# On Arrangements of Orthogonal Circles

• Steven Chaplick
• Henry Förster
• Myroslav Kryven
• Alexander Wolff
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11904)

## 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.

## Notes

### Acknowledgments

We thank Alon Efrat for useful discussions and an anonymous reviewer for pointing us to the Gauss-Bonnet formula.

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