The exact domination number of generalized Petersen graphs $$P(n,k)$$ P ( n , k ) with $$n=2k$$ n = 2 k and $$n=2k+2^{*}$$ n = 2 k + 2 ∗

Abstract

A subset $$S\subseteq V$$ is a dominating set of $$G=(V,E)$$ if each vertex in $$V\backslash S$$ is adjacent to at least one vertex in $$S$$ . The domination number of $$G$$ is the cardinality of a minimum dominating set of $$G$$ . Graph domination numbers and algorithms for finding them have been investigated for numerous classes of graphs, usually for graphs that have some kind of tree-like structure. In this paper, we determine the exact domination number of generalized Petersen graph $$P(n,k)$$ with $$n=2k$$ and $$n=2k+2$$ . $$\begin{aligned} \gamma (P(n,k))= \left\lceil \frac{2n}{3}\right\rceil \quad \hbox {for} \; n=2k \end{aligned}$$ $$\begin{aligned} \gamma (P(n,k))=\left\{ \begin{array}{l@{\quad }l} \frac{n}{2} &{} {{\mathrm{if}}}\;k\;{\mathrm{is\, odd}} \\ \frac{n}{2}+1 &{} {\mathrm{if}}\;k\;{\mathrm{is\,even}} \end{array} \right. \quad {\mathrm{for}}\;n=2k+2. \end{aligned}$$