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Characterizing and Recognizing Twistedness

Authors Oswin Aichholzer , Alfredo García , Javier Tejel , Birgit Vogtenhuber , Alexandra Weinberger

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LIPIcs.GD.2025.25.pdf
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ACM Subject Classification
  • Mathematics of computing → Combinatorics
  • Mathematics of computing → Graph theory
Keywords
  • generalized twisted drawings
  • simple drawings
  • rotation systems
  • recognition
  • combinatorial characterization
  • efficient algorithms

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Abstract

In a simple drawing of a graph, any two edges intersect in at most one point (either a common endpoint or a proper crossing). A simple drawing is generalized twisted if it fulfills certain rather specific constraints on how the edges are drawn. An abstract rotation system of a graph assigns to each vertex a cyclic order of its incident edges. A realizable rotation system is one that admits a simple drawing such that at each vertex, the edges emanate in that cyclic order, and a generalized twisted rotation system can be realized as a generalized twisted drawing. Generalized twisted drawings have initially been introduced to obtain improved bounds on the size of plane substructures in any simple drawing of K_n. They have since gained independent interest due to their surprising properties. However, the definition of generalized twisted drawings is very geometric and drawing-specific.
In this paper, we develop characterizations of generalized twisted drawings that enable a purely combinatorial view on these drawings and lead to efficient recognition algorithms. Concretely, we show that for any n ≥ 7, an abstract rotation system of K_n is generalized twisted if and only if all subrotation systems induced by five vertices are generalized twisted. This implies a drawing-independent and concise characterization of generalized twistedness. Besides, the result yields a simple O(n⁵)-time algorithm to decide whether an abstract rotation system is generalized twisted and sheds new light on the structural features of simple drawings. We further develop a characterization via the rotations of a pair of vertices in a drawing, which we then use to derive an O(n²)-time algorithm to decide whether a realizable rotation system is generalized twisted.

Cite As Get BibTex

Oswin Aichholzer, Alfredo García, Javier Tejel, Birgit Vogtenhuber, and Alexandra Weinberger. Characterizing and Recognizing Twistedness. In 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 357, pp. 25:1-25:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.GD.2025.25

Author Details

Oswin Aichholzer
  • Graz University of Technology, Austria
Alfredo García
  • Departamento de Métodos Estadísticos and IUMA, University of Zaragoza, Spain
Javier Tejel
  • Departamento de Métodos Estadísticos and IUMA, University of Zaragoza, Spain
Birgit Vogtenhuber
  • Graz University of Technology, Austria
Alexandra Weinberger
  • FernUniversität in Hagen, Germany

Funding

  • Tejel, Javier: Partially supported by project E41-23R, funded by Gobierno de Aragón, and project PID2023-150725NB-I00, funded by MICIU/AEI/10.13039/501100011033.

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