Graph polynomials for a class of DI-pathological graphs

Abstract

Let G=V,E be a simple graph. A dominating set D⊆V is a set such that the closed neighborhood of D is the entire vertex set. An independence set of a graph G is a subset of vertices that are pairwise non-adjacent. A DI-pathological graph is a graph where every minimum dominating set intersects every maximal independent set. Let d(G,i) denote the number of dominating sets of G of size i. The domination polynomial of a graph G is defined by D(G,x)=∑i=γ(G)Vd(G,i)xi. Let s(G,i) denote the number of independent sets of size i in a graph G. The independence polynomial is defined by I(G,x)=∑i=0α(G)s(G,i)xi. In this paper, we will examine the domination polynomial and the independence polynomial of a family of extremal DI-pathological graphs. We will further define an independent dominating set and examine the corresponding independent domination polynomial for these graphs.