On domination number of Cartesian product of directed paths

Abstract

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 $\gamma(P_{6}\square P_{n})=2n+\lceil\frac{n+2}{3}\rceil$.