On the hull number of some graph classes

Given a graph G = (V,E), the closed interval of a pair of vertices u, v ∈ V , denoted by I[u, v], is the set of vertices that belongs to some shortest (u, v)-path. For a given S ⊆ V , let I[S] = ⋃ u,v∈S I[u, v]. We say that S ⊆ V is a convex set if I[S] = S. The convex hull Ih[S] of a subset S ⊆ V is the smallest convex set that contains S. We say that S is a hull set if Ih[S] = V . The cardinality of a minimum hull set of G is the hull number of G, denoted by hn(G). We show that deciding if hn(G) ≤ k is an NP-complete problem, even if G is bipartite. We also prove that hn(G) can be computed in polynomial time for cactus and P4-sparse graphs.