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Crossing Numbers of Beyond Planar Graphs Re-Revisited: A Framework Approach

Authors Markus Chimani , Torben Donzelmann, Nick Kloster, Melissa Koch, Jan-Jakob Völlering, Mirko H. Wagner

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  • Mathematics of computing → Graph theory
Keywords
  • Beyond planarity
  • crossing number
  • crossing ratio
  • proof framework

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Abstract

Beyond planarity concepts (prominent examples include k-planarity or fan-planarity) apply certain restrictions on the allowed patterns of crossings in drawings. It is natural to ask, how much the number of crossings may increase over the traditional (unrestricted) crossing number. Previous approaches to bound such ratios, e.g. [Markus Chimani et al., 2022; Nathan van Beusekom et al., 2022], require very specialized constructions and arguments for each considered beyond planarity concept, and mostly only yield asymptotically non-tight bounds. We propose a very general proof framework that allows us to obtain asymptotically tight bounds, and where the concept-specific parts of the proof typically boil down to a couple of lines. We show the strength of our approach by giving improved or first bounds for several beyond planarity concepts.

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Markus Chimani, Torben Donzelmann, Nick Kloster, Melissa Koch, Jan-Jakob Völlering, and Mirko H. Wagner. Crossing Numbers of Beyond Planar Graphs Re-Revisited: A Framework Approach. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 33:1-33:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.GD.2024.33

Author Details

Markus Chimani
  • Theoretical Computer Science, Osnabrück University, Germany
Torben Donzelmann
  • Theoretical Computer Science, Osnabrück University, Germany
Nick Kloster
  • Theoretical Computer Science, Osnabrück University, Germany
Melissa Koch
  • Theoretical Computer Science, Osnabrück University, Germany
Jan-Jakob Völlering
  • Theoretical Computer Science, Osnabrück University, Germany
Mirko H. Wagner
  • Theoretical Computer Science, Osnabrück University, Germany

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