A polynomial time algorithm for geodetic hull number for complementary prisms

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 G G̅ 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 ( G G̅ ) when both G and G̅ are connected and partially when G is disconnected. In this paper, we characterize convex sets in G G̅ and we present equalities and tight lower and upper bounds for h ( G G̅ ). This fills a gap in the literature and allows us to show that h ( G G̅ ) can be determined in polynomial time, for any graph G .