Random graphs from a weighted minor-closed class

TL;DR

This paper extends the study of random graphs from minor-closed classes to a weighted framework, analyzing properties like connectivity and 2-core structure with combinatorial and probabilistic methods.

Contribution

It introduces a general weighted model for random graphs in well-behaved classes, extending previous results and providing new insights into graph properties.

Findings

Results on connectivity probabilities extend to weighted classes.

New insights into the 2-core structure of weighted random graphs.

Framework applies to various graph classes like planar, surface-embeddable, and forests.

Abstract

There has been much recent interest in random graphs sampled uniformly from the n-vertex graphs in a suitable minor-closed class, such as the class of all planar graphs. Here we use combinatorial and probabilistic methods to investigate a more general model. We consider random graphs from a `well-behaved' class of graphs: examples of such classes include all minor-closed classes of graphs with 2-connected excluded minors (such as forests, series-parallel graphs and planar graphs), the class of graphs embeddable on any given surface, and the class of graphs with at most k vertex-disjoint cycles. Also, we give weights to edges and components to specify probabilities, so that our random graphs correspond to the random cluster model, appropriately conditioned. We find that earlier results extend naturally in both directions, to general well-behaved classes of graphs, and to the weighted framework, for example results concerning the probability of a random graph being connected; and we also give results on the 2-core which are new even for the uniform (unweighted) case.