Let $\mathcal{D}=\{D_0,\ldots,D_{n-1}\}$ be a set of $n$ topological disks in the plane and let $\mathcal{A} := \mathcal{A}(\mathcal{D})$ be the arrangement induced by $\mathcal{D}$. For two disks $D_i,D_j\in\mathcal{D}$, let $Δ_{ij}$ be the number of connected components of $D_i\cap D_j$, and let $Δ:= \max_{i,j} Δ_{ij}$. We show that the diameter of $\mathcal{G}^*$, the dual graph of $\mathcal{A}$, can be bounded as a function of $n$ and $Δ$. Thus, any two points in the plane can be connected by a Jordan curve that crosses the disk boundaries a number of times bounded by a function of $n$ and $Δ$. In particular, for the case of two disks, we prove that the diameter of $\mathcal{G}^*$ is at most $\max\{2,2Δ\}$ and this bound is tight. For the general case of $n>2$ disks, we show that the diameter of $\mathcal{G}^*$ is $O(n^3 2^n Δ)$. We achieve this by proving that the number of maximal faces in $\mathcal{A}$ -- faces whose ply is more than the ply of their neighboring faces -- is $O(n^2 2^n Δ)$. To this end, we first show that the number of maximum faces -- faces whose ply is $n$ -- is $O(n^2Δ)$; the latter bound, which is of independent interest, is tight in the worst case.

翻译：令 $\mathcal{D}=\{D_0,\ldots,D_{n-1}\}$ 为平面上 $n$ 个拓扑圆盘的集合，并设 $\mathcal{A} := \mathcal{A}(\mathcal{D})$ 为由 $\mathcal{D}$ 诱导的构型。对于任意两个圆盘 $D_i,D_j\in\mathcal{D}$，令 $Δ_{ij}$ 表示 $D_i\cap D_j$ 的连通分支数，并定义 $Δ:= \max_{i,j} Δ_{ij}$。我们证明了 $\mathcal{A}$ 的对偶图 $\mathcal{G}^*$ 的直径可以被 $n$ 和 $Δ$ 的函数所界定。因此，平面上任意两点均可由一条若尔当曲线连接，且该曲线穿越圆盘边界的次数受限于 $n$ 和 $Δ$ 的某个函数。特别地，对于两个圆盘的情形，我们证明 $\mathcal{G}^*$ 的直径至多为 $\max\{2,2Δ\}$，且该界是紧的。对于 $n>2$ 个圆盘的一般情形，我们证明 $\mathcal{G}^*$ 的直径为 $O(n^3 2^n Δ)$。这一结论的获得基于我们证明了 $\mathcal{A}$ 中极大面（即层数大于其相邻面层数的面）的数量为 $O(n^2 2^n Δ)$。为此，我们首先证明了最大面（即层数为 $n$ 的面）的数量为 $O(n^2Δ)$；后一界在最坏情况下是紧的，且具有独立的研究意义。