From: Martin Mares <mj@ucw.cz>
Date: Mon, 21 Apr 2008 15:37:05 +0000 (+0200)
Subject: Added Edge removal lemma.
X-Git-Tag: printed~47
X-Git-Url: http://mj.ucw.cz/gitweb/?a=commitdiff_plain;h=0d474aeaf519e8117e059bb14c269d0c2a67733d;p=saga.git

Added Edge removal lemma.
---

diff --git a/PLAN b/PLAN
index 6ee8b13..dfd8d68 100644
--- a/PLAN
+++ b/PLAN
@@ -59,10 +59,9 @@ Preface:
 Spanning trees:
 
 - cite Eisner's tutorial \cite{eisner:tutorial}
-- Some algorithms (most notably Fredman-Tarjan) do not need flattening
-* Lemma: deletion of a non-MST edge does not alter the MST
 
-Related:
+Applications:
+
 - K best trees
 - degree-restricted cases and arborescences
 - bounded expansion classes?
diff --git a/mst.tex b/mst.tex
index 0cdab6d..efc4d63 100644
--- a/mst.tex
+++ b/mst.tex
@@ -185,6 +185,18 @@ $T_1$ and $T_2$ must be identical.
 When $G$ is a graph with distinct edge weights, we will use $\mst(G)$ to denote
 its unique minimum spanning tree.
 
+The following trivial lemma will be often invaluable:
+
+\lemman{Edge removal}
+Let~$G$ be a~graph with distinct edge weights and $e$ any its edge
+which does not lie in~$\mst(G)$. Then $\mst(G-e) = \mst(G)$.
+
+\proof
+The tree $T=\mst(G)$ is also a~MST of~$G-e$, because every $T$-light
+edge in~$G-e$ is also $T$-light in~$G$. Then we apply the uniqueness of
+the MST of~$G-e$.
+\qed
+
 \paran{Comparison oracles}\id{edgeoracle}%
 To simplify the description of MST algorithms, we will assume that the weights
 of all edges are distinct and that instead of numeric weights we are given a~comparison oracle.