Graph products of the trivariate total domination polynomial and related polynomials

Abstract

A vertex subset W⊆V of the graph G=(V,E) is a total dominating set if every vertex of the graph is adjacent to at least one vertex in W. The total domination polynomial is the ordinary generating function for the number of total dominating sets in the graph. We investigated some graph products for a generalization of the total domination polynomial, called the trivariate total domination polynomial. We also show that the chromatic polynomial is encoded in the independent domination polynomial of some graph products. These results have a wide applicability to other domination related graph polynomials, e.g. the domination polynomial, the independent domination polynomial or the independence polynomial.