  We investigate polynomial-time preprocessing for the problem of hitting
forbidden minors in a graph, using the framework of kernelization. For a fixed
finite set of connected graphs F, the F-Deletion problem is the following:
given a graph G and integer k, is it possible to delete k vertices from G to
ensure the resulting graph does not contain any graph from F as a minor?
Earlier work by Fomin, Lokshtanov, Misra, and Saurabh [FOCS'12] showed that
when F contains a planar graph, an instance (G,k) can be reduced in polynomial
time to an equivalent one of size $k^{O(1)}$. In this work we focus on
structural measures of the complexity of an instance, with the aim of giving
nontrivial preprocessing guarantees for instances whose solutions are large.
Motivated by several impossibility results, we parameterize the F-Deletion
problem by the size of a vertex modulator whose removal results in a graph of
constant treedepth $\eta$.
  We prove that for each set F of connected graphs and constant $\eta$, the
F-Deletion problem parameterized by the size of a treedepth-$\eta$ modulator
has a polynomial kernel. Our kernelization is fully explicit and does not
depend on protrusion reduction or well-quasi-ordering, which are sources of
algorithmic non-constructivity in earlier works on F-Deletion. Our main
technical contribution is to analyze how models of a forbidden minor in a graph
G with modulator X, interact with the various connected components of G-X. By
bounding the number of different types of behavior that can occur by a
polynomial in |X|, we obtain a polynomial kernel using a recursive
preprocessing strategy. Our results extend earlier work for specific instances
of F-Deletion such as Vertex Cover and Feedback Vertex Set. It also generalizes
earlier preprocessing results for F-Deletion parameterized by a vertex cover,
which is a treedepth-one modulator.
