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Title: The forcing convexity number of a graph (English)
Author: Chartrand, Gary
Author: Zhang, Ping
Language: English
Journal: Czechoslovak Mathematical Journal
ISSN: 0011-4642 (print)
ISSN: 1572-9141 (online)
Volume: 51
Issue: 4
Year: 2001
Pages: 847-858
Summary lang: English
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Category: math
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Summary: For two vertices $u$ and $v$ of a connected graph $G$, the set $I(u, v)$ consists of all those vertices lying on a $u$–$v$ geodesic in $G$. For a set $S$ of vertices of $G$, the union of all sets $I(u,v)$ for $u, v \in S$ is denoted by $I(S)$. A set $S$ is a convex set if $I(S) = S$. The convexity number $\mathop {\mathrm con}(G)$ of $G$ is the maximum cardinality of a proper convex set of $G$. A convex set $S$ in $G$ with $|S| = \mathop {\mathrm con}(G)$ is called a maximum convex set. A subset $T$ of a maximum convex set $S$ of a connected graph $G$ is called a forcing subset for $S$ if $S$ is the unique maximum convex set containing $T$. The forcing convexity number $f(S, \mathop {\mathrm con})$ of $S$ is the minimum cardinality among the forcing subsets for $S$, and the forcing convexity number $f(G, \mathop {\mathrm con})$ of $G$ is the minimum forcing convexity number among all maximum convex sets of $G$. The forcing convexity numbers of several classes of graphs are presented, including complete bipartite graphs, trees, and cycles. For every graph $G$, $f(G, \mathop {\mathrm con}) \le \mathop {\mathrm con}(G)$. It is shown that every pair $a$, $ b$ of integers with $0 \le a \le b$ and $b \ge 3$ is realizable as the forcing convexity number and convexity number, respectively, of some connected graph. The forcing convexity number of the Cartesian product of $H \times K_2$ for a nontrivial connected graph $H$ is studied. (English)
Keyword: convex set
Keyword: convexity number
Keyword: forcing convexity number
MSC: 05C12
idZBL: Zbl 0995.05044
idMR: MR1864046
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Date available: 2009-09-24T10:47:42Z
Last updated: 2016-04-07
Stable URL: http://hdl.handle.net/10338.dmlcz/127690
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