Kernelization for feedback vertex set via elimination distance to a forest

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 NP⁄⊆coNP/poly, we prove that for any minor-closed graph class G, FEEDBACK VERTEX SET parameterized by the size of a modulator to G has a polynomial kernel if and only if G has bounded elimination distance to a forest. This captures and generalizes all existing kernels for structural parameterizations of the FEEDBACK VERTEX SET problem.