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.

中文翻译：

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次闭合图类的k个顶点。一，障碍物

令$ {\ cal G} $为次要封闭图类。我们说图$ G $是$ {\ cal G} $的$ k $顶点，如果$ G $包含最多$ k $个顶点的集合$ S $，使得$ G \ setminus S $属于$ {\ cal G}。$我们用$ {\ cal A} _k（{\ cal G}）$表示所有图的集合，这些图是$ {\ cal G}。$$-元的$。 $ {\ cal A} _k（{\ cal G}）的障碍集中的每个图形，即不属于$ {\ cal A} _k（{\ cal G}）的次要图形组， $的大小最大为$ 2 ^ {2 ^ {2 ^ {2 ^ {{\ sf poly}（k）}}}}，$，其中$ {\ sf poly} $是一个多项式函数，其程度取决于大小$ {\ cal G}。$的次要障碍物。当$ {\ cal G} $排除一些顶点图形作为次要对象时，此界限下降到$ 2 ^ {2 ^ {{\ sf poly}（k）}} $。