Convex Sets in Acyclic Digraphs

A non-empty set X of vertices of an acyclic digraph is called connected if the underlying undirected graph induced by X is connected and it is called convex if no two vertices of X are connected by a directed path in which some vertices are not in X. The set of convex sets (connected convex sets) of an acyclic digraph D is denoted by CO(D) (CC(D)) and its size by co(D) (cc(D)). Gutin, Johnstone, Reddington, Scott, Soleimanfallah, and Yeo (Proc. ACiD’07) conjectured that the sum of the sizes of all (connected) convex sets in D equals Θ(n · co(D)) (Θ(n · cc(D))) where n is the order of D. In this paper we exhibit a family of connected acyclic digraphs with ∑ C∈CO(D) |C| = o(n · co(D)) and ∑ C∈CC(D) |C| = o(n · cc(D)). We also show that the number of connected convex sets of order k in any connected acyclic digraph of order n is at least n− k + 1. This is a strengthening of a theorem by Gutin and Yeo.