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

  • Patrizio Angelini
  • Michael A. BekosEmail author
  • Franz J. Brandenburg
  • Giordano Da Lozzo
  • Giuseppe Di Battista
  • Walter Didimo
  • Giuseppe Liotta
  • Fabrizio Montecchiani
  • Ignaz Rutter
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10520)

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.

Notes

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.

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Patrizio Angelini
    • 1
  • Michael A. Bekos
    • 1
    Email author
  • Franz J. Brandenburg
    • 2
  • Giordano Da Lozzo
    • 3
  • Giuseppe Di Battista
    • 4
  • Walter Didimo
    • 5
  • Giuseppe Liotta
    • 5
  • Fabrizio Montecchiani
    • 5
  • Ignaz Rutter
    • 6
  1. 1.Universität TübingenTübingenGermany
  2. 2.University of PassauPassauGermany
  3. 3.University of CaliforniaIrvineUSA
  4. 4.Roma Tre UniversityRomeItaly
  5. 5.Universitá degli Studi di PerugiaPerugiaItaly
  6. 6.TU EindhovenEindhovenThe Netherlands

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