The hull and geodetic numbers of orientations of graphs

For an oriented graph D, let I-D[u, v] denote the set of all vertices lying on a u-v geodesic or a v-u geodesic. For S subset of V(D), let I-D[S] denote the union of all I-D[u, v] for all u, v is an element of S. Let [S](D) denote the smallest convex set containing S. The geodetic number g(D) of an oriented graph D is the minimum cardinality of a set S with I-D[S] = V(D) and the hull number h(D) of an oriented graph D is the minimum cardinality of a set S with [S](D) = V(D). For a connected graph G, let O(G) be the set of all orientations of G, define g(-)(G) = min{g(D) : D is an element of O(G)}, g(+) (G) = maxi{g(D) : D is an element of O(G)}, h(-)(G) = min{h(D) : D is an element of O(G)}, and h(+) (G) = max{h(D) : D is an element of O(G)}. By the above definitions, h(-) (G) <= g(-)(G) and h(+)(G) <= g(+) (G). In the paper, we prove that g-(G) < h+(G) for a connected graph G of order at least 3, and for any nonnegative integers a and b, there exists a connected graph G such that g(-) (G) - h(-) (G) = a and g(+) (G) - h(+) (G) = b. These results answer a problem of Farrugia in [A. Farrugia, Orientable convexity, geodetic and hull numbers in graphs, Discrete Appl. Math. 148 (2005) 256-262]. (C) 2008 Elsevier B.V. All rights reserved.