# Article

**Keywords:**

convex set; convexity number; forcing convexity number

**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.

References:

[bh:dg] F. Buckley and F. Harary:

**Distance in Graphs**. Addison-Wesley, Redwood City, CA, 1990.

MR 1045632
[cwz:cn] G. Chartrand, C. E. Wall, and P. Zhang:

**The convexity number of a graph**. (to appear).

MR 1913663
[cz:cs] G. Chartrand and P. Zhang:

**$H$-convex graphs**. Math. Bohem. 126 (2001), 209–220.

MR 1826483
[hn:cg] F. Harary and J. Nieminen:

**Convexity in graphs**. J. Differential Geom. 16 (1981), 185–190.

MR 0638785
[n1] L. Nebeský:

**A characterization of the interval function of a connected graph**. Czechoslovak Math. J. 44 (119) (1994), 173–178.

MR 1257943
[n2] L. Nebeský:

**Characterizing of the interval function of a connected graph**. Math. Bohem. 123 (1998), 137–144.

MR 1673965