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arXiv · 2206.04387

Kernelization for Feedback Vertex Set via Elimination Distance to a Forest

David Dekker·Bart M. P. Jansen
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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.

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David Dekker, Bart M. P. Jansen. 2022-06-09. Kernelization for Feedback Vertex Set via Elimination Distance to a Forest. https://arxiv.org/abs/2206.04387

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