Let {\cal G} be a  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 prove that if {\cal  {\cal G} is minor-closed, then there is{\cal  {\cal  {\cal  {\cal  {\cal   an algorithm that (k)}\cdot n^3-time and } either returns a set S certifying that G is a k-apex of {\cal  {\cal G}  or reports that such a set does not exist, in 2^{{\sf poly(k)}\cdot n^3 time} \notin {\cal  {\cal   . Here {\sf poly} is a polynomial function whose degree depends on the maximum size of a minor-obstruction of {\cal G} , i. e., the minor-minimal set of graphs not belonging to%DIFDELCMD < {\cal %%% G . In the special case where {\cal G} excludes some apex graph as a minor, we give an alternative algorithm running in 2^{{\sf poly}(k)}\cdot n^2 time .
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 . In the first paper of this series we obtained upper bounds on the size of the graphs in the minor-obstruction set of{\cal A _k ({\cal G ), i.e., the minor-minimal set of graphs not belonging to{\cal A _k ({\cal G ). In this article we provide an algorithm that , given a graph G on n vertices, runs in 2^{{\sf poly(k)}\cdot n^3-time and } either returns a set S certifying that G \in {\cal A _k ({\cal G} ), or reports that (k)}\cdot n^3 time} G\notin {\cal A _k ({\cal G ) . Here {\sf poly} is a polynomial function whose degree depends on the maximum size of a minor-obstruction of {\cal G} %DIFDELCMD < {\cal %%% . In the special case where {\cal G} excludes some apex graph as a minor, we give an alternative algorithm running in 2^{{\sf poly}(k)}\cdot n^2 -time .
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prove that if
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. In the first paper of this series we obtained upper bounds on the size of the graphs in the minor-obstruction set of
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_k (
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), i.e., the minor-minimal set of graphs not belonging to
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). In this article we provide
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, given a graph G on n vertices, runs in 2^{{\sf poly
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is a k-apex of
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such a set does not exist, in 2^{{\sf poly
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, i. e., the minor-minimal set of graphs not belonging to
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time
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