Bivariate Domination Polynomial

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.