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

Let 𝒢 be a minor-closed 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 denote by 𝒜k (𝒢) the set of all graphs that are k-apices of 𝒢. In the first paper of this series, we obtained upper bounds on the size of the graphs in the minor-obstruction set of 𝒜k (𝒢), i.e., the minor-minimal set of graphs not belonging to 𝒜k (𝒢). In this article, we provide an algorithm that, given a graph G on n vertices, runs in time 2poly(k) ⋅ n3 and either returns a set S certifying that G ∈ 𝒜k (𝒢), or reports that G ∉ 𝒜k (𝒢). Here poly is a polynomial function whose degree depends on the maximum size of a minor-obstruction of 𝒢. In the special case where 𝒢 excludes some apex graph as a minor, we give an alternative algorithm running in 2poly(k) ⋅ n2-time.