Dominating sets and domination polynomials of paths

Abstract

Let G = (V, E) be a simple graph. A set S ⊆ V is a dominating set of G, if every vertex in V\S is adjacent to at least one vertex in S. Let Pni be the family of all dominating sets of a path Pn with cardinality i, and let d(Pn, j) = Dnj . In this paper, we construct Dni, and obtain a recursive formula for d(Pn, i). Using this recursive formula, we consider the polynomial D(Pn, x) = ∑i=[n/3]n d(Pn, i)xi, which we call domination polynomial of paths and obtain some properties of this polynomial.