Bivariate Domination Polynomial

TL;DR

This paper introduces a new bivariate polynomial related to graph domination, provides methods for its efficient computation, and explores graph families sharing the same polynomial, advancing graph theory understanding.

Contribution

The paper presents a novel bivariate domination polynomial and develops methods for its efficient calculation, expanding the tools for graph analysis.

Findings

Identified families of graphs with identical bivariate domination polynomials

Developed algorithms for efficient polynomial computation

Connected the new polynomial to existing domination polynomials

Abstract

We introduce a new bivariate polynomial ${\displaystyle J(G; x,y):=\sum\limits_{W \in V(G)} x^{|W|}y^{|N(W)|}}$ 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.