The Twin Domination Number of Cartesian Product of Directed Cycles

Abstract:

      Let $\gamma^{*}(D)$ denote the twin domination number of digraph $D$ and let $C_{m}\square C_{n}$ denote the Cartesian product of $C_{m}$ and $C_{n}$, the directed cycles of length $m, n\geq 2$. In this paper, we determine the exact values: $\gamma^{*}(C_{2}\square C_{n})=n$; $\gamma^{*}(C_{3}\square C_{n})=n$ if $n\equiv 0~({\rm mod}\,3)$, otherwise, $\gamma^{*}(C_{3}\square C_{n})=n+1$; $\gamma^{*}(C_{4}\square C_{n})=n+\lceil\frac{n}{2}\rceil$ if $n\equiv 0,3,5~({\rm mod}\,8)$, otherwise, $\gamma^{*}(C_{4}\square C_{n})=n+\lceil\frac{n}{2}\rceil+1$; $\gamma^{*}(C_{5}\square C_{n})=2n$; $\gamma^{*}(C_{6}\square C_{n})=2n$ if $n\equiv 0~({\rm mod}\,3)$, otherwise, $\gamma^{*}(C_{6}\square C_{n})=2n+2$.