On the computation of the hull number of a graph

Let G be a graph. If u , v ∈ V ( G ) , a u – v shortest path of G is a path linking u and v with minimum number of edges. The closed interval  I [ u , v ] consists of all vertices lying in some u – v shortest path of G . For S ⊆ V ( G ) , the set I [ S ] is the union of all sets I [ u , v ] for u , v ∈ S . We say that S is a convex set if I [ S ] = S . The convex hull of S , denoted I h [ S ] , is the smallest convex set containing S . A set S is a hull set of G if I h [ S ] = V ( G ) . The cardinality of a minimum hull set of G is the hull number of G , denoted by h n ( G ) . In this work we prove that deciding whether h n ( G ) ≤ k is NP-complete.We also present polynomial-time algorithms for computing h n ( G ) when G is a unit interval graph, a cograph or a split graph.