From 75dd5f6645a8436791094fec6aa2ea67dd36a8e5 Mon Sep 17 00:00:00 2001
From: Martin Mares <mj@ucw.cz>
Date: Sun, 20 Jan 2008 20:27:42 +0100
Subject: [PATCH] Intro to minor-closed classes.

---
 macros.tex |  1 +
 mst.tex    | 43 ++++++++++++++++++++++++++++++++++++-------
 2 files changed, 37 insertions(+), 7 deletions(-)

diff --git a/macros.tex b/macros.tex
index 52b30c5..ca99e44 100644
--- a/macros.tex
+++ b/macros.tex
@@ -302,6 +302,7 @@
 \def\impl{\proclaim{Implementation}}
 \def\cor{\proclaim{Corollary}}
 \def\nota{\proclaim{Notation}}
+\def\example{\proclaim{Example}}
 
 \def\label#1{{\sl (#1)\/}\enspace}
 
diff --git a/mst.tex b/mst.tex
index 9062e2e..848d61a 100644
--- a/mst.tex
+++ b/mst.tex
@@ -480,7 +480,7 @@ formulations and proofs, which we preferred to avoid. We also note that most of
 the algorithms can be run on disconnected multigraphs with little or no
 modifications.
 
-\algn{Contracting version of Bor\o{u}vka's algorithm}
+\algn{Contractive version of Bor\o{u}vka's algorithm}
 \algo
 \algin A~graph~$G$ with an edge comparison oracle.
 \:$T\=\emptyset$.
@@ -510,7 +510,7 @@ $\O(n+m)=\O(m)$.
 \qed
 
 \thm
-The Contracting Bor\o{u}vka's algorithm finds the MST in time $\O(m\log n)$.
+The Contractive Bor\o{u}vka's algorithm finds the MST in time $\O(m\log n)$.
 
 \proof
 As in the original Bor\o{u}vka's algorithm, the number of phases is $\O(\log n)$.
@@ -518,7 +518,7 @@ Then apply the previous lemma.
 \qed
 
 \thmn{\cite{mm:mst}}
-When the input graph is planar, the Contracting Bor\o{u}vka's algorithm runs in
+When the input graph is planar, the Contractive Bor\o{u}vka's algorithm runs in
 time $\O(m)$.
 
 \proof
@@ -646,11 +646,40 @@ edges as every $H_{a,k}$ contains a complete graph on~$a$ vertices.
 
 \section{Minor-closed graph classes}
 
-The contracting algorithm given in the previous section has been found to preform
-well on planar graphs. On the other hand, the bound $\O(m\log n)$ on general graphs
-is tight.
+The contracting algorithm given in the previous section has been found to perform
+well on planar graphs, but in the general case its time complexity was not linear.
+Can we find some broader class of graphs where the algorithm is still efficient?
+The right context turns out to be the minor-closed graph classes, which are
+closed on contractions and have bounded density.
+
+\defn
+A~graph~$H$ is a \df{minor} of a~graph~$G$ iff it can be obtained
+from a subgraph of~$G$ by a sequence of graph contractions (see \thmref{simpcont}).
+
+\defn
+A~class~$\cal C$ of graphs is \df{minor-closed}, when for every $G\in\cal C$ and
+its minor~$H$, the graph~$H$ lies in~$\cal C$ as well. A~class~$\cal C$ is called
+\df{non-trivial} if at least one graph lies in~$\cal C$ and at least one lies outside~$\cal C$.
+
+\example
+Non-trivial minor-closed classes include planar graphs and more generally graphs
+embeddable in any fixed surface. Many nice properties of planar graphs extend
+to these classes, too, most notable the linearity of the number of edges.
+
+\defn
+Let $\cal C$ be a class of graphs. We define its \df{edge density} $\varrho(\cal C)$
+to be the infimum of all~$\varrho$'s such that $\vert E(G) \vert \le \varrho\cdot\vert V(G)\vert$
+holds for every $G\in\cal C$.
+
+\thmn{Density of minor-closed classes}
+A~minor-closed class of graphs has finite edge density if and only if it is
+a non-trivial class.
+
+\proof
+See Theorem 6.1 in \cite{nesetril:minors}, which also lists some other equivalent conditions.
+\qed
+
 
-\cite{nesetril:minors}
 
 \section{Using Fibonacci heaps}
 \secid{fibonacci}
-- 
2.39.5