For two vertices u and v in 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 ∈ S is denoted by I(S). A set S is convex if I(S) = S. The convexity number con(G) is the maximum cardinality of a proper convex set in G. A convex set S is maximum if |S| = con(G). The cardinality of a maximum convex set in a graph G is the convexity number of G. For a nontrivial connected graph H, a connected graph G is an H-convex graph if G contains a maximum convex set S whose induced subgraph is S = H. It is shown that for every positive integer k, there exist k pairwise nonisomorphic graphs H1, H2,...,Hk of the same order and a graph G that is Hi-convex for all i (1 ≤ i ≤ k). Also, for every connected graph H of order k ≥ 3 with convexity number 2, it is shown that there exists an H-convex graph of order n for all n ≥ k + 1. More generally, it is shown that for every nontrivial connected graph H, there exists a positive integer N and an H-convex graph of order n for every integer n ≥ N.
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.