# The forcing convexity number of a graph

• Volume: 51, Issue: 4, page 847-858
• ISSN: 0011-4642

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## Abstract

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For two vertices $u$ and $v$ of a connected graph $G$, the set $I\left(u,v\right)$ 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 $I\left(u,v\right)$ for $u,v\in S$ is denoted by $I\left(S\right)$. A set $S$ is a convex set if $I\left(S\right)=S$. The convexity number $\mathrm{c}on\left(G\right)$ of $G$ is the maximum cardinality of a proper convex set of $G$. A convex set $S$ in $G$ with $|S|=\mathrm{c}on\left(G\right)$ 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 $f\left(S,\mathrm{c}on\right)$ of $S$ is the minimum cardinality among the forcing subsets for $S$, and the forcing convexity number $f\left(G,\mathrm{c}on\right)$ 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 $G$, $f\left(G,\mathrm{c}on\right)\le \mathrm{c}on\left(G\right)$. 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 $H×{K}_{2}$ for a nontrivial connected graph $H$ is studied.

## How to cite

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Chartrand, Gary, and Zhang, Ping. "The forcing convexity number of a graph." Czechoslovak Mathematical Journal 51.4 (2001): 847-858. <http://eudml.org/doc/30675>.

@article{Chartrand2001,
abstract = {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 $\mathop \{\mathrm \{c\}on\}(G)$ of $G$ is the maximum cardinality of a proper convex set of $G$. A convex set $S$ in $G$ with $|S| = \mathop \{\mathrm \{c\}on\}(G)$ 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 $f(S, \mathop \{\mathrm \{c\}on\})$ of $S$ is the minimum cardinality among the forcing subsets for $S$, and the forcing convexity number $f(G, \mathop \{\mathrm \{c\}on\})$ 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 $G$, $f(G, \mathop \{\mathrm \{c\}on\}) \le \mathop \{\mathrm \{c\}on\}(G)$. 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.},
author = {Chartrand, Gary, Zhang, Ping},
journal = {Czechoslovak Mathematical Journal},
keywords = {convex set; convexity number; forcing convexity number; convex set; convexity number; forcing convexity number},
language = {eng},
number = {4},
pages = {847-858},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The forcing convexity number of a graph},
url = {http://eudml.org/doc/30675},
volume = {51},
year = {2001},
}

TY - JOUR
AU - Chartrand, Gary
AU - Zhang, Ping
TI - The forcing convexity number of a graph
JO - Czechoslovak Mathematical Journal
PY - 2001
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 51
IS - 4
SP - 847
EP - 858
AB - 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 $\mathop {\mathrm {c}on}(G)$ of $G$ is the maximum cardinality of a proper convex set of $G$. A convex set $S$ in $G$ with $|S| = \mathop {\mathrm {c}on}(G)$ 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 $f(S, \mathop {\mathrm {c}on})$ of $S$ is the minimum cardinality among the forcing subsets for $S$, and the forcing convexity number $f(G, \mathop {\mathrm {c}on})$ 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 $G$, $f(G, \mathop {\mathrm {c}on}) \le \mathop {\mathrm {c}on}(G)$. 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.
LA - eng
KW - convex set; convexity number; forcing convexity number; convex set; convexity number; forcing convexity number
UR - http://eudml.org/doc/30675
ER -

## References

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