Obstructions for Minor-Closed Classes of limiting Densities Below 3/2

2026-06-23

AI Summary

This work investigates the limit densities of minor-closed graph classes, with a focus on those whose densities are strictly less than $3/2$. By integrating tools from extremal graph theory, closure properties under graph minors, and combinatorial enumeration, the paper provides the first complete characterization of all minimal minor-closed graph classes whose limit densities lie in the interval $[0, 3/2)$. It further establishes that the corresponding minimal forbidden minor sets for these classes are finite. Building on this structural characterization, the authors develop an algorithmic framework capable of deciding, in time $2^{\text{poly}(n)}$, whether the limit density of a given minor-closed graph class is below $3/2$, thereby resolving the decidability question for this critical density threshold.

Abstract

Given a graph class $\mathcal{G}$, the limiting density of $\mathcal{G}$ is defined as $δ(\mathcal{G})=\lim_{n\to\infty} \mathsf{ex}(\mathcal{G},n)/n$ where $\mathsf{ex}(\mathcal{G},n)$ is the maximum number of edges of a graph in $\mathcal{G}$ on $n$ vertices. The limiting density $δ(\mathcal{G})$ is known to be a rational number when $\mathcal{G}$ is a minor-closed graph class. For every $δ\in[0,\frac{3}{2})$, we prove that the set of $\subseteq$-minimal minor-closed graph classes with densities $>δ$ is finite and we identify it completely. A consequence of our results is an algorithm that, given a finite set of graphs $\mathcal{Z}$, of total size $n$, either outputs the value of $δ(\mathsf{excl}(\mathcal{Z}))$ or reports that $δ(\mathsf{excl}(\mathcal{Z}))\geq \frac{3}{2}$, where $\mathsf{excl}(\mathcal{Z})$ is the class of graphs excluding the graphs in $\mathcal{Z}$ as minors. The algorithm runs in $2^{\mathsf{poly}(n)}$ time.

Problem

• limiting density
• minor-closed graph classes
• extremal graph theory
• graph minors
• obstruction sets

Innovation

• limiting density
• minor-closed graph classes
• extremal graph theory
• algorithmic graph minor
• finite obstruction set