The Twin Domination Number of Cartesian Product of Directed Cycles
Received:April 16, 2015  Revised:September 14, 2015
Key Word: twin domination number   Cartesian product   directed cycles
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant Nos.61363020; 11301450; 11226294), the Youth Science and Technology Education Project of Xinjiang (Grant No.2013731011) and China Scholarship Council.
 Author Name Affiliation Hongxia MA College of Mathematics Sciences, Xinjiang Normal University, Xinjiang 830017, P. R. China Juan LIU College of Mathematics Sciences, Xinjiang Normal University, Xinjiang 830017, P. R. China
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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, $\gamma^{*}(C_{6}\square C_{n})=2n+2$.