  Nowhere dense classes of graphs are very general classes of uniformly sparse
graphs with several seemingly unrelated characterisations. From an algorithmic
perspective, a characterisation of these classes in terms of uniform
quasi-wideness, a concept originating in finite model theory, has proved to be
particularly useful. Uniform quasi-wideness is used in many fpt-algorithms on
nowhere dense classes. However, the existing constructions showing the
equivalence of nowhere denseness and uniform quasi-wideness imply a
non-elementary blow up in the parameter dependence of the fpt-algorithms,
making them infeasible in practice.
  As a first main result of this paper, we use tools from logic, in particular
from a subfield of model theory known as stability theory, to establish
polynomial bounds for the equivalence of nowhere denseness and uniform
quasi-wideness.
  A powerful method in parameterized complexity theory is to compute a problem
kernel in a pre-computation step, that is, to reduce the input instance in
polynomial time to a sub-instance of size bounded in the parameter only
(independently of the input graph size). Our new tools allow us to obtain for
every fixed value of $r$ a polynomial kernel for the distance-$r$ dominating
set problem on nowhere dense classes of graphs. This result is particularly
interesting, as it implies that for every class $\mathcal{C}$ of graphs which
is closed under subgraphs, the distance-$r$ dominating set problem admits a
kernel on $\mathcal{C}$ for every value of $r$ if, and only if, it admits a
polynomial kernel for every value of $r$ (under the standard assumption of
parameterized complexity theory that $\mathrm{FPT} \neq W[2]$).
