\s{Lemma:} Pokud existuje zpìtná hrana, kterou algoritmus nenakreslil, graf na~vstupu
není rovinný.
-\s{Dùkaz:} Rozborem pøípadù uká¾eme, ¾e kdykoliv existuje nenakreslená zpìtná hrana,
-pak algoritmus buïto poru¹il Pravidla \#1 a \#2 nebo v~zadaném grafu nalezneme
-minor $K_5$ èi $K_{3,3}$.
+\s{Dùkaz:} Pro spor pøedpokládejme, ¾e pøi zpracování vrcholu~$v$ existuje
+zpìtná hrana~$wv$, kterou algoritmus nenakreslil, èili ¾e pøístup z~$v$ k~$w$
+je v~obou smìrech blokován externì aktivními vrcholy. Rozborem pøípadù uká¾eme,
+¾e tato situace vede ke~sporu buïto s~pravidly \#1 a \#2 nebo s~rovinností grafu.
+
+Oznaème $B$ blok, ve~kterém le¾í na~obou stranách hranice nìjaké externì aktivní
+vrcholy $x$ a~$y$ a pod nimi je pøipojen nìjakou cestou vrchol~$w$. Takový blok
+musí urèitì existovat, proto¾e jinak by algoritmus v¹echny bloky na~cestì z~$v$ do~$w$
+popøeklápìl tak, aby se hrana $wv$ ve¹la. V~grafu se tedy musí vyskytovat jeden
+z~následujících minorù (do~vrcholu~$u$ jsme zkontrahovali celou nenakreslenou èást grafu):
+
+\bigskip
+\centerline{\epsfbox{minor1.eps}\qquad\epsfbox{minor2.eps}}
+\bigskip
+
+Minor~$M$ pøitom odpovídá situaci, kdy $v$ nele¾í v~bloku~$B$. Tento pøípad
+snadno vylouèíme, proto¾e $M$ je isomorfní s~grafem $K_{3,3}$. Víme tedy,
+¾e v~grafu se vyskytuje~$N$. Tento minor je ale rovinný, tak¾e musíme
+ukázat, ¾e vnitøek bloku brání nakreslení hrany~$vw$ dovnitø. V¾dy dojdeme
+k~nìkterému z~následujících nerovinných minorù ($N_1$ a¾ $N_3$ jsou
+isomorfní s~$K_{3,3}$ a $N_4$ s~$K_5$):
+
+\bigskip
+\centerline{\epsfbox{minor3.eps}\qquad\epsfbox{minor4.eps}\qquad\epsfbox{minor5.eps}\qquad\epsfbox{minor6.eps}}
+\bigskip
+
+\>Uva¾me, jak bude $B$ vypadat po~odebrání vrcholu~$v$ a hran z~nìj vedoucích:
+
+{\I a) pøestane být 2-souvislý:}
+
+{\I b) zùstane 2-souvislý:}
\todo{Rozebrat hou¹» pøípadù.}
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+
+
+%%EndResource
+%%BeginResource: font CMR7
+%!PS-AdobeFont-1.1: CMR7 1.0
+%%CreationDate: 1991 Aug 20 16:39:21
+
+% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
+
+11 dict begin
+/FontInfo 7 dict dup begin
+/version (1.0) readonly def
+/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
+/FullName (CMR7) readonly def
+/FamilyName (Computer Modern) readonly def
+/Weight (Medium) readonly def
+/ItalicAngle 0 def
+/isFixedPitch false def
+end readonly def
+/FontName /CMR7 def
+/PaintType 0 def
+/FontType 1 def
+/FontMatrix [0.001 0 0 0.001 0 0] readonly def
+/Encoding 256 array
+0 1 255 {1 index exch /.notdef put} for
+dup 161 /Gamma put
+dup 162 /Delta put
+dup 163 /Theta put
+dup 164 /Lambda put
+dup 165 /Xi put
+dup 166 /Pi put
+dup 167 /Sigma put
+dup 168 /Upsilon put
+dup 169 /Phi put
+dup 170 /Psi put
+dup 173 /Omega put
+dup 174 /ff put
+dup 175 /fi put
+dup 176 /fl put
+dup 177 /ffi put
+dup 178 /ffl put
+dup 179 /dotlessi put
+dup 180 /dotlessj put
+dup 181 /grave put
+dup 182 /acute put
+dup 183 /caron put
+dup 184 /breve put
+dup 185 /macron put
+dup 186 /ring put
+dup 187 /cedilla put
+dup 188 /germandbls put
+dup 189 /ae put
+dup 190 /oe put
+dup 191 /oslash put
+dup 192 /AE put
+dup 193 /OE put
+dup 194 /Oslash put
+dup 195 /suppress put
+dup 196 /dieresis put
