The Monophonic Directed Convexities

Abstract

Graph Convexity is a topic with many papers in the literature, especially on the $P_3$, geodesic, and monophonic convexities in undirected graphs. Regarding directed graphs, Erd{\H o}s et al. in 1972 and Chartrand and Zhang in 2000 obtained the first results on the $P_3$ and geodesic convexities for directed graphs, respectively. In this paper, we start the investigation of monophonic convexity on directed graphs. We define two convexities, the monophonic and the strong monophonic directed convexities, related to the minimal and the induced directed paths, respectively. We show that deciding whether a set is convex is polynomial-time solvable in the monophonic convexity, but coNP-complete in the strong monophonic convexity. For both convexities, we prove that the hull number is NP-complete even in DAGs. Regarding the monophonic convexity, we prove that the hull number is polynomial time solvable in tournaments, but the interval number is NP-complete in tournaments and also in bipartite oriented graphs.