571 4057 a 559 4057 li st 3 setlinewidth np 538 4057 a 526 4057 li st 3 setlinewidth np 505 4057 a 493 4057 li st 3 setlinewidth np 472 4057 a 460 4057 li st 3 setlinewidth np 439 4057 a 427 4057 li st 3 setlinewidth np 406 4057 a 394 4057 li st 3 setlinewidth np 373 4057 a 361 4057 li st 3 setlinewidth np 360 4057 a 360 4045 li st 3 setlinewidth np 360 4024 a 360 4012 li st 3 setlinewidth np 360 3991 a 360 3979 li st 3 setlinewidth np 360 3958 a 360 3946 li st 3 setlinewidth np 360 3925 a 360 3913 li st 3 setlinewidth np 360 3892 a 360 3880 li st 3 setlinewidth np 360 3859 a 360 3847 li st 3 setlinewidth np 360 3826 a 360 3814 li st 3 setlinewidth np 360 3793 a 360 3781 li st 3 setlinewidth np 360 3760 a 360 3748 li st 3 setlinewidth np 360 3727 a 360 3715 li st 3 setlinewidth np 360 3694 a 360 3682 li st 3 setlinewidth np 360 3661 a 360 3649 li st 3 setlinewidth np 360 3628 a 360 3616 li st 3 setlinewidth np 360 3595 a 360 3583 li st 3 setlinewidth np 360 3562 a 360 3550 li st 3 setlinewidth np 360 3529 a 360 3517 li st 3 setlinewidth np 360 3517 a 370.2 3509.8 li st 3 setlinewidth np 387 3498 a 397.2 3490.8 li st 3 setlinewidth np 414 3479 a 424.2 3471.8 li st 3 setlinewidth np 442 3460 a 452.2 3452.8 li st 3 setlinewidth np 469 3441 a 479.2 3433.8 li st 3 setlinewidth np 496 3422 a 506.2 3414.8 li st 3 setlinewidth np 523 3403 a 533.2 3395.8 li st 3 setlinewidth np 550 3384 a 560.2 3376.8 li st 3 setlinewidth np 577 3365 a 587.2 3357.8 li st 3 setlinewidth np 604 3346 a 614.2 3338.8 li st 3 setlinewidth np 631 3327 a 641.2 3319.8 li st 3 setlinewidth np 658 3308 a 668.2 3300.8 li st 3 setlinewidth np 685 3289 a 695.2 3281.8 li st 3 setlinewidth np 712 3270 a 722.2 3262.8 li st 3 setlinewidth np 739 3252 a 749.2 3244.8 li st 3 setlinewidth np 766 3233 a 776.2 3225.8 li st 3 setlinewidth np 793 3214 a 803.2 3206.8 li st 3 setlinewidth np 820 3195 a 830.2 3187.8 li st 3 setlinewidth np 830 3188 a 842 3188 li st 3 setlinewidth np 864 3188 a 876 3188 li st 3 setlinewidth np 897 3188 a 909 3188 li st 3 setlinewidth np 931 3188 a 943 3188 li st 3 setlinewidth np 965 3188 a 977 3188 li st 3 setlinewidth np 998 3188 a 1010 3188 li st 3 setlinewidth np 1032 3188 a 1044 3188 li st 3 setlinewidth np 1066 3188 a 1078 3188 li st 3 setlinewidth np 1100 3188 a 1112 3188 li st 3 setlinewidth np 3953.2 2625 a 3506.8 2883 li st 3 setlinewidth np 3976.6 2648.4 a 3905.8 2766 li st 3 setlinewidth np 4046.8 2625 a 4375.6 2883 li st 3 setlinewidth np 4023.4 2648.4 a 4094.2 2766 li st 3 setlinewidth np 3999.4 2653.2 a 4000 2766 li st 3 setlinewidth np 3413.2 3070.8 a 3131.2 3306 li st 3 setlinewidth np 3108.4 3400.8 a 3061 3493.8 li st 3 setlinewidth np 3130.6 3405 a 3131.2 3493.8 li st 3 setlinewidth np 3154.6 3399.6 a 3201.4 3493.8 li st 3 setlinewidth np 2763 3720 54.6 90.0002 180 arc st 3 setlinewidth np 2763 3642 54.6 180 270.001 arc st 3 setlinewidth np 2889 3642 54.6 270.001 360.001 arc st 3 setlinewidth np 2889 3720 54.6 0 90.0002 arc st 3 setlinewidth np 2708.2 3720.6 a 2708.2 3642.6 li st 3 setlinewidth np 2763.4 3587.4 a 2888.8 3587.4 li st 3 setlinewidth np 2943.4 3642.6 a 2943.4 3720.6 li st 3 setlinewidth np 2888.8 3775.8 a 2763.4 3775.8 li st 3 setlinewidth np 3506.8 3070.8 a 3530.2 3141.6 li st 3 setlinewidth np 3460 3070.8 a 3436.6 3141.6 li st 3 setlinewidth np 3553.6 3070.8 a 3694.6 3306 li st 3 setlinewidth np 3717.4 3405 a 3718 3587.4 li st 3 setlinewidth np 3421 3016 54.6 90.0002 180 arc st 3 setlinewidth np 3421 2937 54.6 180 270.001 arc st 3 setlinewidth np 3546 2937 54.6 270.001 360.001 arc st 3 setlinewidth np 3546 3016 54.6 0 90.0002 arc st 3 setlinewidth np 3365.8 3016.2 a 3365.8 2938.2 li st 3 setlinewidth np 3421 2883 a 3545.8 2883 li st 3 setlinewidth np 3601 2938.2 a 3601 3016.2 li st 3 setlinewidth np 3545.8 3070.8 a 3421 3070.8 li st 3 setlinewidth np 3094.6 3394.2 a 2825.8 3587.4 li st 3 setlinewidth np 3303 3720 54.6 90.0002 180 arc st 3 setlinewidth np 3303 3642 54.6 180 270.001 arc st 3 setlinewidth np 3429 3642 54.6 270.001 360.001 arc st 3 setlinewidth np 3429 3720 54.6 0 90.0002 arc st 3 setlinewidth np 3248.8 3720.6 a 3248.8 3642.6 li st 3 setlinewidth np 3303.4 3587.4 a 3428.8 3587.4 li st 3 setlinewidth np 3483.4 3642.6 a 3483.4 3720.6 li st 3 setlinewidth np 3428.8 3775.8 a 3303.4 3775.8 li st 3 setlinewidth np 3166.6 3389.4 a 3365.8 3587.4 li st 3 setlinewidth np 3656 3720 54.6 90.0002 180 arc st 3 setlinewidth np 3656 3642 54.6 180 270.001 arc st 3 setlinewidth np 3781 3642 54.6 270.001 360.001 arc st 3 setlinewidth np 3781 3720 54.6 0 90.0002 arc st 3 setlinewidth np 3601 3720.6 a 3601 3642.6 li st 3 setlinewidth np 3655.6 3587.4 a 3781 3587.4 li st 3 setlinewidth np 3835.6 3642.6 a 3835.6 3720.6 li st 3 setlinewidth np 3781 3775.8 a 3655.6 3775.8 li st 3 setlinewidth np 4375.6 3070.8 a 4374.4 3298.2 li st 3 setlinewidth np 4375 3399 a 4375.6 3587.4 li st 3 setlinewidth np 4313 3016 54.6 90.0002 180 arc st 3 setlinewidth np 4313 2937 54.6 180 270.001 arc st 3 setlinewidth np 4438 2937 54.6 270.001 360.001 arc st 3 setlinewidth np 4438 3016 54.6 0 90.0002 arc st 3 setlinewidth np 4258.6 3016.2 a 4258.6 2938.2 li st 3 setlinewidth np 4313.2 2883 a 4438.6 2883 li st 3 setlinewidth np 4493.2 2938.2 a 4493.2 3016.2 li st 3 setlinewidth np 4438.6 3070.8 a 4313.2 3070.8 li st 3 setlinewidth np 4313 3720 54.6 90.0002 180 arc st 3 setlinewidth np 4313 3642 54.6 180 270.001 arc st 3 setlinewidth np 4438 3642 54.6 270.001 360.001 arc st 3 setlinewidth np 4438 3720 54.6 0 90.0002 arc st 3 setlinewidth np 4258.6 3720.6 a 4258.6 3642.6 li st 3 setlinewidth np 4313.2 3587.4 a 4438.6 3587.4 li st 3 setlinewidth np 4493.2 3642.6 a 4493.2 3720.6 li st 3 setlinewidth np 4438.6 3775.8 a 4313.2 3775.8 li st 4 setlinewidth np [ 4 27.4783 ] 0 setdash 313 4128 a 313 3399.6 li st [] 0 setdash [] 0 setdash 4 setlinewidth np [ 4 26.7094 ] 0 setdash 313 3399.6 a 1182.4 2812.8 li st [] 0 setdash [] 0 setdash 4 setlinewidth np [ 4 26.7778 ] 0 setdash 1182.4 2812.8 a 1463.8 2812.8 li st [] 0 setdash [] 0 setdash 4 setlinewidth np [ 4 27.1302 ] 0 setdash 1463.8 2812.8 a 1651.6 3141.6 li st [] 0 setdash [] 0 setdash 4 setlinewidth np [ 4 26.6875 ] 0 setdash 1651.6 3141.6 a 1651.6 4128 li st [] 0 setdash [] 0 setdash 4 setlinewidth np [ 4 27.0233 ] 0 setdash 1651.6 4128 a 313 4128 li st [] 0 setdash [] 0 setdash 4 setlinewidth np [ 4 34.1053 ] 0 setdash 2543.8 4128 a 2543.8 3399.6 li st [] 0 setdash [] 0 setdash 4 setlinewidth np [ 4 34.6711 ] 0 setdash 2543.8 3399.6 a 3413.2 2812.8 li st [] 0 setdash [] 0 setdash 4 setlinewidth np [ 4 35.5714 ] 0 setdash 3413.2 2812.8 a 3694.6 2812.8 li st [] 0 setdash [] 0 setdash 4 setlinewidth np [ 4 33.3562 ] 0 setdash 3694.6 2812.8 a 3882.4 3141.6 li st [] 0 setdash [] 0 setdash 4 setlinewidth np [ 4 35.28 ] 0 setdash 3882.4 3141.6 a 3882.4 4128 li st [] 0 setdash [] 0 setdash 4 setlinewidth np [ 4 35.2353 ] 0 setdash 3882.4 4128 a 2543.8 4128 li st [] 0 setdash [] 0 setdash 4 setlinewidth np [ 4 34.2857 ] 0 setdash 4117.6 4128 a 4657.6 4128 li st [] 0 setdash [] 0 setdash 4 setlinewidth np [ 4 34.5588 ] 0 setdash 4657.6 4128 a 4657.6 2812.8 li st [] 0 setdash [] 0 setdash 4 setlinewidth np [ 4 34.2857 ] 0 setdash 4657.6 2812.8 a 4117.6 2812.8 li st [] 0 setdash [] 0 setdash 4 setlinewidth np [ 4 34.5588 ] 0 setdash 4117.6 2812.8 a 4117.6 4128 li st [] 0 setdash [] 0 setdash 3 setlinewidth np 4164 3188 a 4176.6 3188 li st 3 setlinewidth np 4198 3188 a 4210.6 3188 li st 3 setlinewidth np 4231 3188 a 4243.6 3188 li st 3 setlinewidth np 4264 3188 a 4276.6 3188 li st 3 setlinewidth np 4298 3188 a 4310.6 3188 li st 3 setlinewidth np 4331 3188 a 4343.6 3188 li st 3 setlinewidth np 4365 3188 a 4377.6 3188 li st 3 