Skip to main content
SpringerLink
Log in

A Tipping Point for the Planarity of Small and Medium Sized Graphs

  • Conference paper
  • First Online:
Book cover Graph Drawing and Network Visualization (GD 2020)

Part of the book series: Lecture Notes in Computer Science ((LNISA,volume 12590))

Included in the following conference series:

Abstract

This paper presents an empirical study of the relationship between the density of small-medium sized random graphs and their planarity. It is well known that, when the number of vertices tends to infinite, there is a sharp transition between planarity and non-planarity for edge density \(d=0.5\). However, this asymptotic property does not clarify what happens for graphs of reduced size. We show that an unexpectedly sharp transition is also exhibited by small and medium sized graphs. Also, we show that the same “tipping point” behavior can be observed for some restrictions or relaxations of planarity (we considered outerplanarity and near-planarity, respectively).

This research was supported in part by MIUR Project “MODE” under PRIN 20157EFM5C, by MIUR Project “AHeAD” under PRIN 20174LF3T8, and by Roma Tre University Azione 4 Project “GeoView”.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    For the smallest graphs we may not have all densities. For example, there is no graph with 5 vertices and density greater than 2.

  2. 2.

    Function \(\text {Round}()\) rounds a value to the nearest integer, where \(\text {Round}(0.5)=1.0\).

References

  1. Balloni, E., Di Battista, G., Patrignani, M.: A tipping point for the planarity of small and medium sized graphs. Technical report, Cornell University arXiv:2008.09405v2 (2020)

  2. Bannister, M., Cabello, S., Eppstein, D.: Parameterized complexity of 1-planarity. J. Graph Algorithms Appl. 22(1), 23–49 (2018)

    Article  MathSciNet  Google Scholar 

  3. Barrera-Cruz, F., Haxell, P., Lubiw, A.: Morphing Schnyder drawings of planar triangulations. Discrete Comput. Geom. 61(1), 161–184 (2019)

    Article  MathSciNet  Google Scholar 

  4. Bollobás, B.: Random Graph. Academic Press Inc., Harcourt Brace Jovanovich Publishers, London (1985)

    MATH  Google Scholar 

  5. Borrazzo, M., Da Lozzo, G., Di Battista, G., Frati, F., Patrignani, M.: Graph stories in small area. J. Graph Algorithms Appl. 24(3), 269–292 (2020). https://doi.org/10.7155/jgaa.00530

    Article  MathSciNet  MATH  Google Scholar 

  6. Cabello, S., Mohar, B.: Adding one edge to planar graphs makes crossing number and 1-planarity hard. SIAM J. Comput. 42(5), 1803–1829 (2013)

    Article  MathSciNet  Google Scholar 

  7. Chimani, M., Gutwenger, C., Jünger, M., Klau, G.W., Klein, K., Mutzel, P.: Open graph drawing framework (OGDF). In: Tamassia, R. (ed.) Handbook of Graph Drawing and Visualization, chap. 17. CRC Press (2014)

    Google Scholar 

  8. Da Lozzo, G., Di Battista, G., Frati, F., Patrignani, M., Roselli, V.: Upward planar morphs. Algorithmica 82(10), 2985–3017 (2020). https://doi.org/10.1007/s00453-020-00714-6

    Article  MathSciNet  MATH  Google Scholar 

  9. Didimo, W., Liotta, G., Montecchiani, F.: A survey on graph drawing beyond planarity. ACM Comput. Surv. 52(1), 4:1–4:37 (2019). https://doi.org/10.1145/3301281

  10. Erdöos, P., Réenyi, A.: On the evolution of random graphs. Magyar Tud. Akad. Mat. Kutatóo Int. Közl. 5, 17–61 (1960)

    Google Scholar 

  11. de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10(1), 41–51 (1990). https://doi.org/10.1007/BF02122694

    Article  MathSciNet  MATH  Google Scholar 

  12. Janson, S., Łuczak, T., Rucinski, A.: Random Graphs. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley-Interscience, New York (2000)

    Book  Google Scholar 

  13. Korzhik, V.P., Mohar, B.: Minimal obstructions for 1-immersions and hardness of 1-planarity testing. J. Graph Theory 72(1), 30–71 (2013). https://doi.org/10.1002/jgt.21630

    Article  MathSciNet  MATH  Google Scholar 

  14. Łuczak, T., Pittel, B., Wierman, J.C.: The structure of a random graph at the point of the phase transition. Trans. Am. Math. Soc. 341(2), 721–748 (1994)

    Article  MathSciNet  Google Scholar 

  15. Noy, M., Ravelomanana, V., Rué, J.: The probability of planarity of a random graph near the critical point. In: International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), pp. 791–802 (2013)

    Google Scholar 

  16. Schnyder, W.: Embedding planar graphs on the grid. In: Johnson, D.S. (ed.) Proceedings of the First Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 138–148. SIAM (1990)

    Google Scholar 

  17. Tutte, W.T.: How to Draw a Graph. Proc. London Math. Soc. s3–13(1), 743–767 (1963). https://doi.org/10.1112/plms/s3-13.1.743

Download references

Acknowledgments

We thank Carlo Batini for posing us the first question about rapid transitions of graph properties. Sometimes questions are more important than answers. We also thank the anonymous reviewer for pointing out that the smallest not near-planar graph in terms of number of edges is \(K_{3,4}\).

Author information

Authors and Affiliations

  1. Roma Tre University, Rome, Italy

    Emanuele Balloni, Giuseppe Di Battista & Maurizio Patrignani

Authors
  1. Emanuele Balloni
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Giuseppe Di Battista
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Maurizio Patrignani
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Maurizio Patrignani .

Editor information

Editors and Affiliations

  1. LaBRI, University of Bordeaux, Talence, France

    David Auber

  2. Charles University, Prague, Czech Republic

    Pavel Valtr

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Balloni, E., Di Battista, G., Patrignani, M. (2020). A Tipping Point for the Planarity of Small and Medium Sized Graphs. In: Auber, D., Valtr, P. (eds) Graph Drawing and Network Visualization. GD 2020. Lecture Notes in Computer Science(), vol 12590. Springer, Cham. https://doi.org/10.1007/978-3-030-68766-3_15

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-68766-3_15

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-68765-6

  • Online ISBN: 978-3-030-68766-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation