Reconfiguration on nowhere dense graph classes

Let $\mathcal{Q}$ be a vertex subset problem on graphs. In a reconfiguration variant of $\mathcal{Q}$ we are given a graph $G$ and two feasible solutions $Ss, St\subseteq V(G)$ of $\mathcal{Q}$ with $|Ss|=|St|=k$. The problem is to determine whether there exists a sequence $S1,\ldots,Sn$ of feasible solutions, where $S1=Ss$, $Sn=St$, $|Si|\leq k\pm 1$, and each $S{i+1}$ results from $Si$, $1\leq i<n$, by the addition or removal of a single vertex. We prove that for every nowhere dense class of graphs and for every integer $r\geq 1$ there exists a polynomial $pr$ such that the reconfiguration variants of the distance-$r$ independent set problem and the distance-$r$ dominating set problem admit kernels of size $p_r(k)$. If $k$ is equal to the size of a minimum distance-$r$ dominating set, then for any fixed $\epsilon>0$ we even obtain a kernel of almost linear size $\mathcal{O}(k^{{1+\epsilon})$.} We then prove that if a class $\mathcal{C}$ is somewhere dense and closed under taking subgraphs, then for some value of $r\geq 1$ the reconfiguration variants of the above problems on $\mathcal{C}$ are $\mathsf{W}[1]$-hard (and in particular we cannot expect the existence of kernelization algorithms). Hence our results show that the limit of tractability for the reconfiguration variants of the distance-$r$ independent set problem and distance-$r$ dominating set problem on subgraph closed graph classes lies exactly on the boundary between nowhere denseness and somewhere denseness.