setlinewidth np 4398 3188 a 4410.6 3188 li st 3 setlinewidth np 4431 3188 a 4443.6 3188 li st 3 setlinewidth np 4465 3188 a 4477.6 3188 li st 3 setlinewidth np 4498 3188 a 4510.6 3188 li st 3 setlinewidth np 4531 3188 a 4543.6 3188 li st 3 setlinewidth np 4565 3188 a 4577.6 3188 li st 3 setlinewidth np 4598 3188 a 4610.6 3188 li st 3 setlinewidth np 4610 3188 a 4610 3200 li st 3 setlinewidth np 4610 3221 a 4610 3233 li st 3 setlinewidth np 4610 3254 a 4610 3266 li st 3 setlinewidth np 4610 3287 a 4610 3299 li st 3 setlinewidth np 4610 3320 a 4610 3332 li st 3 setlinewidth np 4610 3352 a 4610 3364 li st 3 setlinewidth np 4610 3385 a 4610 3397 li st 3 setlinewidth np 4610 3418 a 4610 3430 li st 3 setlinewidth np 4610 3451 a 4610 3463 li st 3 setlinewidth np 4610 3484 a 4610 3496 li st 3 setlinewidth np 4610 3517 a 4610 3529 li st 3 setlinewidth np 4610 3550 a 4610 3562 li st 3 setlinewidth np 4610 3583 a 4610 3595 li st 3 setlinewidth np 4610 3616 a 4610 3628 li st 3 setlinewidth np 4610 3649 a 4610 3661 li st 3 setlinewidth np 4610 3682 a 4610 3694 li st 3 setlinewidth np 4610 3715 a 4610 3727 li st 3 setlinewidth np 4610 3748 a 4610 3760 li st 3 setlinewidth np 4610 3781 a 4610 3793 li st 3 setlinewidth np 4610 3814 a 4610 3826 li st 3 setlinewidth np 4610 3847 a 4610 3859 li st 3 setlinewidth np 4610 3879 a 4610 3891 li st 3 setlinewidth np 4610 3912 a 4610 3924 li st 3 setlinewidth np 4610 3945 a 4610 3957 li st 3 setlinewidth np 4610 3978 a 4610 3990 li st 3 setlinewidth np 4610 4011 a 4610 4023 li st 3 setlinewidth np 4610 4044 a 4610 4056 li st 3 setlinewidth np 4610 4057 a 4598 4057 li st 3 setlinewidth np 4577 4057 a 4565 4057 li st 3 setlinewidth np 4544 4057 a 4532 4057 li st 3 setlinewidth np 4510 4057 a 4498 4057 li st 3 setlinewidth np 4477 4057 a 4465 4057 li st 3 setlinewidth np 4444 4057 a 4432 4057 li st 3 setlinewidth np 4410 4057 a 4398 4057 li st 3 setlinewidth np 4377 4057 a 4365 4057 li st 3 setlinewidth np 4344 4057 a 4332 4057 li st 3 setlinewidth np 4310 4057 a 4298 4057 li st 3 setlinewidth np 4277 4057 a 4265 4057 li st 3 setlinewidth np 4243 4057 a 4231 4057 li st 3 setlinewidth np 4210 4057 a 4198 4057 li st 3 setlinewidth np 4177 4057 a 4165 4057 li st 3 setlinewidth np 4164 4057 a 4164 4044.4 li st 3 setlinewidth np 4164 4024 a 4164 4011.4 li st 3 setlinewidth np 4164 3991 a 4164 3978.4 li st 3 setlinewidth np 4164 3958 a 4164 3945.4 li st 3 setlinewidth np 4164 3925 a 4164 3912.4 li st 3 setlinewidth np 4164 3892 a 4164 3879.4 li st 3 setlinewidth np 4164 3859 a 4164 3846.4 li st 3 setlinewidth np 4164 3826 a 4164 3813.4 li st 3 setlinewidth np 4164 3793 a 4164 3780.4 li st 3 setlinewidth np 4164 3760 a 4164 3747.4 li st 3 setlinewidth np 4164 3727 a 4164 3714.4 li st 3 setlinewidth np 4164 3694 a 4164 3681.4 li st 3 setlinewidth np 4164 3661 a 4164 3648.4 li st 3 setlinewidth np 4164 3628 a 4164 3615.4 li st 3 setlinewidth np 4164 3595 a 4164 3582.4 li st 3 setlinewidth np 4164 3563 a 4164 3550.4 li st 3 setlinewidth np 4164 3530 a 4164 3517.4 li st 3 setlinewidth np 4164 3497 a 4164 3484.4 li st 3 setlinewidth np 4164 3464 a 4164 3451.4 li st 3 setlinewidth np 4164 3431 a 4164 3418.4 li st 3 setlinewidth np 4164 3398 a 4164 3385.4 li st 3 setlinewidth np 4164 3365 a 4164 3352.4 li st 3 setlinewidth np 4164 3332 a 4164 3319.4 li st 3 setlinewidth np 4164 3299 a 4164 3286.4 li st 3 setlinewidth np 4164 3266 a 4164 3253.4 li st 3 setlinewidth np 4164 3233 a 4164 3220.4 li st 3 setlinewidth np 4164 3200 a 4164 3187.4 li st 4 setlinewidth np [ 4 27.5294 ] 0 setdash 1886.8 4128 a 2426.8 4128 li st [] 0 setdash [] 0 setdash 4 setlinewidth np [ 4 27.2143 ] 0 setdash 2426.8 4128 a 2426.8 2812.8 li st [] 0 setdash [] 0 setdash 4 setlinewidth np [ 4 27.5294 ] 0 setdash 2426.8 2812.8 a 1886.8 2812.8 li st [] 0 setdash [] 0 setdash 4 setlinewidth np [ 4 27.2143 ] 0 setdash 1886.8 2812.8 a 1886.8 4128 li st [] 0 setdash [] 0 setdash 3 setlinewidth np 2115.4 3069.6 a 2098 3141.6 li st 3 setlinewidth np 2098 3070.8 a 2051.2 3141.6 li st 3 setlinewidth np 4346.8 3069.6 a 4328.8 3141.6 li st 3 setlinewidth np 4328.8 3070.8 a 4282 3141.6 li st 3 setlinewidth np 3342 3188 a 3348.6 3198.2 li st 3 setlinewidth np 3359 3215 a 3365.6 3225.2 li st 3 setlinewidth np 3375 3241 a 3381.6 3251.2 li st 3 setlinewidth np 3392 3268 a 3398.6 3278.2 li st 3 setlinewidth np 3408 3295 a 3414.6 3305.2 li st 3 setlinewidth np 3425 3322 a 3431.6 3332.2 li st 3 setlinewidth np 3441 3349 a 3447.6 3359.2 li st 3 setlinewidth np 3458 3375 a 3464.6 3385.2 li st 3 setlinewidth np 3474 3402 a 3480.6 3412.2 li st 3 setlinewidth np 3491 3429 a 3497.6 3439.2 li st 3 setlinewidth np 3507 3456 a 3513.6 3466.2 li st 3 setlinewidth np 3524 3483 a 3530.6 3493.2 li st 3 setlinewidth np 3530 3493 a 3530 3504.4 li st 3 setlinewidth np 3530 3528 a 3530 3539.4 li st 3 setlinewidth np 3530 3562 a 3530 3573.4 li st 3 setlinewidth np 3530 3596 a 3530 3607.4 li st 3 setlinewidth np 3530 3631 a 3530 3642.4 li st 3 setlinewidth np 3530 3665 a 3530 3676.4 li st 3 setlinewidth np 3530 3700 a 3530 3711.4 li st 3 setlinewidth np 3530 3734 a 3530 3745.4 li st 3 setlinewidth np 3530 3769 a 3530 3780.4 li st 3 setlinewidth np 3530 3803 a 3530 3814.4 li st 3 setlinewidth np 3530 3838 a 3530 3849.4 li st 3 setlinewidth np 3530 3872 a 3530 3883.4 li st 3 setlinewidth np 3530 3907 a 3530 3918.4 li st 3 setlinewidth np 3530 3941 a 3530 3952.4 li st 3 setlinewidth np 3530 3975 a 3530 3986.4 li st 3 setlinewidth np 3530 4010 a 3530 4021.4 li st 3 setlinewidth np 3530 4044 a 3530 4055.4 li st 3 setlinewidth np 3530 4057 a 3518 4057 li st 3 setlinewidth np 3497 4057 a 3485 4057 li st 3 setlinewidth np 3464 4057 a 3452 4057 li st 3 setlinewidth np 3431 4057 a 3419 4057 li st 3 setlinewidth np 3398 4057 a 3386 4057 li st 3 setlinewidth np 3365 4057 a 3353 4057 li st 3 setlinewidth np 3332 4057 a 3320 4057 li st 3 setlinewidth np 3299 4057 a 3287 4057 li st 3 setlinewidth np 3266 4057 a 3254 4057 li st 3 setlinewidth np 3232 4057 a 3220 4057 li st 3 setlinewidth np 3199 4057 a 3187 4057 li st 3 setlinewidth np 3166 4057 a 3154 4057 li st 3 setlinewidth np 3133 4057 a 3121 4057 li st 3 setlinewidth np 3100 4057 a 3088 4057 li st 3 setlinewidth np 3067 4057 a 3055 4057 li st 3 setlinewidth np 3034 4057 a 3022 4057 li st 3 setlinewidth np 3001 4057 a 2989 4057 li st 3 setlinewidth np 2968 4057 a 2956 4057 li st 3 setlinewidth np 2934 4057 a 2922 4057 li st 3 setlinewidth np 2901 4057 a 2889 4057 li st 3 setlinewidth np 2868 4057 a 2856 4057 li st 3 setlinewidth np 2835 4057 a 2823 4057 li st 3 setlinewidth np 2802 4057 a 2790 4057 li st 3 setlinewidth np 2769 4057 a 2757 4057 li st 3 setlinewidth np 2736 4057 a 2724 4057 li st 3 setlinewidth np 2703 4057 a 2691 4057 li st 3 setlinewidth np 2670 4057 a 2658 4057 li st 3 setlinewidth np 2637 4057 a 2625 4057 li st 3 setlinewidth np 2603 4057 a 2591 4057 li st 3 setlinewidth np 2591 4057 a 2591 4045 li st 3 setlinewidth np 2591 4024 a 2591 4012 li st 3 setlinewidth np 2591 3991 a 2591 3979 li st 3 setlinewidth np 2591 3958 a 2591 3946 li st 3 setlinewidth np 2591 3925 a 2591 3913 li st 3 setlinewidth np 2591 3892 a 2591 3880 li st 3 setlinewidth np 2591 3859 a 2591 3847 li st 3 setlinewidth np 2591 3826 a 2591 3814 li st 3 setlinewidth np 2591 3793 a 2591 3781 li st 3 setlinewidth np 2591 3760 a 2591 3748 li st 3 setlinewidth np 2591 3727 a 2591 3715 li st 3 setlinewidth np 2591 3694 a 2591 3682 li st 3 setlinewidth np 2591 3661 a 2591 3649 li st 3 setlinewidth np 2591 3628 a 2591 3616 li st 3 setlinewidth np 2591 3595 a 2591 3583 li st 3 setlinewidth np 2591 3562 a 2591 3550 li st 3 setlinewidth np 2591 3529 a 2591 3517 li st 3 setlinewidth np 2591 3517 a 