On the Diameter of Arrangements of Topological Disks

2025-10-20

AI Summary

This paper investigates the diameter upper bound of the dual graph induced by an arrangement of $n$ topological disks in the plane, focusing on its functional dependence on $Delta$, the maximum number of connected components in the intersection of any subcollection of disks. Employing techniques from planar topology, combinatorial geometry, and graph theory, we analyze the evolution of the *ply*—the maximum number of overlapping disks over the plane—to characterize intersection structure, and establish a relationship between face count and dual graph path length. Our main contributions are: (1) the first explicit diameter upper bound in terms of $n$ and $Delta$; for $n=2$, it is tight: $max{2, 2Delta}$; (2) for general $n geq 2$, an upper bound of $2n(Delta+1)^{n(n-1)/2}min{n,Delta+1}$; and (3) a novel, improved upper bound on the maximum number of faces—an advance with theoretical significance. These results provide a key analytical tool for studying connectivity in geometric arrangements.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Social Networks and Social Media: Social media analysis through the lenses of networks
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

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 $D_i,D_jinmathcal{D}$, let $Δ_{ij}$ be the number of connected components of~$D_icap D_j$, 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.

Problem

• Bounding dual graph diameter for topological disk arrangements
• Relating connectivity to intersection complexity parameters
• Establishing tight bounds for two-disk arrangement cases

Innovation

• Bounding dual graph diameter using n and Δ
• Proving tight bound for two-disk case
• Counting maximal faces via combinatorial formula