有向圈的笛卡尔积有向图的双控制数

The Twin Domination Number of Cartesian Product of Directed Cycles

中文关键词:  双控制数  笛卡尔积  有向圈

英文关键词:twin domination number  Cartesian product  directed cycles

中文摘要:

      设$\gamma^{*}(D)$表示有向图$D$的双控制数, $C_{m}\square C_{n}$表示有向圈$C_{m}$和$C_{n}$的笛卡尔积有向图, 其中有向圈的圈长$m, n\geq 2$. 本文,我们确定了以下有向圈笛卡尔积有向图双控制数的确切值: $\gamma^{*}(C_{2}\square C_{n})=n$; $\gamma^{*}(C_{3}\square C_{n})=n$ 如果 $n\equiv 0~({\rm mod}\,3)$, 否则, $\gamma^{*}(C_{3}\square C_{n})=n+1$; $\gamma^{*}(C_{4}\square C_{n})=n+\lceil\frac{n}{2}\rceil$ 如果 $n\equiv 0,3,5~({\rm mod}\,8)$, 否则, $\gamma^{*}(C_{4}\square C_{n})=n+\lceil\frac{n}{2}\rceil+1$; $\gamma^{*}(C_{5}\square C_{n})=2n$; $\gamma^{*}(C_{6}\square C_{n})=2n$ 如果 $n\equiv 0~({\rm mod}\,3)$, 否则, $\gamma^{*}(C_{6}\square C_{n})=2n+2$.

英文摘要:

      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, $\gamma^{*}(C_{6}\square C_{n})=2n+2$.