2601.2 3509.8 li st 3 setlinewidth np 2618 3498 a 2628.2 3490.8 li st 3 setlinewidth np 2645 3479 a 2655.2 3471.8 li st 3 setlinewidth np 2672 3460 a 2682.2 3452.8 li st 3 setlinewidth np 2699 3441 a 2709.2 3433.8 li st 3 setlinewidth np 2726 3422 a 2736.2 3414.8 li st 3 setlinewidth np 2753 3403 a 2763.2 3395.8 li st 3 setlinewidth np 2780 3384 a 2790.2 3376.8 li st 3 setlinewidth np 2807 3365 a 2817.2 3357.8 li st 3 setlinewidth np 2834 3346 a 2844.2 3338.8 li st 3 setlinewidth np 2861 3327 a 2871.2 3319.8 li st 3 setlinewidth np 2889 3308 a 2899.2 3300.8 li st 3 setlinewidth np 2916 3289 a 2926.2 3281.8 li st 3 setlinewidth np 2943 3270 a 2953.2 3262.8 li st 3 setlinewidth np 2970 3252 a 2980.2 3244.8 li st 3 setlinewidth np 2997 3233 a 3007.2 3225.8 li st 3 setlinewidth np 3024 3214 a 3034.2 3206.8 li st 3 setlinewidth np 3051 3195 a 3061.2 3187.8 li st 3 setlinewidth np 3061 3188 a 3073 3188 li st 3 setlinewidth np 3094 3188 a 3106 3188 li st 3 setlinewidth np 3128 3188 a 3140 3188 li st 3 setlinewidth np 3162 3188 a 3174 3188 li st 3 setlinewidth np 3195 3188 a 3207 3188 li st 3 setlinewidth np 3229 3188 a 3241 3188 li st 3 setlinewidth np 3263 3188 a 3275 3188 li st 3 setlinewidth np 3297 3188 a 3309 3188 li st 3 setlinewidth np 3330 3188 a 3342 3188 li st 1675 3678 a Ff(.)31 b(.)g(.)1088 3350 y(.)g(.)g(.)1581 2974 y(.)g(.)g(.)783 3678 y(.)g(.)g(.)2905 b(.)31 b(.)g(.)3319 3350 y(.)g(.)g(.)3812 2974 y(.)g(.)g(.)3014 3678 y(.)g(.)g(.)1675 3350 y(.)g(.)g(.)2013 b(.)31 b(.)g(.)1209 3008 y FB(B)1270 3023 y FA(1)2101 3008 y FB(B)2162 3023 y Fr(k)3439 2993 y FB(B)3500 2957 y Fw(0)3500 3023 y FA(1)4332 2992 y FB(B)4393 2956 y Fw(0)4393 3023 y Fr(k)3859 2624 y FB(a)3909 2587 y Fw(0)3554 3335 y FB(a)3604 3299 y Fw(0)3604 3366 y Fr(l)3622 3376 y Fl(1)4211 3334 y FB(a)4261 3298 y Fw(0)4261 3365 y Fr(l)4279 3375 y Fd(k)2990 3345 y FB(a)3040 3309 y Fw(0)3040 3376 y FA(1)1323 3351 y FB(a)1373 3366 y Fr(l)1391 3376 y Fl(1)760 3361 y FB(a)810 3376 y FA(1)1981 3350 y FB(a)2031 3365 y Fr(l)2049 3375 y Fd(k)1628 2624 y FB(a)3530 2491 y(T)3589 2506 y Fr(a)3622 2487 y Fh(0)1300 2491 y FB(S)1365 2506 y Fr(a)830 3139 y FB(S)895 3154 y Fr(a)928 3164 y Fl(1)3061 3125 y FB(T)3120 3140 y Fr(a)3153 3116 y Fh(0)3153 3164 y Fl(1)2262 3123 y FB(S)2327 3138 y Fr(a)2360 3148 y Fd(l)2374 3164 y(k)4493 3108 y FB(T)4552 3123 y Fr(a)4585 3099 y Fh(0)4585 3148 y Fd(l)4599 3164 y(k)3413 2749 y FB(T)3474 2764 y Fr(B)3515 2740 y Fh(0)3515 2789 y Fl(1)4399 2748 y FB(T)4460 2763 y Fr(B)4501 2740 y Fh(0)4501 2788 y Fd(k)2168 2763 y FB(S)2235 2778 y Fr(B)2276 2788 y Fd(k)1182 2764 y FB(S)1249 2779 y Fr(B)1290 2789 y Fl(1)257 4139 y currentpoint currentpoint translate 1 0.72 div 1 0.72 div scale neg exch neg exch translate 257 4139 a Black 1078 4450 a FF(Figure)25 b(5:)1471 4450 y SDict begin H.S end 1471 4450 a Black Black 1471 4450 a SDict begin H.R end 1471 4450 a 1471 4450 a SDict begin [ /View [/XYZ H.V] /Dest (figure.5) cvn H.B /DEST pdfmark end 1471 4450 a FF(Biconnected)g(component)f (trees.)p Black Black 404 4666 a(W)-8 b(e)21 b(say)g(that)f(tw)o(o)h (biconnected)f(component)g(trees)h(are)h FB(equal)p FF(,)f(denoted)g (by)f FB(S)3157 4681 y Fr(a)3222 4666 y FE(=)3285 4681 y Fg(B)3352 4666 y FB(T)3411 4681 y Fr(a)3444 4662 y Fh(0)3470 4666 y FF(,)257 4786 y(if)j(neither)g(of)h FB(S)813 4801 y Fr(a)877 4786 y FC(<)940 4801 y Fg(B)1007 4786 y FB(T)1066 4801 y Fr(a)1099 4782 y Fh(0)1149 4786 y FF(and)f FB(T)1375 4801 y Fr(a)1408 4782 y Fh(0)1462 4786 y FC(<)1525 4801 y Fg(B)1592 4786 y FB(S)1657 4801 y Fr(a)1717 4786 y FF(holds.)30 b(The)23 b(inducti)n(v)o(e)e(ordering)i (of)g(the)g(subtrees)257 4907 y(of)f FB(S)427 4922 y Fr(a)487 4907 y FF(and)g FB(T)712 4922 y Fr(a)745 4902 y Fh(0)793 4907 y FF(proceeds)g(e)o(xactly)f(as)i(in)e(Lindell)16 b(')-5 b(s)21 b(algorithm,)g(by)h(partitioning)e(them)h(into)257 5027 y(size-classes)k(and)g(comparing)f(the)g(children)h(in)f(the)h (same)g(size-class)f(recursi)n(v)o(ely)-6 b(.)404 5147 y(W)e(e)35 b(summarize)f(no)n(w)-6 b(,)36 b(ho)n(w)e(we)h(can)g (compute)f(the)g(isomorphism)e(order)j(when)g(we)257 5268 y(compare)24 b(subtrees)f(rooted)g(at)g(articulation)g(points,)f (e.g.)h FB(S)2349 5283 y Fr(a)2410 5268 y FF(and)g FB(T)2636 5283 y Fr(a)2669 5263 y Fh(0)2695 5268 y FF(,)h(and)g(when)f(we)g(com-) 257 5388 y(pare)j(subtrees)e(rooted)h(at)f(biconnected)h(components,)e (e.g.)i FB(S)2455 5403 y Fr(B)2496 5413 y Fd(i)2542 5388 y FF(and)g FB(T)2772 5403 y Fr(B)2820 5413 y Fd(j)2849 5384 y Fh(0)2874 5388 y FF(.)p Black Black eop end %%Page: 19 19 TeXDict begin 19 18 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 257 266 a SDict begin H.S end 257 266 a Black Black 257 266 a SDict begin H.R end 257 266 a 257 266 a SDict begin [ /View [/XYZ H.V] /Dest (page.19) cvn H.B /DEST pdfmark end 257 266 a Black 404 573 a FF(Comparing)23 b FB(S)941 588 y Fr(a)1002 573 y FF(and)g FB(T)1228 588 y Fr(a)1261 569 y Fh(0)1311 573 y FF(is)h(similar)e(to)h(the)h(case)g(when)g(we)g(compare)f (subtrees)h(rooted)257 693 y(at)g(separating)f(pairs)h(in)f (triconnected)h(component)e(trees.)31 b(W)-8 b(e)24 b(mak)o(e)f(a)h (cross-comparison)257 814 y(of)h(the)g(children)f(and)h(store)g(the)f (counters)h FB(c)1804 829 y Fs(<)1850 814 y FC(;)17 b FB(c)1936 829 y Fz(=)1982 814 y FC(;)g FB(c)2068 829 y Fs(>)2140 814 y FF(for)25 b(their)f(order)h(pro\002le.)404 940 y(When)j(we)g(compare)h(biconnected)e(components)j FB(B)2269 955 y Fr(i)2319 940 y FF(and)i FB(B)2566 955 y Fr(j)2604 904 y Fw(0)2631 940 y FF(,)d(then)f(we)h(cannot)f(start)257 1060 y(comparing)j(their)g(biconnected)h(canons.)50 b(W)-8 b(e)32 b(e)n(v)o(en)f(cannot)h(compute)e(their)i(canons)f(be-)257 1181 y(cause)26 b(we)f(do)g(not)g(ha)n(v)o(e)g(a)g(unique)g(root)g (separating)g(pair)g(for)g(the)g(trees)h FD(T)p FF(\()s FB(B)2957 1196 y Fr(i)2979 1181 y FF(\))f(and)g FD(T)p FF(\()s FB(B)3371 1196 y Fr(j)3410 1144 y Fw(0)3437 1181 y FF(\).)257 1301 y(The)c(problem)f(occurs)g(when)g(we)h(ha)n(v)o(e)f (only)g(one)h(\002x)o(ed)f(v)o(erte)o(x)g(in)j FB(B)2654 1316 y Fr(i)2676 1301 y FF(,)e(i.e.)29 b(the)20 b(parent)h(artic-)257 1421 y(ulation)f(point.)28 b(Datta)21 b(et.)29 b(al.)g(bound)20 b(the)h(number)f(of)h(candidates)f(of)h(root)f(separating)h(pairs)257 1542 y(of)29 b FB(T)12 b FF(\()s FB(B)533 1557 y Fr(i)555 1542 y FF(\))29 b(and)g FB(T)12 b FF(\()s FB(B)964 1557 y Fr(j)1003 1506 y Fw(0)1030 1542 y FF(\).)42 b(F)o(or)29 b(the)f(detailed)g(case)h(analysis)f(we)h(refere)g(to)g(the)f(paper)-5 b(.)42 b(Basi-)257 1662 y(cally)-6 b(,)22 b(e)o(xcept)g(of)h(some)e (special)i(cases)f(the)o(y)g(sho)n(w)f(that)h(the)g(number)g(of)g (edges)g(is)g(bounded)257 1782 y(by)i FB(k)r FF(,)g(when)g(all)g(the)g (isomorphism)d(classes)j(of)g(the)g(children)f(of)k FB(B)2605 1797 y Fr(i)2651 1782 y FF(and)g FB(B)2893 1797 y Fr(j)2932 1746 y Fw(0)2983 1782 y FF(\(i.e.)k(children)257 1903 y(in)24 b(the)g(biconnected)f(component)g(tree)i(of)f(nodes)g(for)j FB(B)2219 1918 y Fr(i)2265 1903 y FF(and)g FB(B)2507 1918 y Fr(j)2546 1867 y Fw(0)2573 1903 y FF(\))d(are)h(of)f (cardinality)f Fq(>)28 b FB(k)r FF(.)257 2023 y(Hence,)36 b(all)c(the)h(isomorphism)e(classes)h(contain)h(children)d FB(C)38 b FF(such)33 b(that)f Fv(j)p FB(S)2991 2038 y Fr(C)3043 2023 y Fv(j)42 b Fq(6)h Fv(j)p FB(S)3298 2038 y Fr(B)3339 2048 y Fd(i)3361 2023 y Fv(j)p FC(=)p FB(k)r FF(.)257 2144 y(If)29 b(there)g(is)e(one)i(size)f(class)g(of)g (cardinality)g(one,)h(then)f(we)g(treat)g(this)g(child)f(separately)-6 b(.)41 b(If)257 2264 y(there)27 b(are)h(tw)o(o)e(or)h(more)f(such)g (size)h(classes,)g(then)f(we)h(e)n(v)o(en)f(get)g FB(O)p FF(\(1\))h(candidates)g(for)f(the)257 2384 y(root.)31 b(W)-8 b(e)25 b(will)f(need)h(this)f(in)g(the)h(comple)o(xity)d (analysis.)257 