The Geodetic numbers of Graphs and Digraphs

For any two vertices u and v in a graph G (digraph D, respectively), a u − v geodesic is a shortest path between u and v (from u to v, respectively). Let I(u, v) denote the set of all vertices lying on a u− v geodesic. For a vertex subset S, let I(S) denote the union of all I(u, v) for u, v ∈ S. The geodetic number g(G) (g(D), respectively) of a graph G (digraph D, respectively) is the minimum cardinality of a set S with I(S) = V (G) (I(S) = V (D), respectively). The geodetic spectrum of a graph G, denote by S(G), is the set of geodetic numbers of all orientations of graph G. The upper geodetic number is g+(G) = maxS(G). The main purpose of this paper is to study the relations among g(G), S(G) and g+(G) for connected graphs G. In addition, a sufficient and necessary condition for the equality of g(G) and g(G×K2) is presented, which improved a result of Chartrand, Harary and Zhang.