+dup 0 /Gamma put
+dup 1 /Delta put
+dup 2 /Theta put
+dup 3 /Lambda put
+dup 4 /Xi put
+dup 5 /Pi put
+dup 6 /Sigma put
+dup 7 /Upsilon put
+dup 8 /Phi put
+dup 9 /Psi put
+dup 10 /Omega put
+dup 11 /ff put
+dup 12 /fi put
+dup 13 /fl put
+dup 14 /ffi put
+dup 15 /ffl put
+dup 16 /dotlessi put
+dup 17 /dotlessj put
+dup 18 /grave put
+dup 19 /acute put
+dup 20 /caron put
+dup 21 /breve put
+dup 22 /macron put
+dup 23 /ring put
+dup 24 /cedilla put
+dup 25 /germandbls put
+dup 26 /ae put
+dup 27 /oe put
+dup 28 /oslash put
+dup 29 /AE put
+dup 30 /OE put
+dup 31 /Oslash put
+dup 32 /suppress put
+dup 33 /exclam put
+dup 34 /quotedblright put
+dup 35 /numbersign put
+dup 36 /dollar put
+dup 37 /percent put
+dup 38 /ampersand put
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+dup 40 /parenleft put
+dup 41 /parenright put
+dup 42 /asterisk put
+dup 43 /plus put
+dup 44 /comma put
+dup 45 /hyphen put
+dup 46 /period put
+dup 47 /slash put
+dup 48 /zero put
+dup 49 /one put
+dup 50 /two put
+dup 51 /three put
+dup 52 /four put
+dup 53 /five put
+dup 54 /six put
+dup 55 /seven put
+dup 56 /eight put
+dup 57 /nine put
+dup 58 /colon put
+dup 59 /semicolon put
+dup 60 /exclamdown put
+dup 61 /equal put
+dup 62 /questiondown put
+dup 63 /question put
+dup 64 /at put
+dup 65 /A put
+dup 66 /B put
+dup 67 /C put
+dup 68 /D put
+dup 69 /E put
+dup 70 /F put
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+dup 72 /H put
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+dup 75 /K put
+dup 76 /L put
+dup 77 /M put
+dup 78 /N put
+dup 79 /O put
+dup 80 /P put
+dup 81 /Q put
+dup 82 /R put
+dup 83 /S put
+dup 84 /T put
+dup 85 /U put
+dup 86 /V put
+dup 87 /W put
+dup 88 /X put
+dup 89 /Y put
+dup 90 /Z put
+dup 91 /bracketleft put
+dup 92 /quotedblleft put
+dup 93 /bracketright put
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+%%EndResource
+%%BeginResource: font CMR7
+%!PS-AdobeFont-1.1: CMR7 1.0
+%%CreationDate: 1991 Aug 20 16:39:21
+
+% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
+
+11 dict begin
+/FontInfo 7 dict dup begin
+/version (1.0) readonly def
+/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
+/FullName (CMR7) readonly def
+/FamilyName (Computer Modern) readonly def
+/Weight (Medium) readonly def
+/ItalicAngle 0 def
+/isFixedPitch false def
+end readonly def
+/FontName /CMR7 def
+/PaintType 0 def
+/FontType 1 def
+/FontMatrix [0.001 0 0 0.001 0 0] readonly def
+/Encoding 256 array
+0 1 255 {1 index exch /.notdef put} for
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+dup 163 /Theta put
+dup 164 /Lambda put
+dup 165 /Xi put
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+dup 167 /Sigma put
+dup 168 /Upsilon put
+dup 169 /Phi put
+dup 170 /Psi put
+dup 173 /Omega put
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+dup 184 /breve put
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+dup 186 /ring put
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+dup 190 /oe put
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+dup 196 /dieresis put
+dup 0 /Gamma put
+dup 1 /Delta put
+dup 2 /Theta put
+dup 3 /Lambda put
+dup 4 /Xi put
+dup 5 /Pi put
+dup 6 /Sigma put
+dup 7 /Upsilon put
