Asymptotic properties of some minor-closed classes of graphs

TL;DR

This paper explores the asymptotic properties of minor-closed graph classes, especially those excluding non-2-connected minors, revealing diverse behaviors influenced by the generating function's singularity and introducing new insights into the root component size distribution.

Contribution

It extends the understanding of asymptotic graph properties to classes excluding non-2-connected minors and classifies behaviors based on generating function singularities.

Findings

Asymptotic behaviors vary with the nature of the dominant singularity.

Root component size follows non-Gaussian limit laws such as beta and gamma.

Provides a classification framework for minor-closed graph classes.

Abstract

Let A be a minor-closed class of labelled graphs, and let G_n be a random graph sampled uniformly from the set of n-vertex graphs of A. When n is large, what is the probability that G_n is connected? How many components does it have? How large is its biggest component? Thanks to the work of McDiarmid and his collaborators, these questions are now solved when all excluded minors are 2-connected. Using exact enumeration, we study a collection of classes A excluding non-2-connected minors, and show that their asymptotic behaviour may be rather different from the 2-connected case. This behaviour largely depends on the nature of dominant singularity of the generating function C(z) that counts connected graphs of A. We classify our examples accordingly, thus taking a first step towards a classification of minor-closed classes of graphs. Furthermore, we investigate a parameter that has not received any attention in this context yet: the size of the root component. It follows non-gaussian limit laws (beta and gamma), and clearly deserves a systematic investigation.