From bfb63b435b389ea817147e24bc33a887fa6e9ff8 Mon Sep 17 00:00:00 2001
From: Martin Mares <mj@ucw.cz>
Date: Tue, 16 Jan 2007 11:30:00 +0100
Subject: [PATCH] Dokoncen dukaz korektnosti rovinneho kresleni.

---
 11-planar/11-planar.tex |  42 ++++++--
 11-planar/minor1.eps    | 103 ++++++++++---------
 11-planar/minor1.vrr    |  22 ++--
 11-planar/minor2.eps    | 103 ++++++++++---------
 11-planar/minor2.vrr    |  18 ++--
 11-planar/minor3.eps    | 216 ++++++++++++++++++++++------------------
 11-planar/minor3.vrr    |  34 +++----
 11-planar/minor4.eps    | 159 +++++++++++++++--------------
 11-planar/minor4.vrr    |  48 ++++-----
 11-planar/minor5.eps    | 159 +++++++++++++++--------------
 11-planar/minor5.vrr    |  36 +++----
 11-planar/minor6.eps    | 173 +++++++++++++++++---------------
 11-planar/minor6.vrr    |  36 +++----
 13 files changed, 619 insertions(+), 530 deletions(-)

diff --git a/11-planar/11-planar.tex b/11-planar/11-planar.tex
index f44746c..71b2f4c 100644
--- a/11-planar/11-planar.tex
+++ b/11-planar/11-planar.tex
@@ -299,18 +299,18 @@ Ozna
 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):
+z~následujících minorù (do~vrcholu~$u$ jsme zkontrahovali celou dosud nenakreslenou èást grafu;
+vybarvená èást odpovídá vnitøku bloku; hranaté vrcholy jsou externì aktivní):
 
 \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~$K_5$):
+snadno vylouèíme, proto¾e $M$ je isomorfní s~grafem $K_{3,3}$. V~grafu se proto
+musí vyskytovat~$N$. Tento minor je ale rovinný, tak¾e musíme ukázat, ¾e vnitøek
+bloku brání nakreslení hrany~$vw$ dovnitø. V¾dy pak 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}}
@@ -318,11 +318,35 @@ isomorfn
 
 \>Uva¾me, jak bude $B$ vypadat po~odebrání vrcholu~$v$ a hran z~nìj vedoucích:
 
-{\I a) pøestane být 2-souvislý:}
+\numlist\nalpha
+\:{\I pøestane být 2-souvislý} -- tehdy se zamìøíme na~bloky le¾ící na~cestì~$xy$:
 
-{\I b) zùstane 2-souvislý:}
+\numlist\nparen
+\:{\I $w$ je artikulace} na této cestì -- BÚNO je taková artikulace v~DFS prohledána po~bloku obsahujícím~$x$,
+ale pøed~$y$. Tehdy nám jistì $x$ nezabránilo v~tom, abychom do~$w$ do¹li (mù¾e blokovat
+jenom jednu stranu hranice), tak¾e jsme se ve~$w$ museli rozhodnout, ¾e pøednostnì zpracujeme
+pokraèování cesty do~$y$ pøed hranou~$vw$, a~to je spor s~pravidlem~\#1.
 
-\todo{Rozebrat hou¹» pøípadù.}
+\:{\I $w$ je v~bloku pøipojeném pod takovou artikulací} -- aby se pravidlo~\#1 vydalo
+do~$y$ místo podøízených blokù, musí být alespoò jeden z~nich externì aktivní,
+tak¾e v~$G$ je minor~$N_1$.
+
+\:{\I $w$ je v~bloku na~cestì nebo pøipojen pod takový blok} -- opìt si v¹imneme, ¾e do~bloku jsme
+vstoupili mezi~$x$ a~$y$. Abychom se podle pravidla~\#2 rozhodli pro stranu, z~ní¾ nevede
+hrana~$vw$, musela na~druhé stranì být také hrana do~$v$, a~proto se v~grafu vyskytuje minor~$M$.
+\endlist
+
+\:{\I zùstane 2-souvislý} a vznikne z~nìj nìjaký blok~$B'$ -- tehdy rozebereme, jaké hrany vedou mezi $v$ a $B'$:
+
+\itemize\nparen
+\:{\I více ne¾ dvì hrany} -- minor~$N_2$.
+\:{\I alespoò jedna hrana na \uv{horní} cestu} (to jest na~tu, na~ni¾ nele¾í~$w$) -- minor~$N_3$.
+\:{\I dvì hrany do~$x,y$ nebo na \uv{dolní} cestu} -- a» u¾ jsme vstoupili na~hranici bloku~$B'$
+kteroukoliv hranou, pravidlo~\#2 nám øeklo, ¾e máme pokraèovat vrchem, co¾ je mo¾né jedinì tehdy,
+je-li na~spodní cestì je¹tì jeden externì aktivní vrchol, a~to dává minor~$N_4$.
+\endlist
+
+\endlist
 
 \qed
 
diff --git a/11-planar/minor1.eps b/11-planar/minor1.eps
index 7838d1b..426a87b 100644
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+++ b/11-planar/minor1.eps
@@ -1,7 +1,7 @@
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 %%Trailer
diff --git a/11-planar/minor1.vrr b/11-planar/minor1.vrr
index 9f2a15c..f180136 100644
--- a/11-planar/minor1.vrr
+++ b/11-planar/minor1.vrr
@@ -1,25 +1,25 @@
 
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diff --git a/11-planar/minor2.eps b/11-planar/minor2.eps
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-- 
2.39.2