On the disjunctive domination numbers of the torus grid graphs

Let $\Gamma=(V,E)$ be a graph. The disjunctive domination number of $\Gamma$ is the minimum cardinality of a set $S\subseteq V$ such that every vertex not in $S$ is adjacent to a vertex of $S$, or has at least two vertices in $S$ at distance $2$ from it. In this paper, we give bounds for the disjunctive domination numbers of the torus grid graphs $C_m\Box C_n$, and determine the disjunctive domination numbers of $C_3\Box C_n$, $C_4\Box C_{n}$ and $C_8\Box C_{4n}$.