# On Arrangements of Orthogonal Circles

## TL;DR

This paper investigates arrangements of orthogonal circles, proving they have linear complexity and that recognizing their intersection graphs is NP-hard, even for unit circles, revealing computational challenges in this geometric setting.

## Contribution

It introduces the concept of orthogonal circle arrangements, proves their linear face complexity, and establishes NP-hardness of recognition for orthogonal unit circle intersection graphs.

## Key findings

- Orthogonal circle arrangements have linear number of faces.
- Orthogonal circle intersection graphs have linear edges.
- Recognition of orthogonal unit circle graphs is NP-hard.

## Abstract

In this paper, we study arrangements of orthogonal circles, that is, arrangements of circles where every pair of circles must either be disjoint or intersect at a right angle. Using geometric arguments, we show that such arrangements have only a linear number of faces. This implies that orthogonal circle intersection graphs have only a linear number of edges. When we restrict ourselves to orthogonal unit circles, the resulting class of intersection graphs is a subclass of penny graphs (that is, contact graphs of unit circles). We show that, similarly to penny graphs, it is NP-hard to recognize orthogonal unit circle intersection graphs.