A Turing kernelization dichotomy for structural parameterizations of F-Minor-Free Deletion

For a fixed finite family of graphs F, the F-Minor-Free Deletion problem takes as input graph G and integer ℓ asks whether size-ℓ vertex set X exists such that G−X is F-minor-free. {K2}-Minor-Free {K3}-Minor-Free encode Vertex Cover Feedback Set respectively. When parameterized by feedback number these two problems are known to admit polynomial kernelization. We show {P3}-Minor-Free MK[2]-hard. This rules out existence kernel assuming NP⊈coNP/poly. Our hardness result generalizes any F containing only with connected component at least 3 vertices, using parameter vertex-deletion distance treewidth min⁡tw(F), where min⁡tw(F) denotes minimum in F. all other families we present Turing results extend F-Subgraph-Free Deletion.