The edge geodetic number of product graphs

Abstract

For a nontrivial connected graph G = (V (G), E(G)), a set S ⊆ V (G) is called an edge geodetic set of G if every edge of G is contained in a geodesic joining some pair of vertices in S. The edge geodetic number eg(G) of G is the minimum order of its edge geodetic sets. It is observed that the edge geodetic sets and numbers are interesting concepts and possess properties distinct from the vertex geodetic concepts. In this work, we determine some bounds and exact values of the edge geodetic numbers of strong and lexicographic products of graphs.