Asymptotic Properties of Some Minor-Closed Classes of Graphs

Abstract

Let${\cal A}$be a minor-closed class of labelled graphs, and let${\cal G}_{n}$be a random graph sampled uniformly from the set ofn-vertex graphs of${\cal A}$. Whennis large, what is the probability that${\cal G}_{n}$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${\cal A}$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 the dominant singularity of the generating functionC(z) that counts connected graphs of${\cal A}$. 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 merits a systematic investigation.