Domination Number of the Complete Directed Grid Graphs

We use dynamic programming to compute the domination number of the Cartesian product of two directed paths, \( \overrightarrow{P}_m \) and \( \overrightarrow{P}_n \), for \( m \leq 25 \) and all \( n \). This suggests that the domination number for \(\min(m,n) \geq 4\) is \( \left\lfloor \frac{(m+1)(n+1)}{3} \right\rfloor – 1 \), which we then confirm by showing that this is both an upper and a lower bound on the domination number.