Weakly Distinguishing Graph Polynomials on Addable Properties

TL;DR

This paper investigates conditions under which various graph polynomials can distinguish almost all graphs within certain properties, focusing on addable and low-genus graph classes.

Contribution

It provides sufficient conditions for several important graph polynomials to be weakly distinguishing on addable and low-genus graph properties.

Findings

Characteristic, clique, independence, matching, and domination polynomials are weakly distinguishing on certain classes.

The Tutte polynomial and its specializations are weakly distinguishing on addable and low-genus classes.

Addability and genus constraints are key conditions for these properties.

Abstract

A graph polynomial $P$ is weakly distinguishing if for almost all finite graphs $G$ there is a finite graph $H$ that is not isomorphic to $G$ with $P(G)=P(H)$. It is weakly distinguishing on a graph property $\mathcal{C}$ if for almost all finite graphs $G\in\mathcal{C}$ there is $H \in \mathcal{C}$ that is not isomorphic to $G$ with $P(G)=P(H)$. We give sufficient conditions on a graph property $\mathcal{C}$ for the characteristic, clique, independence, matching, and domination and $\xi$ polynomials, as well as the Tutte polynomial and its specialisations, to be weakly distinguishing on $\mathcal{C}$. One such condition is to be addable and small in the sense of C. McDiarmid, A. Steger and D. Welsh (2005). Another one is to be of genus at most $k$.