# 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.

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