On the Approximation Hardness of Geodetic Set and Its Variants

Given a graph, a geodetic set (resp. edge geodetic set) is a subset of its vertices such that every vertex (resp. edge) of the graph is on a shortest path between two vertices of the subset. A strong geodetic set is a subset S of vertices and a choice of a shortest path for every pair of vertices of S such that every vertex is on one of these shortest paths. The geodetic number (resp. edge geodetic number) of a graph is the minimum size of a geodetic set (resp. edge geodetic set) and the strong geodetic number is the minimum size of a strong geodetic set. We first prove that, given a subset of vertices, it is NP-hard to determine whether it is a strong geodesic set. Therefore, it seems natural to study the problem of maximizing the number of covered vertices by a choice of a shortest path for every pair of a provided subset of vertices. We provide a tight 2approximation algorithm to solve this problem. Then, we show that there is no (781)/780 polynomial-time approximation algorithm for edge geodetic number and strong geodetic number on subcubic bipartite graphs with arbitrarily high girth. We also prove that geodetic number and edge geodetic number are both LOG-APX- hard, even on subcubic bipartite graphs with arbitrarily high girth. Finally, we disprove a conjecture of Ir.si.s and Konvalinka by proving that the strong geodetic number can be computed in polynomial time in complete multipartite graphs.