Total Domination of Grid Graphs

The size of a minimum total dominating set in the \(m \times n\) grid graph is denoted by \(\gamma_t(P_m \square P_n)\). Here a dynamic programming algorithm that computes \(\gamma_t(P_m \square P_n)\) for any \(m\) and \(n\) is presented, and it is shown how properties of the algorithm can be used to derive formulae for a fixed, small value of \(m\). Using this method, formulae for \(\gamma_t(P_m \square P_n)\) for \(m \leq 28\) are obtained. Formulae for larger \(m\) are further conjectured, and a new general upper bound on \(\gamma_t(P_m \square P_n)\) is proved.