On domination number of Cartesian product of directed paths

Let γ(G) denote the domination number of a digraph G and let P m □P n denote the Cartesian product of P m and P n , the directed paths of length m and n. In this paper, we give a lower and upper bound for γ(P m □P n ). Furthermore, we obtain a necessary and sufficient condition for P m □P n to have efficient dominating set, and determine the exact values: γ(P 2□P n )=n, $\gamma (P_3\square P_n)=n+\lceil \frac{n}{4}\rceil$ , $\gamma (P_4\square P_n)=n+\lceil \frac{2n}{3}\rceil$ , γ(P 5□P n )=2n+1 and $\gamma (P_6\square P_n)=2n+\lceil \frac{n+2}{3}\rceil$ .