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

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