k -apices of Minor-closed Graph Classes. II. Parameterized Algorithms

Abstract

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