On the Uncrossed Number of Graphs

Authors Martin Balko , Petr Hliněný , Tomáš Masařík , Joachim Orthaber , Birgit Vogtenhuber , Mirko H. Wagner



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Author Details

Martin Balko
  • Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
Petr Hliněný
  • Faculty of Informatics, Masaryk University, Brno, Czech Republic
Tomáš Masařík
  • Institute of Informatics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Poland
Joachim Orthaber
  • Institute of Software Technology, Graz University of Technology, Austria
Birgit Vogtenhuber
  • Institute of Software Technology, Graz University of Technology, Austria
Mirko H. Wagner
  • Institute of Computer Science, Osnabrück University, Germany

Acknowledgements

We would like to thank the organizers of the Crossing Numbers Workshop 2023 where this research was initiated.

Cite AsGet BibTex

Martin Balko, Petr Hliněný, Tomáš Masařík, Joachim Orthaber, Birgit Vogtenhuber, and Mirko H. Wagner. On the Uncrossed Number of Graphs. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 18:1-18:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.GD.2024.18

Abstract

Visualizing a graph G in the plane nicely, for example, without crossings, is unfortunately not always possible. To address this problem, Masařík and Hliněný [GD 2023] recently asked for each edge of G to be drawn without crossings while allowing multiple different drawings of G. More formally, a collection 𝒟 of drawings of G is uncrossed if, for each edge e of G, there is a drawing in 𝒟 such that e is uncrossed. The uncrossed number unc(G) of G is then the minimum number of drawings in some uncrossed collection of G. No exact values of the uncrossed numbers have been determined yet, not even for simple graph classes. In this paper, we provide the exact values for uncrossed numbers of complete and complete bipartite graphs, partly confirming and partly refuting a conjecture posed by Hliněný and Masařík [GD 2023]. We also present a strong general lower bound on unc(G) in terms of the number of vertices and edges of G. Moreover, we prove NP-hardness of the related problem of determining the edge crossing number of a graph G, which is the smallest number of edges of G taken over all drawings of G that participate in a crossing. This problem was posed as open by Schaefer in his book [Crossing Numbers of Graphs 2018].

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Mathematics of computing → Graph theory
Keywords
  • Uncrossed Number
  • Crossing Number
  • Planarity
  • Thickness

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References

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