In this paper, we study the computational complexity of finding the geodetic number of graphs. A set of vertices S of a graph G is a geodetic set if any vertex of G lies in some shortest path between some pair of vertices from S. The Minimum Geodetic Set (MGS) problem is to find a geodetic set with minimum cardinality. In this paper, we prove that solving the MGS problem is NP-hard on planar graphs with a maximum degree six and line graphs. We also show that unless P = N P , there is no polynomial time algorithm to solve the MGS problem with sublogarithmic approximation factor (in terms of the number of vertices) even on graphs with diameter 2. On the positive side, we give an O 3 √ n log n -approximation algorithm for the MGS problem on general graphs of order n. We also give a 3-approximation algorithm for the MGS problem on the family of solid grid graphs which is a subclass of planar graphs.
ACM Subject Classification Theory of computation → Graph algorithms analysis.Hardness and approximation for the geodetic set problem in some graph classes shortest path between u and v. A set of vertices S is a geodetic set if ∪ u,v∈S I(u, v) = V (G). The geodetic number, denoted as g(G), is the minimum integer k such that G has a geodetic set of cardinality k. Given a graph G, the Minimum Geodetic Set (MGS) problem is to compute a geodetic set of G with minimum cardinality. In this paper, we shall study the computational complexity of the MGS problem in various graph classes.The notion of geodetic sets and geodetic number was introduced by Harary et al. [18]. The notion of geodetic number is closely related to convexity and convex hulls in graphs, which have applications in game theory, facility location, information retrieval, distributed computing and communication networks [2,19,15,22,10]. In 2002, Atici [1] proved that finding the geodetic number of arbitrary graphs is Dourado et al. [9,8] strengthened the above result to bipartite graphs, chordal graphs and chordal bipartite graphs. Recently, Bueno et al. [3] proved that the MGS problem remains NP-hard even for subcubic graphs. On the positive side, polynomial time algorithms to solve the MGS problem are known for cographs [8], split graphs [8], ptolemaic graphs [12], outer planar graphs [21] and proper interval graphs [11]. In this paper, we prove the following theorem.
