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On the Relationship Between k-Planar and k-Quasi-Planar Graphs

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Graph-Theoretic Concepts in Computer Science (WG 2017)

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Abstract

A graph is k-planar \((k \ge 1)\) if it can be drawn in the plane such that no edge is crossed \(k+1\) times or more. A graph is k-quasi-planar \((k \ge 2)\) if it can be drawn in the plane with no k pairwise crossing edges. The families of k-planar and k-quasi-planar graphs have been widely studied in the literature, and several bounds have been proven on their edge density. Nonetheless, only trivial results are known about the relationship between these two graph families. In this paper we prove that, for \(k \ge 3\), every k-planar graph is \((k+1)\)-quasi-planar.

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Acknowledgements

The research in this paper started at the Dagstuhl Seminar 16452 “Beyond-Planar Graphs: Algorithmics and Combinatorics”. We thank all participants, and in particular Pavel Valtr and Raimund Seidel, for useful discussions on the topic.

Author information

Authors and Affiliations

  1. Universität Tübingen, Tübingen, Germany

    Patrizio Angelini & Michael A. Bekos

  2. University of Passau, Passau, Germany

    Franz J. Brandenburg

  3. University of California, Irvine, CA, USA

    Giordano Da Lozzo

  4. Roma Tre University, Rome, Italy

    Giuseppe Di Battista

  5. Universitá degli Studi di Perugia, Perugia, Italy

    Walter Didimo, Giuseppe Liotta & Fabrizio Montecchiani

  6. TU Eindhoven, Eindhoven, The Netherlands

    Ignaz Rutter

Authors
  1. Patrizio Angelini
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  2. Michael A. Bekos
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  3. Franz J. Brandenburg
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  4. Giordano Da Lozzo
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  5. Giuseppe Di Battista
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  6. Walter Didimo
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  7. Giuseppe Liotta
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  8. Fabrizio Montecchiani
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  9. Ignaz Rutter
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Corresponding author

Correspondence to Michael A. Bekos .

Editor information

Editors and Affiliations

  1. Utrecht University, Utrecht, The Netherlands

    Hans L. Bodlaender

  2. Rheinisch-Westfälische Technische Hochschule Aachen, Aachen, Germany

    Gerhard J. Woeginger

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Angelini, P. et al. (2017). On the Relationship Between k-Planar and k-Quasi-Planar Graphs. In: Bodlaender, H., Woeginger, G. (eds) Graph-Theoretic Concepts in Computer Science. WG 2017. Lecture Notes in Computer Science(), vol 10520. Springer, Cham. https://doi.org/10.1007/978-3-319-68705-6_5

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  • DOI: https://doi.org/10.1007/978-3-319-68705-6_5

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  • Online ISBN: 978-3-319-68705-6

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