Arrangements of Pseudocircles: On Circularizability
Abstract
An arrangement of pseudocircles is a collection of simple closed curves on the sphere or in the plane such that any two of the curves are either disjoint or intersect in exactly two crossing points. We call an arrangement intersecting if every pair of pseudocircles intersects twice. An arrangement is circularizable if there is a combinatorially equivalent arrangement of circles. In this paper we present the results of the first thorough study of circularizability. We show that there are exactly four noncircularizable arrangements of 5 pseudocircles (one of them was known before). In the set of 2131 digonfree intersecting arrangements of 6 pseudocircles we identify the three noncircularizable examples. We also show noncircularizability of 8 additional arrangements of 6 pseudocircles which have a group of symmetries of size at least 4. Most of our noncircularizability proofs depend on incidence theorems like Miquel's. In other cases we contradict circularizability by considering a continuous deformation where the circles of an assumed circle representation grow or shrink in a controlled way. The claims that we have all noncircularizable arrangements with the given properties are based on a program that generated all arrangements up to a certain size. Given the complete lists of arrangements, we used heuristics to find circle representations. Examples where the heuristics failed were examined by hand.
 Publication:

arXiv eprints
 Pub Date:
 December 2017
 DOI:
 10.48550/arXiv.1712.02149
 arXiv:
 arXiv:1712.02149
 Bibcode:
 2017arXiv171202149F
 Keywords:

 Computer Science  Computational Geometry;
 Mathematics  Combinatorics;
 52Cxx;
 I.3.5;
 G.2.1
 EPrint:
 Appears in the Proceedings of the 26th International Symposium on Graph Drawing and Network Visualization (GD 2018)