From e9863e67eb14173b285f61e0cacb96d8f5766210 Mon Sep 17 00:00:00 2001
From: Martin Mares <mj@ucw.cz>
Date: Wed, 16 Jan 2008 18:26:02 +0100
Subject: [PATCH] Added figures (borrowed from my GA book).

---
 macros.tex   |    9 +
 mst.tex      |   11 +-
 pic/mst1.eps | 3736 ++++++++++++++++++++++++++++++++++++++++++++++++++
 pic/mst1.vrr |   64 +
 pic/mst2.eps | 3602 ++++++++++++++++++++++++++++++++++++++++++++++++
 5 files changed, 7418 insertions(+), 4 deletions(-)
 create mode 100644 pic/mst1.eps
 create mode 100644 pic/mst1.vrr
 create mode 100644 pic/mst2.eps

diff --git a/macros.tex b/macros.tex
index 8af08f2..149b875 100644
--- a/macros.tex
+++ b/macros.tex
@@ -342,6 +342,15 @@
   \bibliography{biblio}
 }
 
+%%% Figures %%%
+
+\def\figure#1#2#3{\midinsert
+\medskip
+\centerline{\epsfxsize=#2\epsfbox{pic/#1}}
+\medskip
+\centerline{#3}
+\endinsert}
+
 %%% Stand-alone chapters %%%
 
 \def\endpart{
diff --git a/mst.tex b/mst.tex
index d6b3af9..91f51d0 100644
--- a/mst.tex
+++ b/mst.tex
@@ -86,6 +86,8 @@ connectivity and $T':=T-f+e$ is another spanning tree with weight $w(T')
 = w(T)-w(f)+w(e) < w(T)$. Hence $T$ could not have been minimal.
 \qed
 
+\figure{mst2.eps}{278pt}{An edge exchange as in the proof of Lemma~\thmref{lightlemma}}
+
 The converse of this lemma is also true and to prove it, we will once again use
 technique of transforming trees by \df{exchanges} of edges. In the proof of the
 lemma, we have made use of the fact that whenever we exchange an edge~$e$ of
@@ -93,7 +95,7 @@ a spanning tree for another edge~$f$ covered by~$e$, the result is again
 a spanning tree. In fact, it is possible to transform any spanning tree
 to any other spanning tree by a sequence of exchanges.
 
-\lemman{Exchange property for trees}
+\lemman{Exchange property for trees}\thmid{xchglemma}%
 Let $T$ and $T'$ be spanning trees of a common graph. Then there exists
 a sequence of edge exchanges which transforms $T$ to~$T'$. More formally,
 there exists a sequence of spanning trees $T=T_0,T_1,\ldots,T_k=T'$ such that
@@ -109,8 +111,9 @@ tree $T^*:=T-e+e'$ such that $d(T^*,T')=d(T,T')-2$ and we can apply the inductio
 hypothesis to $T^*$ and $T'$ to get the rest of the exchange sequence.
 \qed
 
-\lemman{Monotone exchanges}
-\thmid{monoex}
+\figure{mst1.eps}{295pt}{One step of the proof of Lemma~\thmref{xchglemma}}
+
+\lemman{Monotone exchanges}\thmid{monoxchg}%
 Let $T$ be a spanning tree such that there are no $T$-light edges and $T$
 be an arbitrary spanning tree. Then there exists a sequence of edge exchanges
 transforming $T$ to~$T'$ such that the weight does not increase in any step.
@@ -142,7 +145,7 @@ If~$T$ is minimal, then by Lemma~\thmref{lightlemma} there are no $T$-light
 edges.
 Conversely, when $T$ is a spanning tree without $T$-light edges
 and $T_{min}$ is an arbitrary minimal spanning tree, then according to the Monotone
-exchange lemma~\thmref{monoex} there exists a non-decreasing sequence
+exchange lemma~\thmref{monoxchg} there exists a non-decreasing sequence
 of exchanges transforming $T$ to $T_{min}$, so $w(T)\le w(T_{min})$
 and thus $T$~is also minimal.
 \qed
diff --git a/pic/mst1.eps b/pic/mst1.eps
new file mode 100644
index 0000000..b2cc5c9
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+++ b/pic/mst1.eps
@@ -0,0 +1,3736 @@
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+%%BeginResource: font CMSY7
+%!PS-AdobeFont-1.1: CMSY7 1.0
+%%CreationDate: 1991 Aug 15 07:21:52
+
+% 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 (CMSY7) readonly def
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+/Weight (Medium) readonly def
+/ItalicAngle -14.035 def
+/isFixedPitch false def
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+dup 122 /daggerdbl put
+dup 123 /paragraph put
+dup 124 /club put
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+%%CreationDate: 1992 Feb 19 19:54:52
+
+% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.
+
+11 dict begin
+/FontInfo 7 dict dup begin
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+/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
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+%%BeginResource: font CMSY7
+%!PS-AdobeFont-1.1: CMSY7 1.0
+%%CreationDate: 1991 Aug 15 07:21:52
+
+% 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 (CMSY7) readonly def
+/FamilyName (Computer Modern) readonly def
+/Weight (Medium) readonly def
+/ItalicAngle -14.035 def
+/isFixedPitch false def
+end readonly def
+/FontName /CMSY7 def
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+/FontMatrix [0.001 0 0 0.001 0 0] readonly def
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+dup 122 /daggerdbl put
+dup 123 /paragraph put
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+
+
+%%EndResource
+%%BeginResource: font CMR10
+%!PS-AdobeFont-1.1: CMR10 1.00B
+%%CreationDate: 1992 Feb 19 19:54:52
+
+% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.
+
+11 dict begin
+/FontInfo 7 dict dup begin
+/version (1.00B) readonly def
+/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
+/FullName (CMR10) readonly def
+/FamilyName (Computer Modern) readonly def
+/Weight (Medium) readonly def
+/ItalicAngle 0 def
+/isFixedPitch false def
+end readonly def
+/FontName /CMR10 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
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+showpage
+
+%%Trailer
+%%DocumentSuppliedResources: procset VrrEPSProcSet
+%%+ font CMMI10 CMSY7 CMR10
+end restore
+%%EOF
-- 
2.39.5