2583 y SDict begin H.S end 257 2583 a 257 2583 a SDict begin 14.5 H.A end 257 2583 a 257 2583 a SDict begin [ /View [/XYZ H.V] /Dest (section*.9) cvn H.B /DEST pdfmark end 257 2583 a 105 x Fp(Complexity)37 b(according)h(to)f(the)h (biconnected)i(component)e(tr)n(ee.)101 b FF(First,)40 b(when)d(we)257 2808 y(compare)28 b(articulation)e(points)g FB(a)h FF(and)g FB(a)1678 2772 y Fw(0)1732 2808 y FF(in)g(the)g (biconnected)g(component)f(tree,)i(we)g(ha)n(v)o(e)257 2928 y(a)40 b(similar)e(comple)o(xity)f(analysis)h(as)h(in)f(Lindell') -5 b(s)38 b(algorithm.)72 b(F)o(or)39 b(the)g(children)f(of)h FB(a)257 3049 y FF(and)25 b FB(a)476 3013 y Fw(0)503 3049 y FF(,)g(we)g(store)g FB(O)p FF(\(log)16 b FB(k)r FF(\))25 b(bits)f(for)h(isomorphism)d(classes)i(of)h(cardinality)f FB(k)30 b Fq(>)e FF(2.)404 3175 y(Second,)35 b(when)e(we)h(compare)f (biconnected)f(components)j FB(B)e FF(and)j FB(B)2916 3139 y Fw(0)2975 3175 y FF(in)d(the)g(bicon-)257 3295 y(nected)i(component)d(tree)j(then)f(a)g(typical)f(query)h(is)g(of)g (the)g(form)g(\()5 b FB(s)p FC(;)17 b FB(r)r FF(\),)36 b(where)j FB(s)34 b FF(is)f(the)257 3416 y(chosen)26 b(root)f(of)h FD(T)p FF(\()s FB(B)p FF(\))f(and)h FB(r)i FF(is)d(the)g(inde)o(x)g(of)h(the)f(edge)h(in)g(the)f(canon,)h(which)f (is)h(to)f(be)h(re-)257 3536 y(trie)n(v)o(ed.)i(If)22 b(there)f(are)g FB(k)i FF(choices)e(for)g FD(T)p FF(\()s FB(B)p FF(\))f(and)h FB(T)12 b FF(\()s FB(B)2112 3500 y Fw(0)2138 3536 y FF(\),)22 b(the)f(base)g(machine)f(c)o(ycles)g (through)257 3656 y(all)26 b(of)f(them)g(one)h(by)f(one,)g(k)o(eeping)g (track)h(of)g(the)f(minimum)e(canon.)33 b(This)25 b(tak)o(es)g FB(O)p FF(\(log)16 b FB(k)r FF(\))257 3777 y(space.)51 b(In)31 b(both)g(cases,)i(we)f(also)f(consider)f(lar)n(ge)i(children)f (\(i.e.)50 b(children)29 b FB(C)36 b FF(of)f FB(B)c FF(such)257 3897 y(that)26 b Fv(j)p FB(S)516 3912 y Fr(C)567 3897 y Fv(j)k FC(>)g Fv(j)p FB(S)797 3912 y Fr(B)841 3897 y Fv(j)p FC(=)p FF(2\))c(a)g(priori.)34 b(W)-8 b(e)26 b(summarize.)33 b(If)27 b(we)f(consider)g(recursi)n(v)o(ely)e(ho)n(w)h (man)o(y)257 4018 y(bits)30 b(we)h(store)f(for)h(the)g(roots)f(of)g (biconnected)g(components)g(then)g(we)h(get)f(the)h(recursion)257 4138 y(equation)24 b(for)h(the)g(size)g(function.)1196 4431 y Fv(S)p FF(\()r FB(N)6 b FF(\))29 b FE(=)f FF(max)1606 4492 y Fr(j)1712 4291 y Fi(\()1793 4431 y Fv(S)1861 4291 y Fi( )1912 4364 y FB(N)p 1909 4408 77 6 v 1909 4499 a(k)1963 4514 y Fr(j)1996 4291 y Fi(!)2056 4431 y FE(+)22 b FB(O)p FF(\(log)16 b FB(k)2444 4446 y Fr(j)2467 4431 y FF(\))2500 4291 y Fi(\))257 4724 y FF(where)26 b FB(k)580 4739 y Fr(j)630 4724 y Fq(>)i FF(2.)j(Hence,)25 b Fv(S)p FF(\()r FB(N)6 b FF(\))29 b FE(=)f FB(O)p FF(\(log)18 b FB(N)6 b FF(\).)257 4929 y SDict begin H.S end 257 4929 a 257 4929 a SDict begin 14.5 H.A end 257 4929 a 257 4929 a SDict begin [ /View [/XYZ H.V] /Dest (section*.10) cvn H.B /DEST pdfmark end 257 4929 a 98 x Fp(Complexity)44 b(according)i(to)e (the)i(triconnected)g(component)g(tr)n(ees.)101 b FF(W)-8 b(e)45 b(consider)257 5147 y(no)n(w)34 b(the)f(comparison)g(of)i (triconnected)e(component)g(trees)h FD(T)p FF(\()s FB(B)p FF(\))g(and)g FD(T)p FF(\()s FB(B)3030 5111 y Fw(0)3056 5147 y FF(\))h(of)f(bicon-)257 5268 y(nected)d(components)i FB(B)d FF(and)k FB(B)1411 5232 y Fw(0)1437 5268 y FF(.)48 b(In)31 b(the)g(comparison)f(of)g FD(T)p FF(\()s FB(B)p FF(\))h(and)g FD(T)p FF(\()s FB(B)2952 5232 y Fw(0)2978 5268 y FF(\),)h(we)f(still)f(go)257 5388 y(into)f(recursion)h(at)g (separating)f(pairs)g(and)h(when)g(we)g(reach)g(virtual)f(edges)h(in)g (canons)f(for)p Black Black eop end %%Page: 20 20 TeXDict begin 20 19 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 257 266 a SDict begin H.S end 257 266 a Black Black 257 266 a SDict begin H.R end 257 266 a 257 266 a SDict begin [ /View [/XYZ H.V] /Dest (page.20) cvn H.B /DEST pdfmark end 257 266 a Black Black Black Black 884 1451 a currentpoint currentpoint translate 0.80 0.80 scale neg exch neg exch translate 884 1451 a 3 setlinewidth np 2747.4 594.6 a 2865 829.2 li 2724 829.2 li 2888.4 594.6 li 2865 829.2 li st 3 setlinewidth np 1808.4 1299 a 1808.4 1369.2 li 1878.6 1392.6 li 1948.8 1369.2 li 1948.8 1299 li st 3 setlinewidth np 2136.6 594.6 a 2089.8 829.2 li 2277.6 594.6 li 2230.8 829.2 li 2136.6 618 li st 3 setlinewidth np 2982 1299 a 2982 1392.6 li 3123 1392.6 li 3123 1299 li st 3 setlinewidth np 2301 1299 a 2371.8 1392.6 li 2442 1299 li st 3 setlinewidth np 2748 594 a 2747.4 606 li st 3 setlinewidth np 2745 626 a 2744.4 638 li st 3 setlinewidth np 2741 658 a 2740.4 670 li st 3 setlinewidth np 2738 690 a 2737.4 702 li st 3 setlinewidth np 2735 722 a 2734.4 734 li st 3 setlinewidth np 2732 753 a 2731.4 765 li st 3 setlinewidth np 2729 785 a 2728.4 797 li st 3 setlinewidth np 2726 817 a 2725.4 829 li st 3 setlinewidth np 1996.2 1063.8 a 2136.6 1063.8 li st 1902 1110 a FB(a)p 3 setlinewidth np 1910 1080 54.6 90.0002 180 arc st 3 setlinewidth np 1910 1072 54.6 180 270.001 arc st 3 setlinewidth np 2223 1072 54.6 270.001 360.001 arc st 3 setlinewidth np 2223 1080 54.6 0 90.0002 arc st 3 setlinewidth np 1855.2 1079.4 a 1855.2 1071.6 li st 3 setlinewidth np 1909.8 1017 a 2223 1017 li st 3 setlinewidth np 2277.6 1071.6 a 2277.6 1079.4 li st 3 setlinewidth np 2223 1134.6 a 1909.8 1134.6 li st 0 setgray 3 setlinewidth np 2983 1064 9.6 0 360.001 arc fil 0 setgray np 2983 1064 9.6 0 360.001 arc st 3 setlinewidth np 2983 1064 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 2842 1064 9.6 0 360.001 arc fil 0 setgray np 2842 1064 9.6 0 360.001 arc st 3 setlinewidth np 2842 1064 9.6 0 360.001 arc st 3 setlinewidth np 2841.6 1063.8 a 2982 1063.8 li st 0 setgray 3 setlinewidth np 2724 829 9.6 0 360.001 arc fil 0 setgray np 2724 829 9.6 0 360.001 arc st 3 setlinewidth np 2724 829 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 1949 1299 9.6 0 360.001 arc fil 0 setgray np 1949 1299 9.6 0 360.001 arc st 3 setlinewidth np 1949 1299 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 1809 1299 9.6 0 360.001 arc fil 0 setgray np 1809 1299 9.6 0 360.001 arc st 3 setlinewidth np 1809 1299 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 3123 1299 9.6 0 360.001 arc fil 0 setgray np 3123 1299 9.6 0 360.001 arc st 3 setlinewidth np 3123 1299 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 2983 1299 9.6 0 360.001 arc fil 0 setgray np 2983 1299 9.6 0 360.001 arc st 3 setlinewidth np 2983 1299 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 2442 1299 9.6 0 360.001 arc fil 0 setgray np 2442 1299 9.6 0 360.001 arc st 3 setlinewidth np 2442 1299 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 2302 1299 9.6 0 360.001 arc fil 0 setgray np 2302 1299 9.6 0 360.001 arc st 3 setlinewidth np 2302 1299 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 2889 594 9.6 0 360.001 arc fil 0 setgray np 2889 594 9.6 0 360.001 arc st 3 setlinewidth np 2889 594 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 2748 594 9.6 0 360.001 arc fil 0 setgray np 2748 594 9.6 0 360.001 arc st 3 setlinewidth np 2748 594 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 2231 829 9.6 0 360.001 arc fil 0 setgray np 2231 829 9.6 0 360.001 arc st 3 setlinewidth np 2231 829 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 2090 829 9.6 0 360.001 arc fil 0 setgray np 2090 829 9.6 0 360.001 arc st 3 setlinewidth np 2090 829 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 2278 594 9.6 0 360.001 arc fil 0 setgray np 2278 594 9.6 0 360.001 arc st 3 setlinewidth np 2278 594 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 2137 594 9.6 0 360.001 arc fil 0 setgray np 2137 594 9.6 0 360.001 arc st 3 setlinewidth np 2137 594 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 