The geodetic numbers of graphs and digraphs

For every two vertices u and v in a graph G, a u-v geodesic is a shortest path between u and v. 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) of a graph G is the minimum cardinality of a set S with I(S) = V (G). For a digraph D, there is analogous terminology for the geodetic number g(D). The geodetic spectrum of a graph G, denoted by S(G), is the set of geodetic numbers of all orientations of graph G. The lower geodetic number is g ^-(G) = minS(G) and the upper geodetic number is g ⁺(G) = maxS(G). The main purpose of this paper is to study the relations among g(G), g ^-(G) and g ⁺(G) for connected graphs G. In addition, a sufficient and necessary condition for the equality of g(G) and g(G × K ₂) is presented, which improves a result of Chartrand, Harary and Zhang.