A POLYNOMIAL TIME ALGORITHM FOR GEODETIC HULL NUMBER FOR COMPLEMENTARY PRISMS

Let G be a finite, simple, and undirected graph and let S subset of 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 subset of V (G) such that H(S) = V (G). The complementary prism G (G) over bar of a graph G arises from the disjoint union of the graph G and (G) over bar by adding the edges of a perfect matching between the corresponding vertices of G and (G) over bar. Previous works have determined h(GG) when both G and (G) over bar are connected and partially when G is disconnected. In this paper, we characterize convex sets in G (G) over bar and we present equalities and tight lower and upper bounds for h(G (G) over bar). This fills a gap in the literature and allows us to show that h(G (G) over bar) can be determined in polynomial time, for any graph G.