k-apices of minor-closed graph classes. I. Bounding the obstructions

TL;DR

This paper establishes a tower-type upper bound on the size of minimal obstructions for graphs that become part of a minor-closed class after removing at most k vertices, with improved bounds for certain subclasses.

Contribution

It provides a double-exponential bound on the size of obstructions for k-apex graphs in minor-closed classes, refining previous understanding and identifying cases with tighter bounds.

Findings

Obstruction size is at most a quadruple-exponential function of polynomial(k).

Bounds improve to double-exponential when the class excludes an apex graph as a minor.

The results connect minor-closed class properties with the complexity of their obstructions.

Abstract

Let $\mathcal{G}$ be a minor-closed graph class. We say that a graph $G$ is a $k$-apex of $\mathcal{G}$ if $G$ contains a set $S$ of at most $k$ vertices such that $G\setminus S$ belongs to $\mathcal{G}.$ We denote by $\mathcal{A}_k (\mathcal{G})$ the set of all graphs that are $k$-apices of $\mathcal{G}.$ We prove that every graph in the obstruction set of $\mathcal{A}_k (\mathcal{G}),$ i.e., the minor-minimal set of graphs not belonging to $\mathcal{A}_k (\mathcal{G}),$ has size at most $2^{2^{2^{2^{\mathsf{poly}(k)}}}},$ where $\mathsf{poly}$ is a polynomial function whose degree depends on the size of the minor-obstructions of $\mathcal{G}.$ This bound drops to $2^{2^{\mathsf{poly}(k)}}$ when $\mathcal{G}$ excludes some apex graph as a minor.