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arXiv · 1103.2427

On the Domination Number of Generalized Petersen Graphs P(ck,k)

Haoli Wang·Xirong Xu·Yuansheng Yang·Guoqing Wang
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Abstract

Let $G=(V(G),E(G))$ be a simple connected and undirected graph with vertex set $V(G)$ and edge set $E(G)$. A set $S \subseteq V(G)$ is a $dominating$ $set$ if for each $v \in V(G)$ either $v \in S$ or $v$ is adjacent to some $w \in S$. That is, $S$ is a dominating set if and only if $N[S]=V(G)$. The domination number $\gamma(G)$ is the minimum cardinalities of minimal dominating sets. In this paper, we give an improved upper bound on the domination number of generalized Petersen graphs $P(ck,k)$ for $c\geq 3$ and $k\geq 3$. We also prove that $\gamma(P(4k,k))=2k+1$ for even $k$, $\gamma(P(5k,k))=3k$ for all $k\geq 1$, and $\gamma(P(6k,k))=\lceil\frac{10k}{3}\rceil$ for $k\geq 1$ and $k\neq 2$.

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Haoli Wang, Xirong Xu, Yuansheng Yang, Guoqing Wang. 2011-03-12. On the Domination Number of Generalized Petersen Graphs P(ck,k). https://arxiv.org/abs/1103.2427

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