Prof. Meirav Zehavi

In the Directed Feedback Vertex Set (DFVS) problem, we are given as input a directed graph D and an integer k, and the objective is to check whether there exists a set S of at most k vertices such that F = D − S is a directed acyclic graph (DAG). Determining whether DFVS admits a polynomial kernel (parameterized by the solution size) is one of the most important open problems in parameterized complexity. In this article, we give a polynomial kernel for DFVS parameterized by the solution size plus the size of any treewidth-η modulator, for any positive integer η. We also give a polynomial kernel for the problem, which we call Vertex Deletion to treewidth-η DAG, where given as input a directed graph D and a positive integer k, the objective is to decide whether there exists a set of at most k vertices, say S, such that D − S is a DAG and the treewidth¹ of D − S is at most η.