On the Diameter of Arrangements of Topological Disks

TL;DR

This paper investigates the bounds on the diameter of the dual graph of arrangements of topological disks in the plane, providing tight bounds for two disks and exponential bounds for multiple disks based on intersection complexity.

Contribution

It establishes new bounds on the dual graph diameter of disk arrangements, including tight bounds for two disks and exponential bounds for multiple disks based on intersection complexity.

Findings

Diameter of dual graph for two disks is at most max{2, 2Δ} and this is tight.

For n > 2 disks, the diameter is bounded by O(n^3 2^n Δ).

Number of maximal faces in the arrangement is O(n^2 2^n Δ).

Abstract

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 $O(n^3 2^n \Delta)$. 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 $O(n^2 2^n \Delta)$. To this end, we first show that the number of maximum faces -- faces whose ply is $n$ -- is $O(n^2\Delta)$; the latter bound, which is of independent interest, is tight in the worst case.