[Paper Review] On the growth rate of minor-closed classes of graphs

Olivier Bernardi, Marc Noy+1 more|ArXiv.org|Oct 16, 2007

Advanced Graph Theory Research12 references11 citations

TL;DR

This paper investigates the growth rates of minor-closed classes of labelled graphs, establishing that the growth rate of such classes is highly constrained: no minor-closed class exhibits factorial growth with a constant between 0 and 1 or between 1 and approximately 1.76. The authors classify growth rates into distinct categories—factorial, almost-factorial, semi-factorial, exponential, polynomial, and constant—based on the inclusion of specific graph families, and prove that the set of possible growth constants has gaps and limit points, with implications for well-quasi-ordering and smoothness in graph classes.

ABSTRACT

A minor-closed class of graphs is a set of labelled graphs which is closed under isomorphism and under taking minors. For a minor-closed class $C$, we let $c_n$ be the number of graphs in $C$ which have $n$ vertices. A recent result of Norine et al. shows that for all minor-closed class $C$, there is a constant $r$ such that $c_n < r^n n!$. Our main results show that the growth rate of $c_n$ is far from arbitrary. For example, no minor-closed class $C$ has $c_n= r^{n+o(n)} n!$ with $0 < r < 1$ or $1 < r < ξ\approx 1.76$.

Motivation & Objective

• To classify the possible growth rates of minor-closed classes of labelled graphs, extending prior work on hereditary classes.
• To determine which growth rate categories (e.g., factorial, exponential) are achievable and under what structural conditions on the excluded minors.
• To investigate the existence and nature of gaps in the set of possible growth constants for minor-closed graph classes.
• To explore the implications of well-quasi-ordering for the existence of infinite decreasing sequences of growth constants.
• To examine the conditions under which a minor-closed class has a well-defined growth constant and whether such classes are smooth.

Proposed method

• The authors classify minor-closed graph classes based on the inclusion of specific graph families (e.g., paths, star forests, matchings, stars) as a structural criterion for growth rate.
• They use exponential generating functions and singularity analysis to derive asymptotic growth estimates, particularly focusing on the radius of convergence of generating functions.
• The proof relies on known results from graph minor theory, including the finite excluded minor characterization and the well-quasi-ordering conjecture.
• They analyze the growth constants via the inverse of the radius of convergence of generating functions, particularly for families of rooted trees and their variants.
• The authors construct sequences of graph classes with converging growth constants and use convexity and coefficient comparison to bound the limits.
• They relate the existence of infinite decreasing sequences of growth constants to the well-quasi-ordering conjecture for minor-closed classes.

Experimental results

Research questions

• RQ1 Can every minor-closed class of graphs be assigned a well-defined growth constant, defined as the limit of (g_n / n!)^{1/n}?
• RQ2 Are there any gaps in the set of possible growth constants, and if so, which intervals are excluded?
• RQ3 Is the interval (ξ, 2), where ξ ≈ 1.76, a gap in the set of growth constants?
• RQ4 Can any algebraic number other than powers of 2 be a growth constant for a minor-closed graph class?
• RQ5 Under what conditions is a minor-closed class smooth, i.e., does the ratio g_n / (n g_{n-1}) converge?

Key findings

• No minor-closed class has factorial growth with a growth constant c satisfying 0 < c < 1 or 1 < c < ξ ≈ 1.76, establishing two gaps in the spectrum of growth constants.
• The set of growth constants has limit points, with the smallest known limit point being ν ≈ 2.24, the inverse of the smallest positive root of z exp(z/(1−z)) = 1.
• For each nonnegative integer k, the value 2^k is a growth constant, and these are the only known algebraic numbers that are growth constants.
• The growth rate of a minor-closed class is completely classified into six categories based on the inclusion of path forests, star forests, matchings, stars, single-edge graphs, or trivial graphs.
• If the well-quasi-ordering conjecture for minor-closed classes holds, then no infinite decreasing sequence of growth constants can exist, implying that gaps must exist beyond known intervals.
• The growth constant of a minor-closed class is equal to the inverse of the radius of convergence of its exponential generating function, and this constant is well-defined for classes whose excluded minors are 2-connected.