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

Abstract

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 𝒜 k ( 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 𝒜 k ( 𝒢 ), i.e., the minor-minimal set of graphs not belonging to 𝒜 k ( 𝒢 ). In this article, we provide an algorithm that, given a graph G on n vertices, runs in time 2 poly (k) ⋅ n 3 and either returns a set S certifying that G ∈ 𝒜 k ( 𝒢 ), or reports that G ∉ 𝒜 k ( 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 2 poly (k) ⋅ n 2 -time.