1809 1369 9.6 0 360.001 arc fil 0 setgray np 1809 1369 9.6 0 360.001 arc st 3 setlinewidth np 1809 1369 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 1879 1393 9.6 0 360.001 arc fil 0 setgray np 1879 1393 9.6 0 360.001 arc st 3 setlinewidth np 1879 1393 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 1949 1369 9.6 0 360.001 arc fil 0 setgray np 1949 1369 9.6 0 360.001 arc st 3 setlinewidth np 1949 1369 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 2372 1393 9.6 0 360.001 arc fil 0 setgray np 2372 1393 9.6 0 360.001 arc st 3 setlinewidth np 2372 1393 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 2983 1393 9.6 0 360.001 arc fil 0 setgray np 2983 1393 9.6 0 360.001 arc st 3 setlinewidth np 2983 1393 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 3123 1393 9.6 0 360.001 arc fil 0 setgray np 3123 1393 9.6 0 360.001 arc st 3 setlinewidth np 3123 1393 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 2865 829 9.6 0 360.001 arc fil 0 setgray np 2865 829 9.6 0 360.001 arc st 3 setlinewidth np 2865 829 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 2182 717 9.6 0 360.001 arc fil 0 setgray np 2182 717 9.6 0 360.001 arc st 3 setlinewidth np 2182 717 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 2808 712 9.6 0 360.001 arc fil 0 setgray np 2808 712 9.6 0 360.001 arc st 3 setlinewidth np 2808 712 9.6 0 360.001 arc st 3 setlinewidth np 1148 902 49.8 0 360.001 arc st 3 setlinewidth np 1150.8 618 a 1150.8 852.6 li st 3 setlinewidth np 1127.4 946.8 a 1056.6 1063.8 li st 3 setlinewidth np 1056.6 1063.8 a 1009.8 1204.8 li 1103.4 1204.8 li closepath st 3 setlinewidth np 1244.4 1063.8 a 1197.6 1204.8 li 1291.2 1204.8 li closepath st 3 setlinewidth np 1174.2 946.8 a 1244.4 1063.8 li st 3 setlinewidth np 1088 563 54.6 90.0002 180 arc st 3 setlinewidth np 1088 485 54.6 180 270.001 arc st 3 setlinewidth np 1214 485 54.6 270.001 360.001 arc st 3 setlinewidth np 1214 563 54.6 0 90.0002 arc st 3 setlinewidth np 1033.2 562.8 a 1033.2 484.8 li st 3 setlinewidth np 1087.8 430.2 a 1213.2 430.2 li st 3 setlinewidth np 1267.8 484.8 a 1267.8 562.8 li st 3 setlinewidth np 1213.2 618 a 1087.8 618 li st 3 setlinewidth np 939 1346 a 967.2 1346 li st 3 setlinewidth np 996 1346 a 1024.2 1346 li st 3 setlinewidth np 1052 1346 a 1080.2 1346 li st 3 setlinewidth np 1109 1346 a 1137.2 1346 li st 3 setlinewidth np 1165 1346 a 1193.2 1346 li st 3 setlinewidth np 1221 1346 a 1249.2 1346 li st 3 setlinewidth np 1278 1346 a 1306.2 1346 li st 3 setlinewidth np 1334 1346 a 1362.2 1346 li st 3 setlinewidth np 1362 1346 a 1362 1317.8 li st 3 setlinewidth np 1362 1293 a 1362 1264.8 li st 3 setlinewidth np 1362 1239 a 1362 1210.8 li st 3 setlinewidth np 1362 1186 a 1362 1157.8 li st 3 setlinewidth np 1362 1133 a 1362 1104.8 li st 3 setlinewidth np 1362 1080 a 1362 1051.8 li st 3 setlinewidth np 1362 1027 a 1362 998.8 li st 3 setlinewidth np 1362 973 a 1362 944.8 li st 3 setlinewidth np 1362 920 a 1362 891.8 li st 3 setlinewidth np 1362 867 a 1362 838.8 li st 3 setlinewidth np 1362 814 a 1362 785.8 li st 3 setlinewidth np 1362 760 a 1362 731.8 li st 3 setlinewidth np 1362 707 a 1362 678.8 li st 3 setlinewidth np 1362 654 a 1362 625.8 li st 3 setlinewidth np 1362 601 a 1362 572.8 li st 3 setlinewidth np 1362 548 a 1362 519.8 li st 3 setlinewidth np 1362 494 a 1362 465.8 li st 3 setlinewidth np 1362 441 a 1362 412.8 li st 3 setlinewidth np 1362 388 a 1362 359.8 li st 3 setlinewidth np 1362 360 a 1334.4 360 li st 3 setlinewidth np 1306 360 a 1278.4 360 li st 3 setlinewidth np 1250 360 a 1222.4 360 li st 3 setlinewidth np 1193 360 a 1165.4 360 li st 3 setlinewidth np 1137 360 a 1109.4 360 li st 3 setlinewidth np 1081 360 a 1053.4 360 li st 3 setlinewidth np 1024 360 a 996.4 360 li st 3 setlinewidth np 968 360 a 940.4 360 li st 3 setlinewidth np 939 360 a 939 387.6 li st 3 setlinewidth np 939 413 a 939 440.6 li st 3 setlinewidth np 939 466 a 939 493.6 li st 3 setlinewidth np 939 519 a 939 546.6 li st 3 setlinewidth np 939 573 a 939 600.6 li st 3 setlinewidth np 939 626 a 939 653.6 li st 3 setlinewidth np 939 679 a 939 706.6 li st 3 setlinewidth np 939 732 a 939 759.6 li st 3 setlinewidth np 939 785 a 939 812.6 li st 3 setlinewidth np 939 839 a 939 866.6 li st 3 setlinewidth np 939 892 a 939 919.6 li st 3 setlinewidth np 939 945 a 939 972.6 li st 3 setlinewidth np 939 998 a 939 1025.6 li st 3 setlinewidth np 939 1052 a 939 1079.6 li st 3 setlinewidth np 939 1105 a 939 1132.6 li st 3 setlinewidth np 939 1158 a 939 1185.6 li st 3 setlinewidth np 939 1211 a 939 1238.6 li st 3 setlinewidth np 939 1264 a 939 1291.6 li st 3 setlinewidth np 939 1318 a 939 1345.6 li st 3 setlinewidth np 2615 821 54.6 90.0002 180 arc st 3 setlinewidth np 2615 602 54.6 180 270.001 arc st 3 setlinewidth np 3022 602 54.6 270.001 360.001 arc st 3 setlinewidth np 3022 821 54.6 0 90.0002 arc st 3 setlinewidth np 2559.6 821.4 a 2559.6 602.4 li st 3 setlinewidth np 2614.2 547.2 a 3021.6 547.2 li st 3 setlinewidth np 3076.2 602.4 a 3076.2 821.4 li st 3 setlinewidth np 3021.6 876 a 2614.2 876 li st 3 setlinewidth np 2333 352 54.6 90.0002 180 arc st 3 setlinewidth np 2333 344 54.6 180 270.001 arc st 3 setlinewidth np 2646 344 54.6 270.001 360.001 arc st 3 setlinewidth np 2646 352 54.6 0 90.0002 arc st 3 setlinewidth np 2277.6 351.6 a 2277.6 343.8 li st 3 setlinewidth np 2332.8 289.2 a 2645.4 289.2 li st 3 setlinewidth np 2700.6 343.8 a 2700.6 351.6 li st 3 setlinewidth np 2645.4 406.8 a 2332.8 406.8 li st 3 setlinewidth np 2418.6 406.8 a 2325 547.2 li st 3 setlinewidth np 2019.6 1134.6 a 1925.4 1251.6 li st 3 setlinewidth np 2583 406.8 a 2677.2 547.2 li st 3 setlinewidth np 2756 1080 54.6 90.0002 180 arc st 3 setlinewidth np 2756 1072 54.6 180 270.001 arc st 3 setlinewidth np 3069 1072 54.6 270.001 360.001 arc st 3 setlinewidth np 3069 1080 54.6 0 90.0002 arc st 3 setlinewidth np 2700.6 1079.4 a 2700.6 1071.6 li st 3 setlinewidth np 2755.2 1017 a 3068.4 1017 li st 3 setlinewidth np 3123 1071.6 a 3123 1079.4 li st 3 setlinewidth np 3068.4 1134.6 a 2755.2 1134.6 li st 3 setlinewidth np 2160.6 1134.6 a 2254.2 1251.6 li st 3 setlinewidth np 2192 1385 54.6 90.0002 180 arc st 3 setlinewidth np 2192 1307 54.6 180 270.001 arc st 3 setlinewidth np 2505 1307 54.6 270.001 360.001 arc st 3 setlinewidth np 2505 1385 54.6 0 90.0002 arc st 3 setlinewidth np 2136.6 1384.8 a 2136.6 1306.8 li st 3 setlinewidth np 2191.8 1251.6 a 2505 1251.6 li st 3 setlinewidth np 2559.6 1306.8 a 2559.6 1384.8 li st 3 setlinewidth np 2505 1440 a 2191.8 1440 li st 3 setlinewidth np 1699 1385 54.6 90.0002 180 arc st 3 setlinewidth np 1699 1307 54.6 180 270.001 arc st 3 setlinewidth np 2012 1307 54.6 270.001 360.001 arc st 3 setlinewidth np 2012 1385 54.6 0 90.0002 arc st 3 setlinewidth np 1644 1384.8 a 1644 1306.8 li st 3 setlinewidth np 1698.6 1251.6 a 2011.8 1251.6 li st 3 setlinewidth np 2066.4 1306.8 a 2066.4 1384.8 li st 3 setlinewidth np 2011.8 1440 a 1698.6 1440 li st 3 setlinewidth np 2958.6 1134.6 a 3052.8 1251.6 li st 3 setlinewidth np 2184 876 a 2089.8 1017 li st 3 setlinewidth np 2841.6 876 a 2935.2 1017 li st 3 setlinewidth np 2873 1385 54.6 90.0002 180 arc st 3 setlinewidth np 2873 1307 54.6 180 270.001 arc st 3 setlinewidth np 3186 1307 54.6 270.001 360.001 arc st 3 setlinewidth np 3186 1385 54.6 0 90.0002 arc st 3 setlinewidth np 2817.6 1384.8 a 2817.6 1306.8 li st 3 setlinewidth np 2872.8 1251.6 a 3186 1251.6 li st 3 setlinewidth np 3240.6 1306.8 a 3240.6 1384.8 li st 3 setlinewidth np 3186 1440 a 2872.8 1440 li st 3 setlinewidth np 2004 821 54.6 90.0002 180 arc st 3 setlinewidth np 2004 602 54.6 180 270.001 arc st 3 setlinewidth np 2411 602 54.6 270.001 360.001 arc st 3 setlinewidth np 2411 821 54.6 0 90.0002 