On the computation of the hull number of a graph

Let G be a graph. If u,[email protected]?V(G), a u-vshortest 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 [email protected]?V(G), the set I[S] is the union of all sets I[u,v] for u,[email protected]?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 hn(G). In this work we prove that deciding whether hn(G)@?k is NP-complete.We also present polynomial-time algorithms for computing hn(G) when G is a unit interval graph, a cograph or a split graph.