On domination number of Cartesian product of directed paths

Abstract

Let ?(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 ?(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: ?(P 2?P n )=n, $\gamma(P_{3}\square P_{n})=n+\lceil\frac{n}{4}\rceil$ , $\gamma(P_{4}\square P_{n})=n+\lceil\frac{2n}{3}\rceil$ , ?(P 5?P n )=2n+1 and $\gamma(P_{6}\square P_{n})=2n+\lceil\frac{n+2}{3}\rceil$ .