The domination number of Cartesian product of two directed paths

Let γ(P _m □P _n ) be the domination number of the Cartesian product of directed paths P _m and P _n for m,n≥2. Liu et al. in (J. Comb. Optim. 22(4):651–662, 2011) determined the value of γ(P _m □P _n ) for arbitrary n and m≤6. In this work we give the exact value of γ(P _m □P _n ) for any m,n and exhibit dominating sets of minimum cardinality.