On pitfalls in computing the geodetic number of a graph

Given a simple connected graph G = (V, E) the geodetic closure I[S] subset of V of a subset S of V is the union of all sets of nodes lying on some geodesic (or shortest path) joining a pair of nodes v(k), v(l) epsilon S. The geodetic number, denoted by g(G), is the smallest cardinality of a node set S* such that I[S*] = V. In "The geodetic number of a graph", [Harary et al. in Math. Comput. Model. 17: 89-95, 1993] propose an incorrect algorithm to find the geodetic number of a graph G. We provide counterexamples and show why the proposed approach must fail. We then develop a 0-1 integer programming model to find the geodetic number. Computational results are given.