arc st 3 setlinewidth np 1948.8 821.4 a 1948.8 602.4 li st 3 setlinewidth np 2004 547.2 a 2410.8 547.2 li st 3 setlinewidth np 2465.4 602.4 a 2465.4 821.4 li st 3 setlinewidth np 2410.8 876 a 2004 876 li st 3 setlinewidth np 2983 1299 a 2995.6 1299 li st 3 setlinewidth np 3025 1299 a 3037.6 1299 li st 3 setlinewidth np 3068 1299 a 3080.6 1299 li st 3 setlinewidth np 3111 1299 a 3123.6 1299 li st 3 setlinewidth np 2090 829 a 2102.6 829 li st 3 setlinewidth np 2133 829 a 2145.6 829 li st 3 setlinewidth np 2176 829 a 2188.6 829 li st 3 setlinewidth np 2218 829 a 2230.6 829 li st 3 setlinewidth np 2302 1299 a 2314.6 1299 li st 3 setlinewidth np 2344 1299 a 2356.6 1299 li st 3 setlinewidth np 2387 1299 a 2399.6 1299 li st 3 setlinewidth np 2430 1299 a 2442.6 1299 li st 3 setlinewidth np 2748 594 a 2760.6 594 li st 3 setlinewidth np 2790 594 a 2802.6 594 li st 3 setlinewidth np 2833 594 a 2845.6 594 li st 3 setlinewidth np 2876 594 a 2888.6 594 li st 3 setlinewidth np 1809 1299 a 1821.6 1299 li st 3 setlinewidth np 1851 1299 a 1863.6 1299 li st 3 setlinewidth np 1894 1299 a 1906.6 1299 li st 3 setlinewidth np 1937 1299 a 1949.6 1299 li st 3 setlinewidth np 963 829 a 991.8 829 li st 3 setlinewidth np 1021 829 a 1049.8 829 li st 3 setlinewidth np 1079 829 a 1107.8 829 li st 3 setlinewidth np 1137 829 a 1165.8 829 li st 3 setlinewidth np 1194 829 a 1222.8 829 li st 3 setlinewidth np 1252 829 a 1280.8 829 li st 3 setlinewidth np 1310 829 a 1338.8 829 li st 3 setlinewidth np 1339 829 a 1339 856 li st 3 setlinewidth np 1339 882 a 1339 909 li st 3 setlinewidth np 1339 934 a 1339 961 li st 3 setlinewidth np 1339 986 a 1339 1013 li st 3 setlinewidth np 1339 1038 a 1339 1065 li st 3 setlinewidth np 1339 1091 a 1339 1118 li st 3 setlinewidth np 1339 1143 a 1339 1170 li st 3 setlinewidth np 1339 1195 a 1339 1222 li st 3 setlinewidth np 1339 1248 a 1339 1275 li st 3 setlinewidth np 1339 1275 a 1310.8 1275 li st 3 setlinewidth np 1281 1275 a 1252.8 1275 li st 3 setlinewidth np 1223 1275 a 1194.8 1275 li st 3 setlinewidth np 1165 1275 a 1136.8 1275 li st 3 setlinewidth np 1108 1275 a 1079.8 1275 li st 3 setlinewidth np 1050 1275 a 1021.8 1275 li st 3 setlinewidth np 992 1275 a 963.8 1275 li st 3 setlinewidth np 963 1275 a 963 1247.4 li st 3 setlinewidth np 963 1223 a 963 1195.4 li st 3 setlinewidth np 963 1171 a 963 1143.4 li st 3 setlinewidth np 963 1119 a 963 1091.4 li st 3 setlinewidth np 963 1066 a 963 1038.4 li st 3 setlinewidth np 963 1014 a 963 986.4 li st 3 setlinewidth np 963 962 a 963 934.4 li st 3 setlinewidth np 963 909 a 963 881.4 li st 3 setlinewidth np 963 857 a 963 829.4 li st 939 298 a(S)1006 313 y Fr(B)1034 922 y FB(a)1107 548 y(B)2325 382 y(u)p 3 setlinewidth np 2137 594 a 2149.6 594 li st 3 setlinewidth np 2180 594 a 2192.6 594 li st 3 setlinewidth np 2223 594 a 2235.6 594 li st 3 setlinewidth np 2265 594 a 2277.6 594 li st 2654 641 a(u)3006 1110 y FC(w)-320 b FB(u)2889 1345 y(u)208 b FC(w)2912 640 y(v)2583 382 y(v)2302 640 y(v)2161 1110 y FB(b)1973 1345 y(b)443 b(b)2255 829 y(b)-285 b(a)2043 641 y(u)1834 571 y(A)2213 358 y(s)1715 1345 y(a)443 b(a)963 767 y(S)1028 782 y Fr(a)1691 295 y FD(T)p FF(\()s FB(B)p FF(\))p 0 setgray 3 setlinewidth np 1147 901 9.6 0 360.001 arc fil 0 setgray np 1147 901 9.6 0 360.001 arc st 3 setlinewidth np 1147 901 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 2560 336 9.6 0 360.001 arc fil 0 setgray np 2560 336 9.6 0 360.001 arc st 3 setlinewidth np 2560 336 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 2419 336 9.6 0 360.001 arc fil 0 setgray np 2419 336 9.6 0 360.001 arc st 3 setlinewidth np 2419 336 9.6 0 360.001 arc st 3 setlinewidth np 2418.6 336 a 2559.6 336 li st 0 setgray 3 setlinewidth np 2137 1064 9.6 0 360.001 arc fil 0 setgray np 2137 1064 9.6 0 360.001 arc st 3 setlinewidth np 2137 1064 9.6 0 360.001 arc st 0 setgray 3 setlinewidth np 1996 1064 9.6 0 360.001 arc fil 0 setgray np 1996 1064 9.6 0 360.001 arc st 3 setlinewidth np 1996 1064 9.6 0 360.001 arc st 2630 852 a FC(w)884 1451 y currentpoint currentpoint translate 1 0.80 div 1 0.80 div scale neg exch neg exch translate 884 1451 a 257 1763 a FF(Figure)30 b(6:)663 1763 y SDict begin H.S end 663 1763 a Black Black 663 1763 a SDict begin H.R end 663 1763 a 663 1763 a SDict begin [ /View [/XYZ H.V] /Dest (figure.6) cvn H.B /DEST pdfmark end 663 1763 a FF(A)f(biconnected)g(component)e(tree)j FB(S)1992 1778 y Fr(B)2065 1763 y FF(rooted)f(at)g(biconnected)f (component)j FB(B)257 1883 y FF(which)c(has)g(an)g(articulation)f (point)g FB(a)g FF(as)h(child,)g(which)g(occurs)g(in)f(the)h (triconnected)f(com-)257 2003 y(ponent)34 b(tree)g FD(T)p FF(\()s FB(B)p FF(\))g(of)j FB(B)p FF(.)58 b(In)37 b FB(A)d FF(and)g(the)g(other)g(triconnected)g(components)f(the)h(dashed) 257 2124 y(edges)25 b(are)h(separating)e(pairs.)p Black 257 2434 a(triconnected)30 b(components.)44 b(What)30 b(is)f(ne)n(w)-6 b(,)31 b(we)f(go)f(into)g(recursion)h(when)g(we)g (reach)h(ar)n(-)257 2555 y(ticulation)24 b(points.)29 b(F)o(or)c(an)g(e)o(xample,)f(see)h(Figure)2062 2556 y SDict begin H.S end 2062 2556 a Black -1 x FF(6)p Black 2112 2487 a SDict begin H.R end 2112 2487 a 2112 2555 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (figure.6) cvn H.B /ANN pdfmark end 2112 2555 a FF(.)404 2687 y(If)e(an)g(articulation)f (point)g FB(a)h FF(belongs)f(to)g(man)o(y)g(separating)h(pairs,)g(then) g(it)f(can)h(occur)h(in)257 2807 y(man)o(y)g(component)f(nodes)g(in)h FD(T)p FF(\()s FB(B)p FF(\).)30 b(Recall,)25 b(that)f(we)g(ha)n(v)o(e)g (a)g(root)g(for)h(the)f(tree.)31 b(So,)24 b(there)257 2927 y(e)o(xists)g(a)i(unique)f(component)i FB(A)f FF(that)f(is)g (closest)g(to)g(the)g(root,)g(where)h FB(a)g FF(is)f(contained.)32 b(Ob-)257 3048 y(serv)o(e,)24 b(that)f(the)g(set)g(of)g(component)f (nodes)h(where)h FB(a)f FF(is)g(contained)g(is)g(al)o(w)o(ays)g(a)g (connected)257 3168 y(subtree)28 b(in)g FD(T)p FF(\()s FB(B)p FF(\).)41 b(The)28 b(authors)f(sho)n(w)-6 b(,)28 b(that)g(this)f(unique)g(component)g(can)i(be)f(computed)257 3289 y(in)k(log-space)g(and)h(that)f(the)g(\002rst)g(position)f(where)h FB(a)h FF(occurs)f(in)g(the)g(canon)g(of)j FB(A)e FF(can)f(be)257 3409 y(found)g(in)g(log-space.)53 b(Exactly)32 b(there,)i(we)f(go)f (for)g FB(a)h FF(into)e(recursion.)53 b(F)o(or)32 b(all)g(the)g(other) 257 3529 y(occurrences)i(of)f FB(a)f FF(we)h(do)g(not)f(go)h(into)e (recursion.)55 b(Call)33 b(this)e(the)i FB(r)l(efer)l(ence)h(copy)f FF(of)f FB(a)257 3650 y FF(in)25 b FD(T)p FF(\()s FB(B)p FF(\).)404 3782 y(Assume)33 b(we)h(store)f(separately)h(the)g(bits)f (that)g(we)h(need)g(inside)f FD(T)p FF(\()s FB(B)p FF(\))h(for)g(all)g (bicon-)257 3902 y(nected)e(components)h FB(B)p FF(.)51 b(F)o(or)31 b(this)g(part)h(also)f(a)h(log-space)f(bound)g(can)h(be)g (pro)o(v)o(ed.)49 b(The)257 4023 y(size)31 b(function)e(can)i (therefore)g(be)f(re\002ned.)48 b(Let)28 b FB(C)35 b FF(be)30 b(a)h(node)f(in)g FD(T)p FF(\()s FB(B)p FF(\).)47 b(The)30 b(size)g(of)g(the)257 4143 y(subtree)21 b FB(S)629 4158 y Fr(C)702 4143 y FF(rooted)g(at)g(some)f(node)f FB(C)26 b FF(is)20 b(the)h(sum)f(of)h(the)g(size)g(of)g(the)g (triconnected)g(subtree)257 4263 y(rooted)k(at)f FB(C)30 b FF(in)25 b FD(T)p FF(\()s FB(B)p FF(\),)g(say)g Fv(j)p FB(S)1311 4278 y Fr(C)1363 4263 y Fv(j)g FF(plus)g(the)g(size)g(of)h (all)f(the)g(biconnected)g(subtrees)g Fv(j)p FB(S)3252 4278 y Fr(a)3289 4263 y Fv(j)p FF(,)g(if)g FB(a)257 4384 y FF(is)e(a)g(reference)i(cop)o(y)d(of)h(an)g(articulation)f(points)g (in)g FB(S)2153 4399 y Fr(C)2205 4384 y FF(.)30 b(There)23 b(is)g(one)f(more)h(special)g(case.)257 4504 y(If)34 b FB(S)422 4519 y Fr(a)492 4504 y FF(is)f(a)g(lar)n(ge)h(child)e(for)37 b FB(B)32 b FF(in)h(the)g(biconnected)f(component)g(tree)i(and)f(for)e