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+dup 9 /Psi put
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+dup 16 /dotlessi put
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+dup 18 /grave put
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+dup 30 /OE put
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+dup 33 /exclam put
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+dup 35 /numbersign put
+dup 36 /dollar put
+dup 37 /percent put
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+dup 61 /equal put
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+dup 89 /Y put
+dup 90 /Z put
+dup 91 /bracketleft put
+dup 92 /quotedblleft put
+dup 93 /bracketright put
+dup 94 /circumflex put
+dup 95 /dotaccent put
+dup 96 /quoteleft put
+dup 97 /a put
+dup 98 /b put
+dup 99 /c put
+dup 100 /d put
+dup 101 /e put
+dup 102 /f put
+dup 103 /g put
+dup 104 /h put
+dup 105 /i put
+dup 106 /j put
+dup 107 /k put
+dup 108 /l put
+dup 109 /m put
+dup 110 /n put
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+dup 113 /q put
+dup 114 /r put
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+dup 117 /u put
+dup 118 /v put
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+dup 120 /x put
+dup 121 /y put
+dup 122 /z put
+dup 123 /endash put
+dup 124 /emdash put
+dup 125 /hungarumlaut put
+dup 126 /tilde put
+dup 127 /dieresis put
+dup 128 /suppress put
+dup 160 /space put
+readonly def
+/FontBBox{-27 -250 1122 750}readonly def
+/UniqueID 5000790 def
+currentdict end
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+%%EndResource
+%%BeginResource: font CMR7
+%!PS-AdobeFont-1.1: CMR7 1.0
+%%CreationDate: 1991 Aug 20 16:39:21
+
+% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
+
+11 dict begin
+/FontInfo 7 dict dup begin
+/version (1.0) readonly def
+/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
+/FullName (CMR7) readonly def
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+/Weight (Medium) readonly def
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+/isFixedPitch false def
+end readonly def
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+dup 92 /quotedblleft put
+dup 93 /bracketright put
+dup 94 /circumflex put
+dup 95 /dotaccent put
+dup 96 /quoteleft put
+dup 97 /a put
+dup 98 /b put
+dup 99 /c put
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+dup 123 /endash put
+dup 124 /emdash put
+dup 125 /hungarumlaut put
+dup 126 /tilde put
+dup 127 /dieresis put
+dup 128 /suppress put
+dup 160 /space put
+readonly def
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+%%BeginResource: font CMR7
+%!PS-AdobeFont-1.1: CMR7 1.0
+%%CreationDate: 1991 Aug 20 16:39:21
+
+% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
+
+11 dict begin
+/FontInfo 7 dict dup begin
+/version (1.0) readonly def
+/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
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+end readonly def
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+dup 124 /emdash put
+dup 125 /hungarumlaut put
+dup 126 /tilde put
+dup 127 /dieresis put
+dup 128 /suppress put
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\ No newline at end of file
projects / ga.git / commitdiff