Skip to main content
Springer Nature Link
Log in

Strictly-Convex Drawings of 3-Connected Planar Graphs

  • Conference paper
  • First Online:
Graph Drawing and Network Visualization (GD 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13764))

Included in the following conference series:

Abstract

Strictly-convex straight-line drawings of 3-connected planar graphs in small area form a classical research topic in Graph Drawing. Currently, the best-known area bound for such drawings is \(O(n^2) \times O(n^2)\), as shown by Bárány and Rote by means of a sophisticated technique based on perturbing (non-strictly) convex drawings. Unfortunately, the hidden constants in such area bound are in the \(10^4\) order.

We present a new and easy-to-implement technique that yields strictly-convex straight-line planar drawings of 3-connected planar graphs on an integer grid of size \(2(n-1) \times (5n^3-4n^2)\).

Research of FM partially supported by Dip. Ingegneria, Univ. of Perugia, grant RICBA21LG “Algoritmi, modelli e sistemi per la rappresentazione visuale di reti”.

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Bárány, I., Rote, G.: Strictly convex drawings of planar graphs. Doc. Math. 11, 369–391 (2006)

    Article  MathSciNet  Google Scholar 

  2. Bekos, M.A., Gronemann, M., Montecchiani, F., Pálvölgyi, D., Symvonis, A., Theocharous, L.: Grid drawings of graphs with constant edge-vertex resolution. Comput. Geom. 98, 101789 (2021)

    Article  MathSciNet  Google Scholar 

  3. Bekos, M.A., Gronemann, M., Montecchiani, F., Symvonis, A.: Convex grid drawings of planar graphs with constant edge-vertex resolution. In: IWOCA 2022. LNCS, vol. 13270, pp. 157–171. Springer (2022)

    Google Scholar 

  4. Bekos, M.A., Gronemann, M., Montecchiani, F., Symvonis, A.: Strictly-convex drawings of 3-connected planar graphs. CoRR abs/2208.13388 (2022)

    Google Scholar 

  5. Chrobak, M., Goodrich, M.T., Tamassia, R.: Convex drawings of graphs in two and three dimensions (preliminary version). In: Whitesides, S. (ed.) SoCG, pp. 319–328. ACM (1996)

    Google Scholar 

  6. Chrobak, M., Kant, G.: Convex grid drawings of 3-connected planar graphs. Int. J. Comput. Geom. Appl. 7(3), 211–223 (1997)

    Article  MathSciNet  Google Scholar 

  7. Chrobak, M., Payne, T.H.: A linear-time algorithm for drawing a planar graph on a grid. Inf. Process. Lett. 54(4), 241–246 (1995)

    Article  MathSciNet  Google Scholar 

  8. de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10(1), 41–51 (1990)

    Article  MathSciNet  Google Scholar 

  9. Di Battista, G., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing: Algorithms for the Visualization of Graphs. Prentice-Hall (1999)

    Google Scholar 

  10. Di Battista, G., Frati, F.: Drawing trees, outerplanar graphs, series-parallel graphs, and planar graphs in a small area. In: Pach, J. (ed.) Thirty Essays on Geometric Graph Theory, pp. 121–165. Springer, New York (2013)

    Chapter  Google Scholar 

  11. Di Battista, G., Tamassia, R., Vismara, L.: Output-sensitive reporting of disjoint paths. Algorithmica 23(4), 302–340 (1999)

    Article  MathSciNet  Google Scholar 

  12. Felsner, S.: Convex drawings of planar graphs and the order dimension of 3-polytopes. Order 18(1), 19–37 (2001)

    Article  MathSciNet  Google Scholar 

  13. Felsner, S.: Geometric Graphs and Arrangements. Advanced Lectures in Mathematics, Vieweg (2004)

    Google Scholar 

  14. Fáry, I.: On straight lines representation of planar graphs. Acta Sci. Math. (Szeged) 11, 229–233 (1948)

    MathSciNet  Google Scholar 

  15. He, X.: Grid embedding of 4-connected plane graphs. Discret. Comput. Geom. 17(3), 339–358 (1997)

    Article  MathSciNet  Google Scholar 

  16. Kant, G.: Hexagonal grid drawings. In: Mayr, E.W. (ed.) WG 1992. LNCS, vol. 657, pp. 263–276. Springer, Heidelberg (1993). https://doi.org/10.1007/3-540-56402-0_53

    Chapter  Google Scholar 

  17. Kant, G.: Drawing planar graphs using the canonical ordering. Algorithmica 16(1), 4–32 (1996)

    Article  MathSciNet  Google Scholar 

  18. Miura, K., Nakano, S., Nishizeki, T.: Grid drawings of 4-connected plane graphs. Discret. Comput. Geom. 26(1), 73–87 (2001)

    Article  MathSciNet  Google Scholar 

  19. Rote, G.: Strictly convex drawings of planar graphs. In: SODA, pp. 728–734. SIAM (2005)

    Google Scholar 

  20. Royden, H.L., Fitzpatrick, P.M.: Real analysis. NY, fourth. edn, Pearson modern classic, Pearson, New York (2010)

    Google Scholar 

  21. Schnyder, W.: Embedding planar graphs on the grid. In: SODA, pp. 138–148. SIAM (1990)

    Google Scholar 

  22. Schnyder, W., Trotter, W.T.: Convex embeddings of 3-connected plane graphs. Abstracts AMS 13, 502 (1992)

    Google Scholar 

  23. Thomassen, C.: A refinement of Kuratowski’s theorem. J. Comb. Theory, Ser. B D(3), 245–253 (1984)

    Google Scholar 

  24. Tutte, W.T.: How to draw a graph. Proc. London Math. Soc. 13, 743–768 (1963)

    Article  MathSciNet  Google Scholar 

  25. Zhang, H., He, X.: Compact visibility representation and straight-line grid embedding of plane graphs. In: Dehne, F., Sack, J.-R., Smid, M. (eds.) WADS 2003. LNCS, vol. 2748, pp. 493–504. Springer, Heidelberg (2003). https://doi.org/10.1007/978-3-540-45078-8_43

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Ioannina, Ioannina, Greece

    Michael A. Bekos

  2. Algorithms and Complexity Group, TU Wien, Vienna, Austria

    Martin Gronemann

  3. Department of Engineering, University of Perugia, Perugia, Italy

    Fabrizio Montecchiani

  4. School of Applied Mathematical & Physical Sciences, NTUA, Athens, Greece

    Antonios Symvonis

Authors
  1. Michael A. Bekos
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Martin Gronemann
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Fabrizio Montecchiani
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Antonios Symvonis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fabrizio Montecchiani .

Editor information

Editors and Affiliations

  1. John Cabot University, Rome, Italy

    Patrizio Angelini

  2. Kiel University, Kiel, Germany

    Reinhard von Hanxleden

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Bekos, M.A., Gronemann, M., Montecchiani, F., Symvonis, A. (2023). Strictly-Convex Drawings of 3-Connected Planar Graphs. In: Angelini, P., von Hanxleden, R. (eds) Graph Drawing and Network Visualization. GD 2022. Lecture Notes in Computer Science, vol 13764. Springer, Cham. https://doi.org/10.1007/978-3-031-22203-0_11

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-22203-0_11

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-22202-3

  • Online ISBN: 978-3-031-22203-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics