The Forcing Convexity Number of a Graph

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 $\con(G)$ of $G$ is the maximum cardinality of a proper convex set of $G$. A subset $T$ of a maximum convex set $S$ of a graph $G$ is called a forcing subset for $S$ if $S$ is the unique maximum convex set containing $T$. The forcing convexity number $f(S, \con)$ of $S$ is the minimum cardinality among the forcing subsets for $S$, and the forcing convexity number $f(G, \con)$ 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 $G$, $f(G, \con) \leq \con(G)$. It is shown that every pair $a, b$ of integers with $0 \leq a \leq b$ and $b \geq 3$ 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.