The geodetic domination number for the product of graphs

A subset $S$ of vertices in a graph $G$ is called a geodetic set if every vertex not in $S$ lies on a shortest path between two vertices from $S$‎. ‎A subset $D$ of vertices in $G$ is called dominating set if every vertex not in $D$ has at least one neighbor in $D$‎. ‎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 (dominating‎, ‎geodetic dominating) sets in $G$‎. ‎In this paper‎, ‎we show that if a triangle free graph $G$ has minimum degree at least 2 and $g(G) = 2$‎, ‎then $gamma _g(G) = gamma(G)$‎. ‎It is shown‎, ‎for every nontrivial connected graph $G$ with $gamma(G) = 2$ and $diam(G) > 3$‎, ‎that $gamma_g(G) > g(G)$‎. ‎The lower bound for the geodetic domination number of Cartesian product graphs is proved‎. ‎Geodetic domination number of product of cycles (paths) are determined‎. ‎In this work‎, ‎we also determine some bounds and exact values of the geodetic domination number of strong product of graphs‎.