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

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.