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Limits of chordal graphs with bounded tree-width

Jordi Castellví, Benedikt Stufler

TL;DR

This work analyzes random $k$-connected chordal graphs with bounded tree-width, proving a Gromov-Hausdorff-Prokhorov scaling limit to the Brownian tree and a quenched local limit to an infinite rooted chordal graph. The authors develop a probabilistic sampling framework via Boltzmann sampling and a blow-up construction from a two-type Bienaym\'e–Galton–Watson tree with finite exponential moments, yielding precise asymptotics and local limit results. They establish subcritical generating function relations, tail bounds for diameter, and a robust local-to-global limit theory, unifying various models under a Brownian-tree universality class. The methods offer explicit sampling procedures and rigorous limit theorems, with potential implications for understanding the asymptotic geometry and local structure of chordal graphs with constrained tree-width.

Abstract

We study random $k$-connected chordal graphs with bounded tree-width. Our main results are scaling limits and quenched local limits.

Limits of chordal graphs with bounded tree-width

TL;DR

This work analyzes random -connected chordal graphs with bounded tree-width, proving a Gromov-Hausdorff-Prokhorov scaling limit to the Brownian tree and a quenched local limit to an infinite rooted chordal graph. The authors develop a probabilistic sampling framework via Boltzmann sampling and a blow-up construction from a two-type Bienaym\'e–Galton–Watson tree with finite exponential moments, yielding precise asymptotics and local limit results. They establish subcritical generating function relations, tail bounds for diameter, and a robust local-to-global limit theory, unifying various models under a Brownian-tree universality class. The methods offer explicit sampling procedures and rigorous limit theorems, with potential implications for understanding the asymptotic geometry and local structure of chordal graphs with constrained tree-width.

Abstract

We study random -connected chordal graphs with bounded tree-width. Our main results are scaling limits and quenched local limits.
Paper Structure (19 sections, 16 theorems, 103 equations, 3 figures, 1 algorithm)

This paper contains 19 sections, 16 theorems, 103 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Local convergence
  4. Global convergence
  5. Sampling Procedures
  6. Generating functions of labelled chordal graphs with bounded tree-width
  7. Boltzmann sampling procedure
  8. Blow-up sampling procedure
  9. Structural analysis
  10. De-rooting the $k$-clique rooted random graphs
  11. Proof of the local limit
  12. Local convergence of random trees
  13. Local convergence of random decorated trees
  14. Local convergence of random graphs
  15. Proof of the scaling limit
  16. ...and 4 more sections

Key Result

Theorem 1

There is a constant $\kappa_{t, k}>0$ such that in the Gromov--Hausdorff--Prokhorov sense as $n \to \infty$.

Figures (3)

  • Figure 1: Example of a $2$-type Bienaymé--Galton--Watson tree.
  • Figure 2: The non-empty decorations of the tree in Figure \ref{['fig:tree']}. The thick oriented edges are the ordered roots.
  • Figure 3: Blow-up of the tree in Figure \ref{['fig:tree']} with the decoration in Figure \ref{['fig:tree_decorations']}.

Theorems & Definitions (32)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 4
  • proof
  • Proposition 5
  • proof
  • Proposition 6
  • proof
  • Lemma 7
  • ...and 22 more