FB(C)38 b FF(in)33 b FB(S)3419 4519 y Fr(C)3470 4504 y FF(,)257 4624 y(then)23 b(we)g(still)e(go)h(only)g(once)h(into)f (recursion)g(for)h FB(S)2071 4639 y Fr(a)2131 4624 y FF(a)g(priori)f(and)g(store)h(the)f(result.)30 b(In)22 b(this)257 4745 y(case,)34 b(if)d FB(a)h FF(has)f(a)h(reference)h(cop)o (y)f(in)f(the)g(subtree)g(of)h FD(T)p FF(\()s FB(B)p FF(\))f(rooted)g(at)f FB(C)36 b FF(then)31 b FB(S)3201 4760 y Fr(a)3270 4745 y FF(is)g(not)257 4865 y(included)c(in)g(the)g (size)h(of)g FB(S)1239 4880 y Fr(C)1290 4865 y FF(.)39 b(Hence,)29 b(we)e(get)h(the)f(same)g(recursion)h(equation)e(as)i (before.)257 4986 y(This)c(\002nishes)h(the)f(comple)o(xity)f(analysis) h(and)h(leads)f(to)h(the)g(follo)n(wing)d(theorem.)257 5007 y SDict begin H.S end 257 5007 a 257 5007 a SDict begin 14.5 H.A end 257 5007 a 257 5007 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.4.2) cvn H.B /DEST pdfmark end 257 5007 a Black 240 x Fp(Theor)n(em)41 b(4.2.)p Black 49 w FB([)920 5248 y SDict begin H.S end 920 5248 a Black -1 x FB(6)p Black 970 5177 a SDict begin H.R end 970 5177 a 970 5247 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.DLNTW08) cvn H.B /ANN pdfmark end 970 5247 a FB(])e(The)h(isomorphism)d (or)l(der)j(between)f(two)h(planar)e(gr)o(aphs)g(can)h(be)257 5368 y(computed)25 b(in)f(lo)o(g-space)o(.)p Black Black eop end %%Page: 21 21 TeXDict begin 21 20 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 257 266 a SDict begin H.S end 257 266 a Black Black 257 266 a SDict begin H.R end 257 266 a 257 266 a SDict begin [ /View [/XYZ H.V] /Dest (page.21) cvn H.B /DEST pdfmark end 257 266 a Black 257 473 a SDict begin H.S end 257 473 a 257 473 a SDict begin 14.5 H.A end 257 473 a 257 473 a SDict begin [ /View [/XYZ H.V] /Dest (section*.11) cvn H.B /DEST pdfmark end 257 473 a 100 x Fp(The)32 b(canon.)100 b FF(The)31 b(canonization)f(of)g(planar)h(graphs)f(proceeds)h(e)o(xactly)f(as)g (in)g(the)h(case)257 693 y(of)j(biconnected)f(planar)h(graphs.)57 b(A)34 b(log-space)g(procedure)g(tra)n(v)o(erses)f(the)h(biconnected) 257 814 y(component)i(tree,)41 b(mak)o(es)c(oracle)g(queries)g(to)g (the)g(isomorphism)d(order)k(algorithm)d(and)257 934 y(outputs)f(a)i(canonical)f(list)f(of)h(edges,)j(along)d(with)f (delimiters)g(to)h(separate)h(the)f(lists)f(for)257 1054 y(siblings.)e(A)26 b(log-space)g(transducer)f(then)h(renames)g(the)f(v) o(ertices)h(according)f(to)h(their)f(\002rst)257 1175 y(occurrence)k(in)e(this)g(list,)g(to)g(get)g(the)h(\002nal)g(canon)f (for)h(the)g(biconnected)f(component)f(tree.)257 1295 y(This)19 b(canon)f(depends)h(upon)f(the)h(choice)g(of)g(the)g(root)f (of)h(the)g(biconnected)f(component)g(tree.)257 1416 y(Further)29 b(log-space)f(transducers)g(c)o(ycle)g(through)g(all)g (the)g(articulation)f(points)g(as)h(roots)g(to)257 1536 y(\002nd)d(the)f(minimum)e(canon)i(among)g(them,)f(then)h(rename)h(the) f(v)o(ertices)g(according)g(to)g(their)257 1656 y(\002rst)29 b(occurrence)h(in)e(the)g(canon)h(and)f(\002nally)-6 b(,)29 b(remo)o(v)o(e)e(the)i(virtual)f(edges)g(and)h(delimiters)257 1777 y(to)c(obtain)f(a)h(canon)g(for)g(the)f(planar)h(graph.)31 b(This)24 b(pro)o(v)o(es)g(the)g(main)g(theorem.)257 1798 y SDict begin H.S end 257 1798 a 257 1798 a SDict begin 14.5 H.A end 257 1798 a 257 1798 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.4.3) cvn H.B /DEST pdfmark end 257 1798 a Black 189 x Fp(Theor)n(em)j(4.3.)p Black 41 w FB([)898 1988 y SDict begin H.S end 898 1988 a Black -1 x FB(6)p Black 947 1917 a SDict begin H.R end 947 1917 a 947 1987 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.DLNTW08) cvn H.B /ANN pdfmark end 947 1987 a FB(])e(A)g(planar)e(gr)o(aph)h (can)h(be)g(canonized)e(in)i(lo)o(g-space)o(.)257 2322 y FG(Refer)m(ences)257 2419 y SDict begin H.S end 257 2419 a 257 2419 a SDict begin 13.6 H.A end 257 2419 a 257 2419 a SDict begin [ /View [/XYZ H.V] /Dest (section*.12) cvn H.B /DEST pdfmark end 257 2419 a 257 2381 a SDict begin H.S end 257 2381 a 257 2381 a SDict begin 13.6 H.A end 257 2381 a 257 2381 a SDict begin [ /View [/XYZ H.V] /Dest (cite.ADK08) cvn H.B /DEST pdfmark end 257 2381 a Black 581 2532 a FK([1])p Black 50 w(V)-5 b(ikraman)33 b(Arvind,)j(Biresw)o(ar)d(Das,) i(and)e(Johannes)j(K\366bler)-5 b(.)64 b(A)31 b(logspace)k(algo-)736 2645 y(rithm)25 b(for)f(partial)i(2-tree)f(canonization.)39 b(In)25 b FI(CSR)e(2008:)31 b(Computer)25 b(Science)h(Sym-)736 2758 y(posium)f(in)e(Russia)p FK(,)h(pages)h(40\22651,)f(2008.)257 2780 y SDict begin H.S end 257 2780 a 257 2780 a SDict begin 13.6 H.A end 257 2780 a 257 2780 a SDict begin [ /View [/XYZ H.V] /Dest (cite.ADR05) cvn H.B /DEST pdfmark end 257 2780 a Black 581 2914 a FK([2])p Black 50 w(Eric)e(Allender)l(,)h (Samir)e(Datta,)h(and)h(Samb)n(uddha)g(Ro)o(y)-6 b(.)30 b(The)21 b(directed)j(planar)f(reach-)736 3027 y(ability)32 b(problem.)55 b(In)30 b FI(Pr)l(oc.)g(25th)g(annual)i(Confer)m(ence)g (on)e(F)-10 b(oundations)34 b(of)29 b(Soft-)736 3140 y(war)m(e)c(T)-8 b(ec)o(hnolo)o(gy)27 b(and)e(Theor)m(etical)i (Computer)f(Science)h(\(FSTTCS\))p FK(,)c(pages)j(238\226)736 3253 y(249.,)e(2005.)257 3268 y SDict begin H.S end 257 3268 a 257 3268 a SDict begin 13.6 H.A end 257 3268 a 257 3268 a SDict begin [ /View [/XYZ H.V] /Dest (cite.AM00) cvn H.B /DEST pdfmark end 257 3268 a Black 581 3409 a FK([3])p Black 50 w(Eric)36 b(Allender)i(and)e(Meena)h(Mahajan.)75 b(The)36 b(comple)o(xity)i(of)e(planarity)j(testing.)736 3522 y(In)30 b FI(ST)-5 b(A)m(CS)30 b('00:)42 b(Pr)l(oceedings)33 b(of)d(the)g(17th)h(Annual)g(Symposium)g(on)f(Theor)m(etical)736 3634 y(Aspects)25 b(of)e(Computer)i(Science)p FK(,)g(pages)f(87\22698,) h(2000.)257 3657 y SDict begin H.S end 257 3657 a 257 3657 a SDict begin 13.6 H.A end 257 3657 a 257 3657 a SDict begin [ /View [/XYZ H.V] /Dest (cite.BTV07) cvn H.B /DEST pdfmark end 257 3657 a Black 581 3791 a FK([4])p Black 50 w(Chris)g(Bourk)o(e,)h(Raghunath)g(T)-6 b(e)n(w)o(ari,)24 b(and)h(N)f(V)f(V)-5 b(inodchandran.)40 b(Directed)26 b(planar)736 3904 y(reachability)38 b(is)d(in)f(unambiguous)k (logspace.)71 b(In)34 b FI(to)h(appear)h(in)e(Pr)l(oceedings)j(of)736 4016 y(IEEE)22 b(Confer)m(ence)j(on)f(Computational)i(Comple)n(xity)f (CCC)p FK(,)d(pages)i(\226,)f(2007.)257 4039 y SDict begin H.S end 257 4039 a 257 4039 a SDict begin 13.6 H.A end 257 4039 a 257 4039 a SDict begin [ /View [/XYZ H.V] /Dest (cite.DLN08) cvn H.B /DEST pdfmark end 257 4039 a Black 581 4173 a FK([5])p Black 50 w(Samir)k(Datta,)i(Nutan)f(Limaye,)h(and)f (Prajakta)h(Nimbhorkar)-5 b(.)48 b(3-connected)31 b(planar)736 4285 y(graph)d(isomorphism)g(is)e(in)g(log-space.)44 b(In)26 b FI(IARCS)f(Annual)i(Confer)m(ence)h(on)f(F)-10 b(oun-)736 4398 y(dations)22 b(of)f(Softwar)m(e)g(T)-8 b(ec)o(hnolo)o(gy)22 b(and)f(Theor)m(etical)h(Computer)f(Science)h (\(FSTTCS)736 4511 y(2008\))p FK(,)29 b(Dagstuhl,)g(German)o(y)-6 b(,)28 b(2008.)f(Schloss)h(Dagstuhl)g(-)f(Leibniz-Zentrum)i(fuer)736 4624 y(Informatik,)d(German)o(y)-6 b(.)257 4647 y SDict begin H.S end 257 4647 a 257 4647 a SDict begin 13.6 H.A end 257 4647 a 257 4647 a SDict begin [ /View [/XYZ H.V] /Dest (cite.DLNTW08) cvn H.B /DEST pdfmark end 257 4647 a Black 581 4780 a FK([6])p Black 50 w(Samir)34 b(Datta,)j(Nutan)d(Limaye,)j (Prajakta)e(Nimbhorkar)l(,)k(Thomas)34 b(Thierauf,)k(and)736 4893 y(F)o(abian)32 b(W)-7 b(agner)i(.)61 b(A)31 b(log-space)k (algorithm)e(for)f(canonization)k(of)c(planar)h(graphs.)736 5006 y(2008.)i(in)23 b(arXi)n(v:0809.2319.)257 5011 y SDict begin H.S end 257 5011 a 257 5011 a SDict begin 13.6 H.A end 257 5011 a 257 5011 a SDict begin [ /View [/XYZ H.V] /Dest (cite.GrVe06) cvn H.B /DEST pdfmark end 257 5011 a Black 581 5162 a FK([7])p Black 50 w(M.)h(Grohe)h(and)g(O.)e(V)-10 b(erbitsk)o(y)k(.)39 