A single-exponential FPT algorithm for the $K_4$-minor cover problem

TL;DR

This paper presents a single-exponential fixed-parameter tractable algorithm for the K_4-minor cover problem, solving an open question about its computational complexity.

Contribution

It introduces the first single-exponential FPT algorithm for the K_4-minor cover problem, advancing understanding of graph minor deletion problems.

Findings

The problem can be solved in c^k * n^{O(1)} time.

Provides a new algorithmic approach for K_4-minor cover.

Answers an open question in parameterized complexity.

Abstract

Given an input graph G 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. This problem is inspired by two well-studied parameterized vertex deletion problems, Vertex Cover and Feedback Vertex Set, which can also 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 \textsc{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 problem 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.