On the geodetic domination and domination numbers of some Cartesian product graphs

In a graph G = (V, E), a geodetic set of G is a subset S subset of V such that every vertex not in S lies on a shortest path between two vertices from S. A dominating set of G is a subset S subset of V such that every vertex in V \ S has at least one neighbor in S. A geodetic dominating set S is both a geodetic and a dominating set. The geodetic(domination, geodetic domination) number g(G)(gamma(G), gamma(g)(G)) of G is the minimum cardinality among all geodetic(dorninating, geodetic dominating) sets in G. A.Hansherg and L.Volkmann[Discrete Mathematics 310(2010)] have proved that if a graph G with minimum degree delta >= 2 has girth at least 6, then gamma(g)(G) = gamma(G). In this paper, we show that almost all Cartesian product graphs of paths and cycles, except P-2 rectangle C-3, have the equal geodetic domination number and domination number.