On the Diameter of Arrangements of Topological Disks

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 $\Delta_{ij}$ be the number of connected components of~$D_i\cap D_j$, and let $\Delta := \max_{i,j} \Delta_{ij}$. We show that the diameter of $\mathcal{G}^*$, the dual graph of~$\mathcal{A}$, can be bounded as a function of $n$ and $\Delta$. 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~$\Delta$. In particular, for the case of two disks we prove that the diameter of $\mathcal{G}^*$ is at most $\max\{2,2\Delta\}$ and this bound is tight. % For the general case of $n>2$ disks, we show that the diameter of $\mathcal{G}^*$ is at most $2 n(\Delta+1)^{n(n-1)/2} \min\{n,\Delta+1\}$. We achieve this by proving that the number of maximal faces in $\mathcal{A}$ -- the faces whose ply is more than the ply of their neighboring faces -- is at most $n(\Delta+1)^{n(n-1)/2}$, which is interesting in its own right.