Lossy kernels for connected distance-$r$ domination on nowhere dense graph classes

For $\alpha \text{\hspace{0.17em}⁣}:\mathbb{N}\to \mathbb{R}$, an $\alpha$-approximate bi-kernel is a polynomial-time algorithm that takes as input an instance $(I, k)$ofaproblem$Q$andoutputsaninstance$(I’,k’)$ofaproblem$Q’$ of size bounded by a function of $k$ such that, for every $c\ge 1$, a $c$-approximate solution for the new instance can be turned into a $c\cdot \alpha (k)$-approximate solution of the original instance in polynomial time. This framework of \emph{lossy kernelization} was recently introduced by Lokshtanov et al. We prove that for every nowhere dense class of graphs, every $\alpha >1$ and $r\in \mathbb{N}$ there exists a polynomial $p$ (whose degree depends only on $r$ while its coefficients depend on $\alpha$) such that the connected distance-$r$ dominating set problem with parameter $k$ admits an $\alpha$-approximate bi-kernel of size $p(k)$. Furthermore, we show that this result cannot be extended to more general classes of graphs which are closed under taking subgraphs by showing that if a class $C$ is somewhere dense and closed under taking subgraphs, then for some value of $r\in \mathbb{N}$ there cannot exist an $\alpha$-approximate bi-kernel for the (connected) distance-$r$ dominating set problem on $C$ for any function $\alpha \text{\hspace{0.17em}⁣}:\mathbb{N}\to \mathbb{R}$ (assuming the Gap Exponential Time Hypothesis).