Proving properties of the edge elimination polynomial using equivalent graph polynomials

TL;DR

This paper introduces two equivalent graph polynomials related to the edge elimination polynomial, demonstrating its reconstructibility from polynomial decks and its encoding of degree sequences, thereby advancing understanding of graph invariants.

Contribution

It establishes the equivalence of two new graph polynomials with the edge elimination polynomial and proves its reconstructibility and encoding capabilities.

Findings

Edge elimination polynomial is reconstructible from its polynomial deck.

The polynomial encodes the degree sequence of the graph.

Introduction of two equivalent graph polynomials related to edge elimination.

Abstract

Averbouch, Godlin and Makowsky define the edge elimination polynomial of a graph by a recurrence relation with respect to the deletion, contraction and extraction of an edge. It generalizes some well-known graph polynomials such as the chromatic polynomial and the matching polynomial. By introducing two equivalent graph polynomials, one enumerating subgraphs and the other enumerating colorings, we show that the edge elimination polynomial of a simple graph is reconstructible from its polynomial deck and that it encodes the degree sequence of an arbitrary graph.