The Strong Convexity Spectra of Grids

Abstract

Let D be a connected oriented graph. A set S ⊆ V (D) is convex in D if, for every pair of vertices x, y ∈ S, the vertex set of every xy-geodesic, (xy shortest directed path) and every yx-geodesic in D is contained in S. The convexity number, con(D), of a non-trivial oriented graph, D, is the maximum cardinality of a proper convex set of D. The strong convexity spectrum of the graph G, S SC (G), is the set {con(D) : D is a strong orientation of G}. In this paper we prove that the problem of determining the convexity number of an oriented graph is N P-complete, even for bipartite oriented graphs of arbitrary large girth, extending previous known results for graphs. We also determine S SC (P n 2P m ), for every pair of integers n, m ≥ 2.