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Asymptotic Properties of Some Minor-Closed Classes of Graphs

Published online by Cambridge University Press:  09 July 2014

MIREILLE BOUSQUET-MÉLOU  and
KERSTIN WELLER
MIREILLE BOUSQUET-MÉLOU
Affiliation:
MBM: CNRS, LaBRI, UMR 5800, Université de Bordeaux, 351 Cours de la Libération, 33405 Talence Cedex, France (e-mail: mireille.bousquet@labri.fr)
KERSTIN WELLER
Affiliation:
KW: Department of Statistics, University of Oxford, 1 South Parks Road, Oxford, OX1 3TG, UK (e-mail: weller@stats.ox.ac.uk)
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Abstract

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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 of n-vertex graphs of ${\cal A}$. When n is 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 function C(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.

Keywords

Primary 05A1505C30Secondary 05C8360C05

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 23 , Issue 5: Honouring the Memory of Philippe Flajolet - Part 1 , September 2014 , pp. 749 - 795
Copyright
Copyright © Cambridge University Press 2014 

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