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 (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.