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 \(G\) such that every vertex of \(G\) is either in \(S\) or adjacent to a vertex in \(S\). The domination polynomial of \(G\) is defined as \(D(G, x) = \sum_{i=0}^{n} d(G, i)x^i\), where \(d(G, i)\) denotes the number of dominating sets of \(G\) of size \(i\). In this paper, we demonstrate that cycles are uniquely determined by their domination polynomials.