Neighborhood complexity of planar graphs

Reidl, Sánchez Villaamil, and Stravopoulos (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 $f:\mathbb{N}\to \mathbb{N}$ 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 $f(r)\mathrm{\mid }A\mathrm{\mid }$. 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 (2021) 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 $O(r^4\mathrm{\mid }A\mathrm{\mid })$. We also show that a polynomial bound holds more generally for every proper minor-closed class of graphs.