Neighborhood complexity and kernelization for nowhere dense classes of graphs

We prove that whenever G is a graph from a nowhere dense graph class 𝒞, and A is a subset of vertices of G, then the number of subsets of A that are realized as intersections of A with r-neighborhoods of vertices of G is at most f(r,ϵ)· |A|^1+ϵ, where r is any positive integer, ϵ is any positive real, and f is a function that depends only on the class 𝒞. This yields a characterization of nowhere dense classes of graphs in terms of neighborhood complexity, which answers a question posed by Reidl et al. As an algorithmic application of the above result, we show that for every fixed r, the parameterized Distance-r Dominating Set problem admits an almost linear kernel on any nowhere dense graph class. This proves a conjecture posed by Drange et al., and shows that the limit of parameterized tractability of Distance-r Dominating Set on subgraph-closed graph classes lies exactly on the boundary between nowhere denseness and somewhere denseness.