Tractability of König edge deletion problems

Abstract

A graph is said to be a König graph if the size of its maximum matching is equal to the size of its minimum vertex cover. The König Edge Deletion problem asks if in a given graph there exists a set of at most k edges whose deletion results in a König graph. While the vertex version of the problem (König vertex deletion) has been shown to be fixed-parameter tractable more than a decade ago, the fixed-parameter-tractability of the König Edge Deletion problem has been open since then, and has been conjectured to be W[1]-hard in several papers. In this paper, we settle the conjecture by proving it W[1]-hard. We prove that a variant of this problem, where we are given a graph G and a maximum matching M and we want a k-sized König edge deletion set that is disjoint from M, is fixed-parameter-tractable.