Reidl, S\'anchez 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 $f(r) |A|$. 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 Soko{\l}owski (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 $O(r^4 |A|)$. We also show that a polynomial bound holds more generally for every proper minor-closed class of graphs.

翻译：Reidl、Sánchez Villaamil 和 Stravopoulos (2019) 对具有有界扩张的图类进行了如下刻画：一个关于子图封闭的图类 $\mathcal{C}$ 具有有界扩张当且仅当存在一个函数 $f:\mathbb{N} \to \mathbb{N}$，使得对于任意图 $G \in \mathcal{C}$、任意非空顶点子集 $A \subseteq V(G)$ 及任意非负整数 $r$，$A$ 与 $G$ 中半径为 $r$ 的球的交集种类数至多为 $f(r) |A|$。当 $\mathcal{C}$ 具有有界扩张时，现有证明所得函数 $f(r)$ 通常呈指数增长。对于平面图这一特例，Sokołowski (2021) 猜想 $f(r)$ 可取为多项式函数。本文证明了该猜想：对于任意平面图 $G$ 的任意非空顶点子集 $A$ 及任意非负整数 $r$，$A$ 与 $G$ 中半径为 $r$ 的球的交集种类数为 $O(r^4 |A|)$。我们进一步证明，对于每个真子图闭图类，多项式上界同样成立。