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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 $k$ verticessuch 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 $\mathcal{G}.$ Weprove 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 ofthe minor-obstructions of $\mathcal{G}.$ This bound drops to$2^{2^{\mathsf{poly}(k)}}$ when $\mathcal{G}$ excludes some apex graph as aminor.

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