  We prove a number of results around kernelization of problems parameterized
by the size of a given vertex cover of the input graph. We provide three sets
of simple general conditions characterizing problems admitting kernels of
polynomial size. Our characterizations not only give generic explanations for
the existence of many known polynomial kernels for problems like q-Coloring,
Odd Cycle Transversal, Chordal Deletion, Eta Transversal, or Long Path,
parameterized by the size of a vertex cover, but also imply new polynomial
kernels for problems like F-Minor-Free Deletion, which is to delete at most k
vertices to obtain a graph with no minor from a fixed finite set F.
  While our characterization captures many interesting problems, the
kernelization complexity landscape of parameterizations by vertex cover is much
more involved. We demonstrate this by several results about induced subgraph
and minor containment testing, which we find surprising. While it was known
that testing for an induced complete subgraph has no polynomial kernel unless
NP is in coNP/poly, we show that the problem of testing if a graph contains a
complete graph on t vertices as a minor admits a polynomial kernel. On the
other hand, it was known that testing for a path on t vertices as a minor
admits a polynomial kernel, but we show that testing for containment of an
induced path on t vertices is unlikely to admit a polynomial kernel.
