Journal of Combinatorial Theory, Series B 76, 155169 (1999)
The Connectivities of Leaf Graphs of
2-Connected Graphs
Atsushi Kaneko
Department of Computer Science and Communication Engineering, Kogakuin University,
1-24-2 Nishi-Shinjuku, Shinjuku-ku, Tokyo 163-8677, Japan
E-mail: kanekoÄee.kogakuin.ac.jp
and
Kiyoshi Yoshimoto
Department of Mathematics, College of Science and Technology, Nihon University,
1-8 Kanda-Surugadai, Chiyoda-ku, Tokyo 101-8308, Japan
E-mail: yosimotoÄmath.cst.nihon-u.ac.jp
Received December 15, 1997
Given a connected graph G, denote by V the family of all the spanning trees of
G. Define an adjacency relation in V as follows: the spanning trees t and t$ are said
to be adjacent if for some vertex u # V, t&u is connected and coincides with t$&u.
The resultant graph G is called the leaf graph of G. The purpose of this paper is to
show that if G is 2-connected with minimal degree $, then G is (2$&2)-connected.
1999 Academic Press
1. INTRODUCTION
Let G=(V, E) be a connected graph. Let t be a spanning tree of G.If
a vertex u # V is adjacent to only one vertex in t, then the vertex u is called
an outer vertex of the spanning tree t. A vertex which is not outer is called
inner. An edge is called ``outer'' if it is incident to an outer vertex and
``inner'' otherwise.
Let u be an outer vertex of a spanning tree t and assume that the edge
uu
1
is included in the set E(t) of all the edges in t.LetN(u)=
[u
i
# V | uu
i
# E]. For any vertex u
i
in N(u), t
i
=(t&uu
1
) _ uu
i
is also a
spanning tree of G. Since the vertex u is adjacent to only u
1
in t, u is also
adjacent to only u
i
in t
i
. Thus the vertex u is an outer vertex in t
i
.
The subgraph t&uu
1
includes two connected components because any
edge in a spanning tree is a bridge. One is a singleton u and the other is
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Copyright 1999 by Academic Press
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