Neighborhood Complexity of Planar Graphs

Reidl et al. (Eur J Comb 75:152–168, 2019) characterized graph classes of bounded expansion as follows: A class \({\mathcal {C}}\) closed under subgraphs has bounded expansion if and only if there exists a function \(f:{\mathbb {N}} \rightarrow {\mathbb {N}}\) such that for every graph \(G \in {\mathcal {C}}\), every nonempty subset A of vertices in G and every nonnegative integer r, the number of distinct intersections between A and a ball of radius r in G is at most f(r) |A|. When \({\mathcal {C}}\) has bounded expansion, the function f(r) coming from existing proofs is typically exponential. In the special case of planar graphs, it was conjectured by Sokołowski (Electron J Comb 30(2):P2.3, 2023) that f(r) could be taken to be a polynomial. In this paper, we prove this conjecture: For every nonempty subset A of vertices in a planar graph G and every nonnegative integer r, the number of distinct intersections between A and a ball of radius r in G is \({{\,\mathrm{{\mathcal {O}}}\,}}(r^4 |A|)\). We also show that a polynomial bound holds more generally for every proper minor-closed class of graphs.