Triangle-free polyconvex graphs

The notion of convexity in graphs is based on the one in topology: a set of vertices S is convex if an interval is entirely contained in S when its endpoints belong to S. The order of the largest proper convex subset of a graph G is called the convexity number of the graph and is denoted con(G). A graph containing a convex subset of one order need not contain convex subsets of all smaller orders. If G has convex subsets of order m for all 1 less than or equal to m less than or equal to con(G), then G is called polyconvex. In response to a question of Chartand and Zhang [3], we show that, given any pair of integers n and k with 2 less than or equal to k less than or equal to n, there is a connected triangle-free polyconvex graph G of order n with convexity number k.