A single-exponential FPT algorithm for the K 4-minor cover problem

Abstract

Given an input graph G on n vertices and an integer k, the parameterized K 4 -minor cover problem asks whether there is a set S of at most k vertices whose deletion results in a K 4-minor free graph or, equivalently, in a graph of treewidth at most 2. The problem can thus also be called Treewidth-2 Vertex Deletion. This problem is inspired by two well-studied parameterized vertex deletion problems, Vertex Cover and Feedback Vertex Set, which can be expressed as Treewidth- t Vertex Deletion problems: t = 0 for Vertex Cover and t∈=∈1 for Feedback Vertex Set. While a single-exponential FPT algorithm has been known for a long time for Vertex Cover, such an algorithm for Feedback Vertex Set was devised comparatively recently. While it is known to be unlikely that Treewidth- t Vertex Deletion can be solved in time c o(k)•n O(1), it was open whether the K 4 -minor cover could be solved in single-exponential FPT time, i.e. in c k •n O(1) time. This paper answers this question in the affirmative. © 2012 Springer-Verlag.