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Block-weighted random graphs: planar and beyond

Mihyun Kang, Zéphyr Salvy, Ronen Wdowinski

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.

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.
Paper Structure (20 sections, 8 theorems, 19 equations, 1 figure, 2 algorithms)

This paper contains 20 sections, 8 theorems, 19 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Motivation and background
  3. Main results
  4. Phase transition in block-weighted block-stable graphs
  5. Enumeration and block sizes in block-weighted planar graphs
  6. Key proof techniques and related work
  7. Plan of the paper
  8. Preliminaries
  9. Notation
  10. Labelled classes
  11. Graphs and their generating functions
  12. Block decomposition of a connected graph
  13. Decorated block trees of connected block-stable graphs
  14. Construction of decorated block tree
  15. Reverse construction
  16. ...and 5 more sections

Key Result

Theorem 1

Let $\mathcal{C}$ be a block-stable class of connected graphs, let $\mathcal{B}$ be its class of $2$-connected components whose EGF we denote by $B(y)$, and let $\rho_B$ be the radius of convergence of $B$. Let Then the following hold.

Figures (1)

  • Figure 1: The image on the left is a connected labelled rooted planar graph $\mathfrak{g}$, with root highlighted in grey. The image on the right is the associated decorated block tree ${T}_\mathfrak{g}$.

Theorems & Definitions (12)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 4
  • Proposition 5
  • Proposition 6
  • Remark 7
  • Remark 8
  • Theorem 9
  • Remark 10
  • ...and 2 more