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

48 pages and 12 figures. arXiv admin note: text overlap with arXiv:2004.12692