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, g(P3square Pn)=n+éfracn4ùgamma(P_{3}square P_{n})=n+lceilfrac{n}{4}rceil, g(P4square Pn)=n+éfrac2n3ùgamma(P_{4}square P_{n})=n+lceilfrac{2n}{3}rceil, γ(P 5□P n )=2n+1 and g(P6square Pn)=2n+éfracn+23ùgamma(P_{6}square P_{n})=2n+lceilfrac{n+2}{3}rceil.
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机译：设γ（G）表示有向图G的支配数，设P m □P n 表示P m 和P的笛卡尔积 n ，长度为m和n的有向路径。在本文中，我们给出了γ（P m □P n ）的上下限。此外，我们获得了P m □P n 具有有效控制集的充要条件，并确定了精确值：γ（P 2 □P n ）= n，g（P 3 平方P n ）= n +éfracn4ùgamma（P_ {3}平方P_ { n}）= n + lceilfrac {n} {4} rceil，g（P 4 square P n ）= n +éfrac2n3ùgamma（P_ {4} square P_ {n }）= n + lceilfrac {2n} {3} rceil，γ（P 5 □P n ）= 2n + 1和g（P 6 square P n ）= 2n +éfracn+23ùgamma（P_ {6} square P_ {n}）= 2n + lceilfrac {n + 2} {3} rceil。
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