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Abstract

Given an input graph G on \(n\) vertices and an integer k, the parameterized \textscK_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 \textsc{Treewidth-2 Vertex Deletion}. This problem is inspired by two well-studied parameterized vertex deletion problems, \textsc{Vertex Cover} and \textsc{Feedback Vertex Set}, which can be expressed as \textsc{Treewidth-t Vertex Deletion} problems: t=0 for {\sc Vertex Cover} and t=1 for {\sc Feedback Vertex Set}. While a single-exponential FPT algorithm has been known for a long time for \textsc{Vertex Cover}, such an algorithm for \textsc{Feedback Vertex Set} was devised comparatively recently. While it is known to be unlikely that \textsc{Treewidth-t Vertex Deletion} can be solved in time c^{o(k)}⋅ n^{O(1)}, it was open whether the \textsc{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.