Block-weighted random graphs: planar and beyond

TL;DR

This paper studies weighted random graphs based on their block structure, revealing phase transitions and typical block sizes, especially in planar graphs, and extends understanding of their enumeration and structure.

Contribution

It introduces a decorated block tree framework and analyzes phase transitions in block-weighted classes, providing new enumeration results and insights into block sizes.

Findings

Identifies phase transitions in the singular behaviour of generating functions.

Determines typical sizes of largest blocks in different regimes.

Strengthens known results on block sizes in random planar graphs.

Abstract

We investigate random connected graphs from a block-stable class whose distribution is weighted based on the number of $2$-connected components, or blocks. This includes the class of planar graphs. For this, we develop a notion of a decorated block tree. Following similar ideas to Fleurat and the second author on block-weighted planar maps, we find a phase transition in the singular behaviour of the appropriate generating function and in the typical structure of the block tree. Moreover, for certain block-stable classes (including planar graphs), we obtain precise enumeration results and determine also the typical sizes of the largest blocks in subcritical, critical, and supercritical regimes. It strengthens previously known results on block sizes in uniform random planar graphs.