A polynomial time algorithm for geodetic hull number for complementary prisms

Abstract

Let G be a finite, simple, and undirected graph and let S ⊆ V (G). In the geodetic convexity, S is convex if all vertices belonging to any shortest path between two vertices of S lie in S. The convex hull H(S) of S is the smallest convex set containing S. The hull number h(G) is the minimum cardinality of a set S ⊆ V (G) such that H(S) = V (G). The complementary prism GG̅ of a graph G arises from the disjoint union of the graph G and G̅ by adding the edges of a perfect matching between the corresponding vertices of G and G̅. Previous works have determined h(GG̅) when both G and G̅ are connected and partially when G is disconnected. In this paper, we characterize convex sets in GG̅ and we present equalities and tight lower and upper bounds for h(GG̅). This fills a gap in the literature and allows us to show that h(GG̅) can be determined in polynomial time, for any graph G.