The Minimality Theorem has a nice name.

diff --git a/PLAN b/PLAN
index c768f978ed09a7724b89c2d631a7c3410c1c2ee5..3600b29997111ed4cf328c6678ef1357fd66a03f 100644 (file)
--- a/PLAN
+++ b/PLAN
@@ -63,7 +63,6 @@ Spanning trees:
 - use the notation for contraction by a set
 - mention bugs in Valeria's verification paper
 - more references on decision trees
-- rename theorem on Minimality by order
 - introduce Cut rule and Cycle rule earlier
 - Lemma: deletion of a non-MST edge does not alter the MST
 

diff --git a/adv.tex b/adv.tex
index 12c634677f84d6d312b61d2c355e3a4aeda097ec..b161b8af4a6aa1b86f792c23aa235b7fb77fa801 100644 (file)
--- a/adv.tex
+++ b/adv.tex
@@ -558,7 +558,7 @@ needed, saving the actual efficient implementation for later.
 
 \para
 To verify that a~spanning~$T$ is minimum, it is sufficient to check that all
-edges outside~$T$ are $T$-heavy (by Theorem \ref{mstthm}). In fact, we will be
+edges outside~$T$ are $T$-heavy (by the Minimality Theorem, \ref{mstthm}). In fact, we will be
 able to find all $T$-light edges efficiently. For each edge $uv\in E\setminus T$,
 we will find the heaviest edge of the tree path $T[u,v]$ and compare its weight
 to $w(uv)$. It is therefore sufficient to solve the following problem:
@@ -1259,7 +1259,7 @@ spanning tree problem which frequently arise in practice.
 When the edges are already sorted by their weights, we can use the Kruskal's
 algorithm to find the MST in time $\O(m\timesalpha(n))$ (Theorem \ref{kruskal}).
 We however can do better: As the minimality of a~spanning tree depends only on the
-order of weights and not on the actual values (Theorem \ref{mstthm}), we can
+order of weights and not on the actual values (The Minimality Theorem, \ref{mstthm}), we can
 renumber the weights to $1, \ldots, m$ and find the MST using the Fredman-Willard
 algorithm for integer weights. According to Theorem \ref{intmst} it runs in
 time $\O(m)$ on the Word-RAM.

diff --git a/dyn.tex b/dyn.tex
index 41a5e2de52da9cd1962aeec65e7fcb5dad9267c9..a2bb4cc5fab3a3fc889aa978f6bcf6099b9c4056 100644 (file)
--- a/dyn.tex
+++ b/dyn.tex
@@ -68,8 +68,8 @@ therefore also of~$F$), we have to add~$e$ to~$F$. Otherwise, one of the followi
 Either $e$~is $F$-heavy and so the forest~$F$ is also the MSF of the new graph. Or it is $F$-light
 and we have to modify~$F$ by exchanging the heaviest edge~$f$ of the path $F[e]$ with~$e$.
 
-Correctness of the former case follows immediately from the Theorem on Minimality by order
-(\ref{mstthm}), because any $F'$-light would be also $F$-light, which is impossible as $F$~was
+Correctness of the former case follows immediately from the Minimality Theorem (\ref{mstthm}),
+because any $F'$-light would be also $F$-light, which is impossible as $F$~was
 minimum. In the latter case, the edge~$f$ is not contained in~$F'$ because it is the heaviest
 on the cycle $F[e]+e$ (by the Red lemma, \ref{redlemma}). We can now use the Blue lemma
 (\ref{bluelemma}) to prove that it should be replaced with~$e$. Consider the tree~$T$

diff --git a/mst.tex b/mst.tex
index 820ba51afd3bd4be0aa95a636f6059983ba1095d..8d3f18a21e6f09daef274a8999a111dcaf85142e 100644 (file)
--- a/mst.tex
+++ b/mst.tex
@@ -149,7 +149,7 @@ $w(f)\ge w(e')$ due to the choice of $e'$ and all other edges on~$C$ are lighter
 than~$e'$ as $e'$ was not $T$-light.
 \qed
 
-\thmn{Minimality by order}\id{mstthm}%
+\thmn{Minimality of spanning trees}\id{mstthm}%
 A~spanning tree~$T$ is minimum iff there is no $T$-light edge.
 
 \proof
@@ -214,8 +214,8 @@ their monotone isomorphism. Then $\mst(G_2) = \pi[\mst(G_1)]$.
 The isomorphism~$\pi$ maps spanning trees onto spanning trees and it preserves
 the relation of covering. Since it is monotone, it preserves the property of
 being a light edge (an~edge $e\in E(G_1)$ is $T$-light $\Leftrightarrow$
-the edge $\pi[e]\in E(G_2)$ is~$f[T]$-light). Therefore by Theorem~\ref{mstthm}, $T$
-is the MST of~$G_1$ if and only if $\pi[T]$ is the MST of~$G_2$.
+the edge $\pi[e]\in E(G_2)$ is~$f[T]$-light). Therefore by the Minimality Theorem
+(\ref{mstthm}), $T$ is the MST of~$G_1$ if and only if $\pi[T]$ is the MST of~$G_2$.
 \qed
 
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