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Dominating sets of centipedes. (English) Zbl 1180.05078
Summary: Let $G = (V,E)$ 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 ${\mathcal D}(G,i)$ 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 ${\mathcal D}(P^*_n,i) = {\mathcal D}(C^*_n,i) ={\mathcal D}(C^*_n,i)$, where $C_n$ and $G_n$ are respectively, cycle and arbitrary graph of order $n$.

MSC:
 05C69 Dominating sets, independent sets, cliques