Résumé
Given an input graph G on n vertices and an integer k, the parameterized K4-minor cover problem asks whether there is a set S of at most k vertices whose deletion results in a K4-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 K4-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