| The Twin Domination Number of Cartesian Product of Directed Cycles |
| Received:April 16, 2015 Revised:September 14, 2015 |
| Key Words:
twin domination number Cartesian product directed cycles
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| Fund Project: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. |
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| 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, $\gamma^{*}(C_{6}\square C_{n})=2n+2$. |
| Citation: |
| DOI:10.3770/j.issn:2095-2651.2016.02.005 |
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