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.

里德尔等人。 (Eur J Comb 75:152–168, 2019) 将有界扩展的图类描述如下：子图下封闭的类 \({\mathcal {C}}\) 具有有界扩展当且仅当存在函数 \( f:{\mathbb {N}} \rightarrow {\mathbb {N}}\) 使得对于每个图 \(G \in {\mathcal {C}}\)，G 中顶点的每个非空子集 A 和每个非负整数 r，A 与 G 中半径为 r 的球之间的不同交点的数量最多为 f(r) |A|。当 \({\mathcal {C}}\) 具有有界扩展时，来自现有证明的函数 f(r) 通常是指数函数。在平面图的特殊情况下，Sokołowski (Electron J Comb 30(2):P2.3, 2023) 推测 f(r) 可以被视为多项式。在本文中，我们证明了这个猜想：对于平面图 G 中顶点的每个非空子集 A 和每个非负整数 r，A 与 G 中半径为 r 的球之间的不同交集的数量为 \({{\,\ mathrm{{\mathcal {O}}}\,}}(r^4 |A|)\)。我们还表明，多项式界限更普遍地适用于每个真次闭类图。