Cycles are determined by their domination polynomials

Abstract

Let $G$ be a simple graph of order $n$. A dominating set of $G$ is a set $S$ of vertices of $G$ so that every vertex of $G$ is either in $S$ or adjacent to a vertex in $S$. The domination polynomial of $G$ is the polynomial $D(G,x)={\sum }_{i=1}^{n}d(G,i)x^i$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$. In this paper we show that cycles are determined by their domination polynomials.

To appear in Ars Combinatoria