The covered components polynomial: A new representation of the edge elimination polynomial

TL;DR

This paper introduces the covered components polynomial, a new graph polynomial that counts spanning subgraphs based on components, edges, and covered components, and explores its relation to the edge elimination polynomial.

Contribution

It defines the covered components polynomial, establishes its recurrence relation, and demonstrates its connection to the edge elimination polynomial and other graph polynomials.

Findings

The covered components polynomial can be expressed as a substitution of the edge elimination polynomial.

Both polynomials are interrelated through a recurrence relation.

The paper discusses properties and relations to other graph polynomials.

Abstract

Motivated by the definition of the edge elimination polynomial of a graph we define the covered components polynomial counting spanning subgraphs with respect to their number of components, edges and covered components. We prove a recurrence relation, which shows that both graph polynomials are substitution instances of each other. We give some properties of the covered components polynomial and some results concerning relations to other graph polynomials.