Kernelization for Feedback Vertex Set via Elimination Distance to a Forest

TL;DR

This paper characterizes when the Feedback Vertex Set problem admits polynomial kernels based on the elimination distance to a forest, generalizing previous results and linking kernelization to graph structural parameters.

Contribution

It establishes a complete characterization of polynomial kernel existence for Feedback Vertex Set parameterized by modulators to minor-closed classes, based on elimination distance to a forest.

Findings

Polynomial kernels exist if and only if the graph class has bounded elimination distance to a forest.

Generalizes all known kernels for structural parameterizations of Feedback Vertex Set.

Links kernelization feasibility to the structural property of elimination distance.

Abstract

We study efficient preprocessing for the undirected Feedback Vertex Set problem, a fundamental problem in graph theory which asks for a minimum-sized vertex set whose removal yields an acyclic graph. More precisely, we aim to determine for which parameterizations this problem admits a polynomial kernel. While a characterization is known for the related Vertex Cover problem based on the recently introduced notion of bridge-depth, it remained an open problem whether this could be generalized to Feedback Vertex Set. The answer turns out to be negative; the existence of polynomial kernels for structural parameterizations for Feedback Vertex Set is governed by the elimination distance to a forest. Under the standard assumption that NP is not a subset of coNP/poly, we prove that for any minor-closed graph class $\mathcal G$, Feedback Vertex Set parameterized by the size of a modulator to $\mathcal G$ has a polynomial kernel if and only if $\mathcal G$ has bounded elimination distance to a forest. This captures and generalizes all existing kernels for structural parameterizations of the Feedback Vertex Set problem.