Abstract:

Let $\gamma^{*}(D)$ denote the twin domination number of digraph $D$ and let $C_{m}\square C_{n}$ denote the Cartesian product of the directed cycle $C_{m}$ and $C_{n}$, for $m, n\geq 2$. In this paper, we give a lower bound for $\gamma^{*}(C_{m}\square C_{n})$ and we determine the exact values of $\gamma^{*}(C_{m}\square C_{n})$ when $m,~n\equiv 0~ ({\rm mod} ~3)$ and when $m\equiv 2~({\rm mod}~3)$.

Key words: Digraphs ,  Twin domination number ,  Cartesian product ,  Directed cycle