  We give the first linear kernels for the (Connected) Dominating Set problems
on H-topological minor free graphs. We prove the existence of polynomial time
algorithms that, for a given H-topological-minor-free graph G and a positive
integer k, output an H-topological-minor-free graph G' on O(k) vertices such
that G has a (connected) dominating set of size k iff G' has one. Our results
extend the known classes of graphs on which the Dominating Set and Connected
Dominating Set problems admit linear kernels. Prior to our work, it was known
that these problems admit linear kernels on graphs excluding a fixed apex graph
H as a minor. Moreover, for Dominating Set, a kernel of size kc(H), where c(H)
is a constant depending on the size of H, follows from a more general result on
the kernelization of Dominating Set on graphs of bounded degeneracy. Alon and
Gutner explicitly asked whether one can obtain a linear kernel for Dominating
Set on H-minor-free graphs. We answer this question in the affirmative and in
fact prove a more general result. For Connected Dominating Set no polynomial
kernel even on H-minor-free graphs was known prior to our work. On the negative
side, it is known that Connected Dominating Set on 2-degenerated graphs does
not admit a polynomial kernel unless coNP $\subseteq$ NP/poly. Our
kernelization algorithm is based on a non-trivial combination of the following
ingredients The structural theorem of Grohe and Marx [STOC 2012] for graphs
excluding a fixed graph H as a topological minor; A novel notion of
protrusions, different than the one defined in [FOCS 2009]; Our results are
based on a generic reduction rule that produces an equivalent instance (in case
the input graph is H-minor-free) of the problem, with treewidth $O(\sqrt{k})$.
The application of this rule in a divide-and-conquer fashion, together with the
new notion of protrusions, gives us the linear kernels.
