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Asymptotic properties of some minor-closed classes of graphs

Mireille Bousquet-Mélou, Kerstin Weller

TL;DR

This work advances the asymptotic theory of random graphs drawn from minor-closed classes by systematically studying how the nature of excluded minors—especially when some are not 2-connected—shapes connectivity, component counts, and root-component sizes through generating-function singularities. The authors develop and apply a rich toolkit comprising singularity analysis and Hayman-admissibility, including a new uniform extension for exp(C(z)) under parameterization, to derive detailed limit laws (Gaussian, Beta, Gamma, Poisson-Dirichlet) for N_n, S_n, and L_n across diverse regimes (tree-dominated, logarithmic, simple-pole, and algebraic singularities). Key contributions include explicit asymptotics for classes with bounded, path/caterpillar forests, and bowtie/diamond exclusions, plus a comprehensive framework linking singularity type to component structure. The results offer a first-step classification of minor-closed classes by asymptotic behaviour and motivate systematic study of the root component and other parameters, with practical implications for random generation and probabilistic combinatorics of graph families.

Abstract

Let A be a minor-closed class of labelled graphs, and let G_n be a random graph sampled uniformly from the set of n-vertex graphs of A. When n is large, what is the probability that G_n is connected? How many components does it have? How large is its biggest component? Thanks to the work of McDiarmid and his collaborators, these questions are now solved when all excluded minors are 2-connected. Using exact enumeration, we study a collection of classes A excluding non-2-connected minors, and show that their asymptotic behaviour may be rather different from the 2-connected case. This behaviour largely depends on the nature of dominant singularity of the generating function C(z) that counts connected graphs of A. We classify our examples accordingly, thus taking a first step towards a classification of minor-closed classes of graphs. Furthermore, we investigate a parameter that has not received any attention in this context yet: the size of the root component. It follows non-gaussian limit laws (beta and gamma), and clearly deserves a systematic investigation.

Asymptotic properties of some minor-closed classes of graphs

TL;DR

This work advances the asymptotic theory of random graphs drawn from minor-closed classes by systematically studying how the nature of excluded minors—especially when some are not 2-connected—shapes connectivity, component counts, and root-component sizes through generating-function singularities. The authors develop and apply a rich toolkit comprising singularity analysis and Hayman-admissibility, including a new uniform extension for exp(C(z)) under parameterization, to derive detailed limit laws (Gaussian, Beta, Gamma, Poisson-Dirichlet) for N_n, S_n, and L_n across diverse regimes (tree-dominated, logarithmic, simple-pole, and algebraic singularities). Key contributions include explicit asymptotics for classes with bounded, path/caterpillar forests, and bowtie/diamond exclusions, plus a comprehensive framework linking singularity type to component structure. The results offer a first-step classification of minor-closed classes by asymptotic behaviour and motivate systematic study of the root component and other parameters, with practical implications for random generation and probabilistic combinatorics of graph families.

Abstract

Let A be a minor-closed class of labelled graphs, and let G_n be a random graph sampled uniformly from the set of n-vertex graphs of A. When n is large, what is the probability that G_n is connected? How many components does it have? How large is its biggest component? Thanks to the work of McDiarmid and his collaborators, these questions are now solved when all excluded minors are 2-connected. Using exact enumeration, we study a collection of classes A excluding non-2-connected minors, and show that their asymptotic behaviour may be rather different from the 2-connected case. This behaviour largely depends on the nature of dominant singularity of the generating function C(z) that counts connected graphs of A. We classify our examples accordingly, thus taking a first step towards a classification of minor-closed classes of graphs. Furthermore, we investigate a parameter that has not received any attention in this context yet: the size of the root component. It follows non-gaussian limit laws (beta and gamma), and clearly deserves a systematic investigation.

Paper Structure

This paper contains 15 sections, 39 theorems, 188 equations, 18 figures, 1 table.

Table of Contents

  1. Introduction
  2. "Generatingfunctionology" for graphs
  3. Classes defined by 2-connected excluded minors
  4. When trees dominate: $\boldsymbol {C(z)}$ converges at $\boldsymbol {\rho}$
  5. Excluding the diamond and the bowtie: a logarithmic singularity
  6. Hayman admissibility and extensions
  7. Graphs with bounded components: $\boldsymbol {C(z)}$ is a polynomial
  8. Forests of paths or caterpillars: a simple pole in $\boldsymbol {C(z)}$
  9. Graphs with maximum degree 2: a simple pole and a logarithm in $\boldsymbol { C(z)}$
  10. Excluding the bowtie: a singularity in $\boldsymbol{(1-z/\rho)^{-1/2}}$
  11. Final comments and further questions
  12. Random generation
  13. The nature of the dominant singularities of $\boldsymbol{C(z)}$
  14. More examples and predictions
  15. Other parameters

Key Result

Proposition 1

The generating functions $C(z)$ and $A(z)=e^{C(z)}$ are finite at their (positive) radius of convergence $\rho$. Moreover, the sequence $(a_n/n!)_n$ is smooth. The probability that $\mathcal{G}_n$ is connected tends to $1/A(\rho)$, which is clearly in $(0,1)$. In fact, this limit is also larger than

Figures (18)

  • Figure 1: A zoo of graphs. Top: the 3-star, the triangle $K_3$, the bowtie and the diamond. Bottom: A caterpillar and the 4-spoon (a $k$-spoon consists of a "handle" formed of $k$ edges, to which a triangle is attached).
  • Figure 2: A random forest of size $n=1165$. It has two connected components.
  • Figure 3: A core having several cycles and avoiding the $k$-spoon cannot contain a path of length $3k-1$.
  • Figure 4: A random graph of size $n=541$ avoiding the diamond, the bowtie and the 20-spoon.
  • Figure 5: Graphs with several cycles avoiding the $2$-spoon.
  • ...and 13 more figures

Theorems & Definitions (73)

  • Proposition 1: The number of graphs --- when excluded minors are 2-connected
  • Example 2
  • Proposition 3: Number of components --- when excluded minors are 2-connected
  • Lemma 4
  • proof
  • Proposition 5: The root component and the largest component --- when excluded minors are 2-connected
  • proof
  • Proposition 6: The number of graphs --- when trees dominate
  • proof
  • Proposition 7: Number of components --- when trees dominate
  • ...and 63 more