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.