An FPT-Algorithm for Recognizing k-Apices of Minor-Closed Graph Classes

Abstract : Let G be a graph class. We say that a graph G is a k-apex of G if G contains a set S of at most k vertices such that G \ S belongs to G. We prove that if G is minor-closed, then there is an algorithm that either returns a set S certifying that G is a k-apex of G or reports that such a set does not exist, in 2 poly(k) n 3 time. Here poly is a polynomial function whose degree depends on the maximum size of a minor-obstruction of G, i.e., the minor-minimal set of graphs not belonging to G. In the special case where G excludes some apex graph as a minor, we give an alternative algorithm running in 2 poly(k) n 2 time.