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Tangling and Untangling Trees on Point-Sets

Authors Giuseppe Di Battista , Giuseppe Liotta , Maurizio Patrignani , Antonios Symvonis , Ioannis G. Tollis

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ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Mathematics of computing → Graph theory
Keywords
  • Tree drawings
  • Prescribed edge crossings
  • Thrackle
  • Curve complexity
  • Point-set embeddings
  • RAC drawings
  • Topological linear embeddings

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Abstract

We study a question that lies at the intersection of classical research subjects in Topological Graph Theory and Graph Drawing: Computing a drawing of a graph with a prescribed number of crossings on a given set S of points, while ensuring that its curve complexity (i.e., maximum number of bends per edge) is bounded by a constant. We focus on trees: Let T be a tree, ϑ(T) be its thrackle number, and χ be any integer in the interval [0,ϑ(T)]. In the tangling phase we compute a topological linear embedding of T with ϑ(T) edge crossings and a constant number of spine traversals. In the untangling phase we remove edge crossings without increasing the spine traversals until we reach χ crossings. The computed linear embedding is used to construct a drawing of T on S with χ crossings and constant curve complexity. Our approach gives rise to an O(n²)-time algorithm for general trees and an O(n log n)-time algorithm for paths. We also adapt the approach to compute RAC drawings, i.e. drawings where the angles formed at edge crossings are π/2.

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Giuseppe Di Battista, Giuseppe Liotta, Maurizio Patrignani, Antonios Symvonis, and Ioannis G. Tollis. Tangling and Untangling Trees on Point-Sets. In 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 357, pp. 8:1-8:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.GD.2025.8

Author Details

Giuseppe Di Battista
  • Roma Tre University, Italy
Giuseppe Liotta
  • University of Perugia, Italy
Maurizio Patrignani
  • Roma Tre University, Italy
Antonios Symvonis
  • School of Applied Mathematical & Physical Sciences, National Technical University of Athens, Greece
Ioannis G. Tollis
  • Department of Computer Science, University of Crete, Heraklion, Greece

Funding

  • Di Battista, Giuseppe: Research supported by MUR PRIN Project no. 2022TS4Y3N.
  • Liotta, Giuseppe: Research supported by MUR PON Proj. ARS01 00540 and by MUR PRIN Project no. 2022TS4Y3N.
  • Patrignani, Maurizio: Research supported by MUR PRIN Project no. 2022TS4Y3N.
  • Symvonis, Antonios: Supported by HFRI Grant No: 26320.

Acknowledgements

This work started at the Bertinoro Workshop on Graph Drawing BWGD 2024.

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