b(T)-6 b(esting)25 b(graph)h(isomorphism)g(in)f (parallel)i(by)d(play-)736 5275 y(ing)g(a)e(game.)32 b(In)23 b FI(33r)m(d)g(International)k(Colloquium)e(on)e(A)n(utomata,)h (Langua)o(g)o(es)h(and)736 5388 y(Pr)l(o)o(gr)o(amming)o(.)p FK(,)f(pages)h(3\22614,)f(2006.)p Black Black eop end %%Page: 22 22 TeXDict begin 22 21 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 257 266 a SDict begin H.S end 257 266 a Black Black 257 266 a SDict begin H.R end 257 266 a 257 266 a SDict begin [ /View [/XYZ H.V] /Dest (page.22) cvn H.B /DEST pdfmark end 257 266 a Black 257 473 a SDict begin H.S end 257 473 a 257 473 a SDict begin 13.6 H.A end 257 473 a 257 473 a SDict begin [ /View [/XYZ H.V] /Dest (cite.HoTa71) cvn H.B /DEST pdfmark end 257 473 a Black 581 573 a FK([8])p Black 50 w(J.)21 b(E.)f(Hopcroft)j(and)f(R.E.)e(T)-7 b(arjan.)29 b(A)20 b Fc(v)2001 540 y FA(2)2059 573 y FK(algorithm)j(for)f(determining)i (isomorphism)736 686 y(of)g(planar)h(graphs.)34 b FI(Information)27 b(Pr)l(ocessing)e(Letter)o(s)p FK(,)f(1\(1\):32\22634,)j(1971.)257 705 y SDict begin H.S end 257 705 a 257 705 a SDict begin 13.6 H.A end 257 705 a 257 705 a SDict begin [ /View [/XYZ H.V] /Dest (cite.HT73) cvn H.B /DEST pdfmark end 257 705 a Black 581 836 a FK([9])p Black 50 w(John)i(E.)d(Hopcroft)i(and)g (Robert)g(E.)e(T)-7 b(arjan.)46 b(Di)n(viding)29 b(a)e(graph)i(into)f (triconnected)736 949 y(components.)36 b FI(SIAM)23 b(J)n(ournal)i(on)f (Computing)p FK(,)g(2\(3\):135\226158,)k(1973.)257 969 y SDict begin H.S end 257 969 a 257 969 a SDict begin 13.6 H.A end 257 969 a 257 969 a SDict begin [ /View [/XYZ H.V] /Dest (cite.HoTa74) cvn H.B /DEST pdfmark end 257 969 a Black 536 1099 a FK([10])p Black 50 w(John)34 b(E.)d(Hopcroft)j(and)g(Robert) f(T)-7 b(arjan.)63 b(E)p FJ(\016)p FK(cient)34 b(planarity)h(testing.) 65 b FI(J)n(.)31 b(A)m(CM)p FK(,)736 1212 y(21\(4\):549\226568,)d (1974.)257 1228 y SDict begin H.S end 257 1228 a 257 1228 a SDict begin 13.6 H.A end 257 1228 a 257 1228 a SDict begin [ /View [/XYZ H.V] /Dest (cite.HW74) cvn H.B /DEST pdfmark end 257 1228 a Black 536 1362 a FK([11])p Black 50 w(John)23 b(E.)d(Hopcroft)j(and)f(J.)e(K.)h(W)-7 b(ong.)29 b(Linear)22 b(time)f(algorithm)i(for)f(isomorphism)h(of)736 1475 y(planar)31 b(graphs)f(\(preliminary)h(report\).)52 b(In)29 b FI(ST)n(OC)f('74:)40 b(Pr)l(oceedings)32 b(of)c(the)h(sixth) 736 1588 y(annual)d(A)m(CM)21 b(symposium)k(on)f(Theory)g(of)g (computing)p FK(,)h(pages)g(172\226184,)g(1974.)257 1608 y SDict begin H.S end 257 1608 a 257 1608 a SDict begin 13.6 H.A end 257 1608 a 257 1608 a SDict begin [ /View [/XYZ H.V] /Dest (cite.JKMT03) cvn H.B /DEST pdfmark end 257 1608 a Black 536 1738 a FK([12])p Black 50 w(Bir)n(git)i(Jenner)l(,)i (Johannes)g(K\366bler)l(,)e(Pierre)g(McK)n(enzie,)g(and)g(Jacobo)h(T)-7 b(or\341n.)42 b(Com-)736 1851 y(pleteness)32 b(results)e(for)f(graph)h (isomorphism.)53 b FI(J)n(.)28 b(Comput.)h(Syst.)g(Sci.)p FK(,)h(66\(3\):549\226)736 1964 y(566,)24 b(2003.)257 1977 y SDict begin H.S end 257 1977 a 257 1977 a SDict begin 13.6 H.A end 257 1977 a 257 1977 a SDict begin [ /View [/XYZ H.V] /Dest (cite.KHC04) cvn H.B /DEST pdfmark end 257 1977 a Black 536 2115 a FK([13])p Black 50 w(Jacek)32 b(P)-10 b(.)28 b(K)o(ukluk,)k(La)o(wrence)f(B.)e(Holder)l(,)j(and)f (Diane)f(J.)g(Cook.)55 b(Algorithm)31 b(and)736 2227 y(e)o(xperiments)c(in)e(testing)h(planar)h(graphs)f(for)f(isomorphism.) 39 b FI(J)n(.)24 b(Gr)o(aph)h(Algorithms)736 2340 y(Appl.)p FK(,)e(8\(2\):313\226356,)28 b(2004.)257 2359 y SDict begin H.S end 257 2359 a 257 2359 a SDict begin 13.6 H.A end 257 2359 a 257 2359 a SDict begin [ /View [/XYZ H.V] /Dest (cite.Lin92) cvn H.B /DEST pdfmark end 257 2359 a Black 536 2491 a FK([14])p Black 50 w(Ste)n(v)o(en)33 b(Lindell.)62 b(A)31 b(logspace)j(algorithm)g(for)f(tree)g(canonization)j(\(e)o (xtended)e(ab-)736 2604 y(stract\).)f(In)23 b FI(ST)n(OC)f('92:)29 b(Pr)l(oceedings)c(of)e(the)g(twenty-fourth)j(annual)f(A)m(CM)c(sympo-) 736 2716 y(sium)j(on)f(Theory)i(of)e(computing)p FK(,)i(pages)g (400\226404,)h(1992.)257 2736 y SDict begin H.S end 257 2736 a 257 2736 a SDict begin 13.6 H.A end 257 2736 a 257 2736 a SDict begin [ /View [/XYZ H.V] /Dest (cite.M37) cvn H.B /DEST pdfmark end 257 2736 a Black 536 2867 a FK([15])p Black 50 w(Saunders)39 b(Maclane.)77 b(A)36 b(structural)j (characterization)j(of)37 b(planar)h(combinatorial)736 2980 y(graphs.)d FI(Duk)o(e)24 b(Mathematical)h(J)n(ournal)p FK(,)g(3:460\226472,)i(1937.)257 2999 y SDict begin H.S end 257 2999 a 257 2999 a SDict begin 13.6 H.A end 257 2999 a 257 2999 a SDict begin [ /View [/XYZ H.V] /Dest (cite.MR91) cvn H.B /DEST pdfmark end 257 2999 a Black 536 3130 a FK([16])p Black 50 w(Gary)32 b(L.)f(Miller)i(and)f(John)i(H.)c(Reif.) 61 b(P)o(arallel)33 b(tree)f(contraction)k(part)d(2:)46 b(further)736 3243 y(applications.)37 b FI(SIAM)23 b(J)n(.)f(Comput.)p FK(,)h(20\(6\):1128\2261147)q(,)28 b(1991.)257 3263 y SDict begin H.S end 257 3263 a 257 3263 a SDict begin 13.6 H.A end 257 3263 a 257 3263 a SDict begin [ /View [/XYZ H.V] /Dest (cite.RA00) cvn H.B /DEST pdfmark end 257 3263 a Black 536 3393 a FK([17])p Black 50 w(Klaus)e(Reinhardt)g(and)g(Eric) f(Allender)-5 b(.)39 b(Making)26 b(nondeterminism)j(unambiguous.)736 3506 y FI(SIAM)23 b(J)n(ournal)i(on)f(Computing)p FK(,)g (29\(4\):1118\2261131)q(,)k(2000.)257 3525 y SDict begin H.S end 257 3525 a 257 3525 a SDict begin 13.6 H.A end 257 3525 a 257 3525 a SDict begin [ /View [/XYZ H.V] /Dest (cite.Rei05) cvn H.B /DEST pdfmark end 257 3525 a Black 536 3656 a FK([18])p Black 50 w(Omer)20 b(Reingold.)30 b(Undirected)23 b(st-connecti)n(vity)i(in)c(log-space.)30 b(In)21 b FI(ST)n(OC)e('05:) 29 b(Pr)l(o-)736 3769 y(ceedings)23 b(of)e(the)f(thirty-se)o(venth)25 b(annual)d(A)m(CM)d(symposium)j(on)f(Theory)g(of)f(comput-)736 3882 y(ing)p FK(,)k(pages)h(376\226385,)g(2005.)257 3902 y SDict begin H.S end 257 3902 a 257 3902 a SDict begin 13.6 H.A end 257 3902 a 257 3902 a SDict begin [ /View [/XYZ H.V] /Dest (cite.TW08) cvn H.B /DEST pdfmark end 257 3902 a Black 536 4032 a FK([19])p Black 50 w(Thomas)g(Thierauf)h(and)e(F)o (abian)h(W)-7 b(agner)i(.)37 b(The)24 b(isomorphism)j(problem)e(for)g (planar)736 4145 y(3-connected)i(graphs)e(is)e(in)h(unambiguous)i (logspace.)36 b(In)23 b FI(ST)-5 b(A)m(CS)p FK(,)23 b(pages)h (633\226644,)736 4258 y(2008.)257 4259 y SDict begin H.S end 257 4259 a 257 4259 a SDict begin 13.6 H.A end 257 4259 a 257 4259 a SDict begin [ /View [/XYZ H.V] /Dest (cite.Ve07) cvn H.B /DEST pdfmark end 257 4259 a Black 536 4408 a FK([20])p Black 50 w(Ole)o(g)19 b(V)-10 b(erbitsk)o(y)k(.)24 b(Planar)19 b(graphs:)28 b(Logical)20 b(comple)o(xity)h(and)e(parallel) h(isomorphism)736 4521 y(tests.)34 b(In)24 b FI(ST)-5 b(A)m(CS)p FK(,)23 b(pages)h(682\226693,)i(2007.)257 4541 y SDict begin H.S end 257 4541 a 257 4541 a SDict begin 13.6 H.A end 257 4541 a 257 4541 a SDict begin [ /View [/XYZ H.V] /Dest (cite.Wei66) cvn H.B /DEST pdfmark end 257 4541 a Black 536 4672 a FK([21])p Black 50 w(H.)20 b(W)-7 b(einber)n(g.)31 b(A)20 b(simple)i(and)g(e)p FJ(\016)p FK(cient)h(algorithm)g(for)e(determining)j(isomorphism)736 4785 y(of)g(planar)h(triply)f(connected)j(graphs.)34 b FI(Cir)m(cuit)24 b(Theory)p FK(,)g(13:142148,)j(1966.)257 4804 y SDict begin H.S end 257 4804 a 257 4804 a SDict begin 13.6 H.A end 257 4804 a 257 4804 a SDict begin [ /View [/XYZ H.V] /Dest (cite.Whi33) cvn H.B /DEST pdfmark end 257 4804 a Black 536 4935 a FK([22])p Black 50 w(H.)22 b(Whitne)o(y)-6 b(.)33 b(A)22 b(set)h(of)g(topological)j(in)l(v)n(ariants)g(for)d (graphs.)33 b FI(American)24 b(J)n(ournal)g(of)736 5048 y(Mathematics)p FK(,)h(55:235\226321,)i(1933.)p Black Black eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF