From 1d5c163092871674da74b56ff08f9d7ad38dc746 Mon Sep 17 00:00:00 2001
From: Martin Mares <mj@ucw.cz>
Date: Mon, 17 Mar 2008 14:29:14 +0100
Subject: [PATCH] Remark on disconnected graphs.

---
 PLAN       |  4 ++--
 macros.tex |  1 +
 mst.tex    | 10 +++++++++-
 3 files changed, 12 insertions(+), 3 deletions(-)

diff --git a/PLAN b/PLAN
index 05c1bc1..2a6ae25 100644
--- a/PLAN
+++ b/PLAN
@@ -50,7 +50,7 @@ Spanning trees:
 - cite Eisner's tutorial \cite{eisner:tutorial}
 - \cite{pettie:onlineverify} online lower bound
 - mention matroids
-- mention disconnected graphs
+- move the remark on disconnected graphs? separate section?
 - Some algorithms (most notably Fredman-Tarjan) do not need flattening
 - reference to mixed Boruvka-Jarnik
 - use the notation for contraction by a set
@@ -58,7 +58,7 @@ Spanning trees:
 - parallel algorithms: p243-cole (are there others?)
 - bounded expansion classes?
 - restricted cases and arborescences
-- mention randomized algorithms (see remarks in Karger)
+- mention parallel algorithms (see remarks in Karger)
 
 Models:
 
diff --git a/macros.tex b/macros.tex
index c238860..2bf71d3 100644
--- a/macros.tex
+++ b/macros.tex
@@ -358,6 +358,7 @@
 \def\notan{\nota\labelx}
 \def\examplen{\example\labelx}
 \def\problemn{\problem\labelx}
+\def\remn{\rem\labelx}
 
 \def\paran#1{\para {\sl #1:}}
 
diff --git a/mst.tex b/mst.tex
index 9f212a6..421fbc5 100644
--- a/mst.tex
+++ b/mst.tex
@@ -59,7 +59,7 @@ In this section, we will examine the basic properties of spanning trees and prov
 several important theorems to base the algorithms upon. We will follow the theory
 developed by Tarjan in~\cite{tarjan:dsna}.
 
-For the whole section, we will fix a graph~$G$ with edge weights~$w$ and all
+For the whole section, we will fix a~connected graph~$G$ with edge weights~$w$ and all
 other graphs will be spanning subgraphs of~$G$. We will use the same notation
 for the subgraphs as for the corresponding sets of edges.
 
@@ -669,4 +669,12 @@ to finish on the remaining complete graph. Each iteration runs on a graph with $
 edges as every $H_{a,k}$ contains a complete graph on~$a$ vertices.
 \qed
 
+\remn{Disconnected graphs}
+The basic properties of minimum spanning trees and the algorithms presented in
+this chapter apply to minimum spanning forests of disconnected graphs, too.
+The proofs of our theorems and the steps of our algorithms are based on adjacency
+of vertices and existence of paths, so they are always local to a~single
+connected component. The Bor\o{u}vka's and Kruskal's algorithm need no changes,
+the Jarn\'\i{}k's algorithm has to be invoked separately for each component.
+
 \endpart
-- 
2.39.2