Bivariate Domination Polynomial

Abstract

We introduce a new bivariate polynomial
$$J(G;x,y):=\sum _{W\subseteq V(G)}{x}^{|W|}{y}^{|N[W]|–|W|}$$
which contains the standard domination polynomial of the graph $G$ in two different ways. We build methods for efficient calculation of this polynomial and prove that there are still some families of graphs which have the same bivariate polynomial.