Document Open Access Logo

The Number of Edges in Maximal 2-Planar Graphs

Authors Michael Hoffmann , Meghana M. Reddy

PDF
Thumbnail PDF

File

LIPIcs.SoCG.2023.39.pdf
  • Filesize: 0.8 MB
  • 15 pages

Document Identifiers

Author Details

Michael Hoffmann
  • Department of Computer Science, ETH Zürich, Switzerland
Meghana M. Reddy
  • Department of Computer Science, ETH Zürich, Switzerland

Funding

Supported by the Swiss National Science Foundation within the collaborative DACH project Arrangements and Drawings as SNSF Project 200021E-171681.

Cite AsGet BibTex

Michael Hoffmann and Meghana M. Reddy. The Number of Edges in Maximal 2-Planar Graphs. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 39:1-39:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.SoCG.2023.39

Abstract

A graph is 2-planar if it has local crossing number two, that is, it can be drawn in the plane such that every edge has at most two crossings. A graph is maximal 2-planar if no edge can be added such that the resulting graph remains 2-planar. A 2-planar graph on n vertices has at most 5n-10 edges, and some (maximal) 2-planar graphs - referred to as optimal 2-planar - achieve this bound. However, in strong contrast to maximal planar graphs, a maximal 2-planar graph may have fewer than the maximum possible number of edges. In this paper, we determine the minimum edge density of maximal 2-planar graphs by proving that every maximal 2-planar graph on n ≥ 5 vertices has at least 2n edges. We also show that this bound is tight, up to an additive constant. The lower bound is based on an analysis of the degree distribution in specific classes of drawings of the graph. The upper bound construction is verified by carefully exploring the space of admissible drawings using computer support.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorics
  • Mathematics of computing → Graph theory
  • Human-centered computing → Graph drawings
Keywords
  • k-planar graphs
  • local crossing number
  • saturated graphs
  • beyond-planar graphs

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

Supplementary Materials

References

  1. Eyal Ackerman. On topological graphs with at most four crossings per edge. Computational Geometry, 85:101574, 2019. URL: https://doi.org/10.1016/j.comgeo.2019.101574.
  2. Patrizio Angelini, Michael A. Bekos, Michael Kaufmann, and Thomas Schneck. Efficient generation of different topological representations of graphs beyond-planarity. J. Graph Algorithms Appl., 24(4):573-601, 2020. URL: https://doi.org/10.7155/jgaa.00531.
  3. Christopher Auer, Franz-Josef Brandenburg, Andreas Gleißner, and Kathrin Hanauer. On sparse maximal 2-planar graphs. In Proc. 20th Int. Sympos. Graph Drawing (GD 2012), volume 7704 of Lecture Notes Comput. Sci., pages 555-556. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-36763-2_50.
  4. János Barát and Géza Tóth. Improvements on the density of maximal 1-planar graphs. J. Graph Theory, 88(1):101-109, 2018. URL: https://doi.org/10.1002/jgt.22187.
  5. János Barát and Géza Tóth. Saturated 2-planar drawings with few edges. CoRR, abs/2110.12781, 2021. URL: https://arxiv.org/abs/2110.12781.
  6. Michael A. Bekos, Michael Kaufmann, and Chrysanthi N. Raftopoulou. On optimal 2- and 3-planar graphs. In Proc. 33rd Internat. Sympos. Comput. Geom. (SoCG 2017), volume 77 of LIPIcs, pages 16:1-16:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/LIPIcs.SoCG.2017.16.
  7. Rainer Bodendiek, Heinz Schumacher, and Klaus Wagner. Bemerkungen zu einem Sechsfarbenproblem von G. Ringel. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 53:41-52, 1983. URL: https://doi.org/10.1007/BF02941309.
  8. Franz-Josef Brandenburg, David Eppstein, Andreas Gleißner, Michael T. Goodrich, Kathrin Hanauer, and Josef Reislhuber. On the density of maximal 1-planar graphs. In Proc. 20th Int. Sympos. Graph Drawing (GD 2012), volume 7704 of Lecture Notes Comput. Sci., pages 327-338. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-36763-2_29.
  9. Steven Chaplick, Fabian Klute, Irene Parada, Jonathan Rollin, and Torsten Ueckerdt. Edge-minimum saturated k-planar drawings. In Proc. 29th Int. Sympos. Graph Drawing (GD 2021), volume 12868 of Lecture Notes Comput. Sci., pages 3-17. Springer, 2021. URL: https://doi.org/10.1007/978-3-030-92931-2_1.
  10. Mark de Berg, Marc van Kreveld, Mark Overmars, and Otfried Schwarzkopf. Computational Geometry: Algorithms and Applications. Springer, Berlin, Germany, 3rd edition, 2008. URL: https://doi.org/10.1007/978-3-540-77974-2.
  11. Walter Didimo, Giuseppe Liotta, and Fabrizio Montecchiani. A survey on graph drawing beyond planarity. ACM Comput. Surv., 52(1):1-37, 2020. URL: https://doi.org/10.1145/3301281.
  12. Péter Hajnal, Alexander Igamberdiev, Günter Rote, and André Schulz. Saturated simple and 2-simple topological graphs with few edges. J. Graph Algorithms Appl., 22(1):117-138, 2018. URL: https://doi.org/10.7155/jgaa.00460.
  13. Michael Hoffmann and Meghana M Reddy. The number of edges in maximal 2-planar graphs - code repository. URL: https://github.com/meghanamreddy/Sparse-maximal-2-planar-graphs.
  14. Michael Hoffmann and Meghana M. Reddy. The number of edges in maximal 2-planar graphs, 2023. URL: https://doi.org/10.48550/ARXIV.2303.08726.
  15. Seok-Hee Hong and Takeshi Tokuyama, editors. Beyond Planar Graphs. Springer, Singapore, 2020. URL: https://doi.org/10.1007/978-981-15-6533-5.
  16. Dávid Hudák, Tomaš Madaras, and Yusuke Suzuki. On properties of maximal 1-planar graphs. Discussiones Mathematicae, 32:737-747, 2012. URL: https://doi.org/10.7151/dmgt.1639.
  17. Vladimir P. Korzhik. Minimal non-1-planar graphs. Discrete Math., 308(7):1319-1327, 2008. URL: https://doi.org/10.1016/j.disc.2007.04.009.
  18. Jan Kynčl, János Pach, Radoš Radoišić, and Géza Tóth. Saturated simple and k-simple topological graphs. Computational Geometry, 48(4):295-310, 2015. URL: https://doi.org/10.1016/j.comgeo.2014.10.008.
  19. János Pach, Radoš Radoičić, Gábor Tardos, and Géza Tóth. Improving the crossing lemma by finding more crossings in sparse graphs. Discrete Comput. Geom., 36(4):527-552, 2006. URL: https://doi.org/10.1007/s00454-006-1264-9.
  20. János Pach and Géza Tóth. Graphs drawn with few crossings per edge. Combinatorica, 17(3):427-439, 1997. URL: https://doi.org/10.1007/BF01215922.
  21. Gerhard Ringel. Ein Sechsfarbenproblem auf der Kugel. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 29:107-117, 1965. URL: https://doi.org/10.1007/BF02996313.
  22. Marcus Schaefer. The graph crossing number and its variants: A survey. The Electronic Journal of Combinatorics, 20, 2013. Version 7 (April 8, 2022). URL: https://doi.org/10.37236/2713.
  23. Yusuke Suzuki. Optimal 1-planar graphs which triangulate other surfaces. Discr. Math., 310(1):6-11, 2010. URL: https://doi.org/10.1016/j.disc.2009.07.016.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact