On the hull number of some graph classes

Abstract

Abstract 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 I h [ S ] of a subset S ⊆ V is the smallest convex set that contains S. We say that S is a hull set if I h [ S ] = V . The cardinality of a minimum hull set of G is the hull number of G, denoted by h n ( G ) . We show that deciding if h n ( G ) ⩽ k is an NP-complete problem, even if G is bipartite. We also prove that h n ( G ) can be computed in polynomial time for cactus and P 4 -sparse graphs.