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Arrangements of Pseudocircles: On Circularizability

  • Ricky Pollack Memorial Issue
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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 non-circularizable arrangements of 5 pseudocircles (one of them was known before). In the set of 2131 digon-free intersecting arrangements of 6 pseudocircles we identify the three non-circularizable examples. We also show non-circularizability of eight additional arrangements of 6 pseudocircles which have a group of symmetries of size at least 4. Most of our non-circularizability 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 non-circularizable 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.

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Notes

  1. This name refers to the logo of the Krupp AG, a German steel company. Krupp was the largest company in Europe at the beginning of the 20th century.

  2. We recommend the Sage Reference Manual on Graph Theory [30] and its collection of excellent examples.

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  4. If the three planes \(E_i,E_j,E_k\) intersect in a common line, we still take the expression as a definition for \(I_{ijk}\), i.e., the homogeneous coordinates are all zero.

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Authors and Affiliations

  1. Institut für Mathematik, Technische Universität Berlin, 10623, Berlin, Germany

    Stefan Felsner & Manfred Scheucher

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  1. Stefan Felsner
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  2. Manfred Scheucher
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Correspondence to Manfred Scheucher.

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Editor in Charge: Kenneth Clarkson

Dedicated to the memory of Ricky Pollack.

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Partially supported by the DFG Grants FE 340/11-1 and FE 340/12-1. Manfred Scheucher was partially supported by the ERC Advanced Research Grant No. 267165 (DISCONV). We gratefully acknowledge the computing time granted by TBK Automatisierung und Messtechnik GmbH and by the Institute of Software Technology, Graz University of Technology. We also thank the anonymous reviewers for valuable comments.

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Felsner, S., Scheucher, M. Arrangements of Pseudocircles: On Circularizability. Discrete Comput Geom 64, 776–813 (2020). https://doi.org/10.1007/s00454-019-00077-y

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