Given a minor-closed graph class ${\cal G}$, the (minor) obstruction of ${\cal G}$ is the set of all minor-minimal graphs not in ${\cal G}$. Given a non-negative integer $k$, we define the $k$-apex of ${\cal A}$ as the class containing every graph $G$ with a set $S$ of vertices whose removal from $G$ gives a graph on ${\cal G}$. We prove that every obstruction of the $k$-apex of ${\cal G}$ has size bounded by some 4-fold exponential function of $p(k)$ where p is a polynomial function whose degree depends on the size of the minor-obstructions of ${\cal G}$. This bound drops to a 2-fold exponential one when ${\cal G}$ excludes some apex graph as a minor (i.e., a graph in the $1$-apex of planar graphs).

Joint work with Ignasi Sau and Giannos Stamoulis.