The Strong Convexity Spectra of Grids

Abstract

Let D be a connected oriented graph. A set $$S \subseteq V(D)$$ is convex in D if, for every pair of vertices $$x, y \in S$$ , the vertex set of every xy-geodesic, (xy shortest directed path) and every yx-geodesic in D is contained in S. The convexity number, $$\hbox {con}(D)$$ , 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 $$\{\hbox {con}(D) : D \hbox { is a strong orientation of G} \}$$ . In this paper we prove that the problem of determining the convexity number of an oriented graph is $$\mathcal {NP}$$ -complete, even for bipartite oriented graphs of arbitrary large girth, extending previous known results for graphs. We also determine $$S_{SC} (P_n \Box P_m)$$ , for every pair of integers $$n,m \ge 2$$ .