The Hull Number of an Oriented Graph

We present characterizations of connected graphs G of order n ≥ 2 for which h + (G) = n. It is shown that for every two integers n and m with 1 ≤ n − 1 ≤ m ≤ n 2 , there exists a connected graph G of order n and size m such that for each integer k with 2 ≤ k ≤ n, there exists an orientation of G with hull number k. 1. Introduction. The (directed) distance d(u, v) from a vertex u to a vertex v in an oriented graph D is the length of a shortest directed u−v path in D. A directed u−v path of length d(u, v) is referred to as a u−v geodesic. A vertex w is said to lie in a u − v geodesic P if w is an internal vertex of P , that is, w is a vertex of P distinct from u and v. The closed interval I[u, v] consists of u and v together with all vertices lying in a u−v geodesic or in a v −u geodesic in D. Hence, if there is neither a u−v geodesic nor a v −u geodesic in D, then I[u, v] = {u, v}. For a nonempty subset S of V (D), define