On the disjunctive domination numbers of the torus grid graphs

Let $\mathrm{\Gamma }=(V,E)$ be a graph. The disjunctive domination number of $\mathrm{\Gamma }$ 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 $C_m\mathrm{\square }{C}_n$, and determine the disjunctive domination numbers of $C_3\mathrm{\square }{C}_n$, $C_4\mathrm{\square }{C}_n$ and $C_8\mathrm{\square }{C}_{4n}$.