Block-weighted random graphs: planar and beyond
Abstract
We investigate random connected graphs from a block-stable class whose distribution is weighted based on the number of -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.
Cite
@article{arxiv.2602.23912,
title = {Block-weighted random graphs: planar and beyond},
author = {Mihyun Kang and Zéphyr Salvy and Ronen Wdowinski},
journal= {arXiv preprint arXiv:2602.23912},
year = {2026}
}
Comments
The authors have been informed that some of the results in the preprint were discussed in [Stufler 2020, Probability Surveys]