Uniform kernelization complexity of hitting forbidden minors

The F-MINOR-FREE DELETION problem asks, for a fixed set F and an input consisting of a graph G and integer k, whether κ vertices can be removed from G such that the resulting graph does not contain any member of F as a minor. At FOCS 2012, Fomin et al. showed that the special case when F contains at least one planar graph has a kernel of size f (F) · κ^g(F) for some functions f and g. They left open whether this PLANAR F-MINOR-FREE DELETION problem has kernels whose size is uniformly polynomial, of the form f (F) · κ^c for some universal constant c. We prove that some PLANAR F-MINOR-FREE DELETION problems do not have uniformly polynomial kernels (unless NP ⊆ coNP/poly), not even when parameterized by the vertex cover number. On the positive side, we consider the problem of determining whether κ vertices can be removed to obtain a graph of treedepth at most η. We prove that this problem admits uniformly polynomial kernels with O(κ⁶) vertices for every fixed η.