We revisit the topic of polynomial kernels for Vertex Cover relative to
structural parameters. Our starting point is a recent paper due to Fomin and
Str{\o}mme [WG 2016] who gave a kernel with O(∣X∣12) vertices
when X is a vertex set such that each connected component of G−X contains
at most one cycle, i.e., X is a modulator to a pseudoforest. We strongly
generalize this result by using modulators to d-quasi-forests, i.e., graphs
where each connected component has a feedback vertex set of size at most d,
and obtain kernels with O(∣X∣3d+9) vertices. Our result relies
on proving that minimal blocking sets in a d-quasi-forest have size at most
d+2. This bound is tight and there is a related lower bound of
O(∣X∣d+2−ϵ) on the bit size of kernels.
In fact, we also get bounds for minimal blocking sets of more general graph
classes: For d-quasi-bipartite graphs, where each connected component can be
made bipartite by deleting at most d vertices, we get the same tight bound of
d+2 vertices. For graphs whose connected components each have a vertex cover
of cost at most d more than the best fractional vertex cover, which we call
d-quasi-integral, we show that minimal blocking sets have size at most
2d+2, which is also tight. Combined with existing randomized polynomial
kernelizations this leads to randomized polynomial kernelizations for
modulators to d-quasi-bipartite and d-quasi-integral graphs. There are
lower bounds of O(∣X∣d+2−ϵ) and
O(∣X∣2d+2−ϵ) for the bit size of such kernels