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) 将具有有界扩张性的图类刻画如下：一个在子图下封闭的图类 $\mathcal{C}$ 具有有界扩张性当且仅当存在一个函数 $f:\mathbb{N} \to \mathbb{N}$，使得对于每个图 $G \in \mathcal{C}$，每个非空顶点子集 $A \subseteq V(G)$ 以及每个非负整数 $r$，$A$ 与 $G$ 中半径为 $r$ 的球的不同交集的数目至多为 $f(r) |A|$。当 $\mathcal{C}$ 具有有界扩张性时，现有证明中得到的函数 $f(r)$ 通常是指数级的。在平面图的特殊情形下，Sokołowski (2021) 猜想 $f(r)$ 可以取为多项式函数。本文证明了这一猜想：对于平面图 $G$ 中的任意非空顶点子集 $A$ 以及任意非负整数 $r$，$A$ 与 $G$ 中半径为 $r$ 的球的不同交集的数目为 $O(r^4 |A|)$。我们还表明，对于每个真子式封闭的图类，多项式界同样成立。