Two classes of graphs in which some problems related to convexity are efficiently solvable

Abstract

Monophonic, geodesic and 2-geodesic convexities ([Formula: see text]-convexity, [Formula: see text]-convexity and [Formula: see text]-convexity, for short) on graphs are based on the families of induced paths, shortest paths and shortest paths of length [Formula: see text], respectively. We introduce a class of graphs, the class of cross-cyclicgraphs, in which every connected [Formula: see text]-convex set is also [Formula: see text]-convex and [Formula: see text]-convex. We show that this class is properly contained in the class, say [Formula: see text], of graphs in which geodesic and monophonic convexities are equivalent and properly contains the class of distance-hereditary graphs. Moreover, we show that (1) an [Formula: see text]-hull set (i.e., a subset of vertices, with minimum cardinality, whose [Formula: see text]-convex hull equals the whole vertex set) and, hence, the m-hull number and the g-hull number of a graph in [Formula: see text] can be computed in polynomial time and that (2) both the geodesic-convex hull and the monophonic-convex hull can be computed in linear time in a cross-cyclic graph without cycles of length [Formula: see text] and, hence, in a bipartite distance-hereditary graph.