On the Diameter of Arrangements of Topological Disks

TL;DR

The paper studies the arrangement $\mathcal{A}(\mathcal{D})$ of $n$ topological disks with overlap number $\Delta=\max_{i,j} \Delta_{ij}$ and bounds the diameter of the dual graph $\mathcal{G}^*$ as a function of $n$ and $\Delta$, even when boundaries intersect arbitrarily. It proves a tight bound for two disks: $\diam(\mathcal{G}^*)=\max\{2,2\Delta\}$, and establishes general bounds for $n$ disks by first bounding the number of maximal faces $\mu(\mathcal{A})$ via $\mu(n,\Delta) \le n(\Delta+1)^{n(n-1)/2}$, then deriving $\xi(n,\Delta) \le 2n \cdot \min\{n,\Delta+1\} \cdot (\Delta+1)^{n(n-1)/2}$. The methods combine a detailed analysis of boundary intervals, component sets, and connectivity lemmas to control the arrangement's combinatorial structure, yielding results relevant to barrier-like path-crossing problems. The work leaves open whether $\mu(\mathcal{A})$ can be bounded polynomially in $n$ and $\Delta$, noting an almost-quadratic construction and conjecturing near-tightness.

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 $Δ_{ij}$ 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 at most $2 n(Δ+1)^{n(n-1)/2} \min\{n,Δ+1\}$. 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 $n(Δ+1)^{n(n-1)/2}$, which is interesting in its own right.

Figures (5)

• Figure 1: (i) A set of two topological disks with $\Delta=1$ can induce an arrangement with arbitrarily many faces. (ii) A set of three topological disks and the dual graph of their arrangement. The arrangement has four maximal faces, whose dual nodes are shown in black. The edges incident to the node corresponding to the unbounded face are only drawn partially, to avoid cluttering the figure. If the blue disk is $D_0$, the red disk is $D_1$, and the green disk is $D_2$, then $\Delta_{01}=1$ and $\Delta_{02}=\Delta_{12}=3$, so $\Delta=3$.
• Figure 2: The dual graph of an arrangement of two disks, with white, blue, red, and purple nodes.
• Figure 4: Left: Two spiral-shaped topological disks with overlap number $\Delta=6$. The path shown in the figure is a shortest path in the dual graph between the nodes corresponding to the innermost blue face and the outermost blue face. Its length is $2\Delta=12$. Right: Any path between the innermost and the outermost blue face must cross each of the black and gray rings.
• Figure 5: The partitioning of $\partial D_0$ into elementary intervals (indicated in black). The colored intervals (which are drawn slightly outside $\partial D_0$ for clarity) indicate the intersections of $\partial D_0$ with the various components in $\mathcal{C}(D_0,D_i)$, where intervals of the same color belong to the same component. The different shades of green indicate that the intervals belong to different components of the green disk and the blue disk $D_0$.
• Figure 7: An example with $\Theta(n^2\Delta)$ maximal faces: in $n\times n$ grid for every grid cell on the main diagonal we introduce a disk inscribed in the interior of the cell having $k$ parallel strains extended to the four sides of the grid. By construction, there are $n$ disks and $\Delta=2k^2$ while the number of maximal faces is $n(n-1)k^2=n(n-1)\Delta/2$.

Theorems & Definitions (8)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 4
• Lemma 5
• Lemma 6
• Lemma 7
• Lemma 8