Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Kang, Mihyun, Salvy, Zéphyr, Wdowinski, Ronen
Institutions: Graz University of Technology, TU Wien
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Austrian Science Fund
OpenAlex ID: W7133235356
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