Title:

Connectivity and related properties for graph classes

There has been much recent interest in random graphs sampled uniformly from the set of (labelled) graphs on n vertices in a suitably structured class A. An important and wellstudied example of such a suitable structure is bridgeaddability, introduced in 2005 by McDiarmid et al. in the course of studying random planar graphs. A class A is bridgeaddable when the following holds: if we take any graph G in A and any pair u,v of vertices that are in different components in G, then the graph G′ obtained by adding the edge uv to G is also in A. It was shown that for a random graph sampled from a bridgeaddable class, the probability that it is connected is always bounded away from 0, and the number of components is bounded above by a Poisson law. What happens if ’bridgeaddable’ is replaced by something weaker? In this thesis, this question is explored in several different directions. After an introductory chapter and a chapter on generating function methods presenting standard techniques as well as some new technical results needed later, we look at minorclosed, labelled classes of graphs. The excluded minors are always assumed to be connected, which is equivalent to the class A being decomposable  a graph is in A if and only if every component of the graph is in A. When A is minorclosed, decomposable and bridgeaddable various properties are known (McDiarmid 2010), generalizing results for planar graphs. A minorclosed class is decomposable and bridgeaddable if and only if all excluded minors are 2connected. Chapter 3 presents a series of examples where the excluded minors are not 2connected, analysed using generating functions as well as techniques from graph theory. This is a step towards a classification of connectivity behaviour for minorclosed classes of graphs. In contrast to the bridgeaddable case, different types of behaviours are observed. Chapter 4 deals with a new, more general vari ant of bridgeaddability related to edgeexpander graphs. We will see that as long as we are allowed to introduce ’sufficiently many’ edges between components, the number of components of a random graph can still be bounded above by a Pois son law. In this context, random forests in Kn,n are studied in detail. Chapter 5 takes a different approach, and studies the class of labelled forests where some vertices belong to a specified stable set. A weighting parameter y for the vertices belonging to the stable set is introduced, and a graph is sampled with probability proportional to y*s where s is the size of its stable set. The behaviour of this class is studied for y tending to ∞. Chapters 6 concerns random graphs sampled from general decomposable classes. We investigate the minimum size of a component, in both the labelled and the unlabelled case.
