Kernelization for Feedback Vertex Set via Elimination Distance to a Forest

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