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Dominating sets of centipedes. (English) Zbl 1180.05078
Summary: Let $G=\left(V,E\right)$ be a simple graph. A set $S\subseteq V$ is a dominating set of $G$, if every vertex in $V-S$ is adjacent to at least one vertex in $S$. Let $𝒟\left(G,i\right)$ be the family of all dominating sets of a graph $G$ with cardinality $i$, and ${G}^{*}$ be the graph obtained by appending a single $i$ pendant edge to each vertex of graph $G$. We call ${P}_{n}^{*}$ a centipede, where ${P}_{n}$ is a path with $n$ vertices. In this paper we study the dominating sets of centipedes and obtain the number of dominating sets of ${P}_{n}^{*}$. We show that $𝒟\left({P}_{n}^{*},i\right)=𝒟\left({C}_{n}^{*},i\right)=𝒟\left({C}_{n}^{*},i\right)$, where ${C}_{n}$ and ${G}_{n}$ are respectively, cycle and arbitrary graph of order $n$.
##### MSC:
 05C69 Dominating sets, independent sets, cliques
##### Keywords:
dominating set; domination number; centipede; unimodal