Skip to main content
SpringerLink
Log in

Drawing Partial 2-Trees with Few Slopes

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

The planar slope number of a planar graph G is the minimum integer k such that G admits a planar drawing with vertices as points and edges as straight-line segments with k distinct slopes. Similarly, a plane slope number is defined for a plane graph, where a fixed combinatorial embedding of the graph is given and the output must respect the given embedding. We prove tight bounds (up to a small multiplicative or additive constant) for the plane and the planar slope numbers of partial 2-trees of bounded degree. We also answer a long standing question by Garg and Tamassia (In: van Leeuwen J (eds) Proceedings of the Second Annual European Symposium on Algorithms (ESA), LNCS, vol 855, pp 12–23, Springer, 1994) on the angular resolution of the planar straight-line drawings of series-parallel graphs of bounded degree.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

Data Availibility

All the data in this paper are available from the authors upon reasonable request.

References

  1. Lenhart, W., Liotta, G., Mondal, D., Nishat, R.I.: Graph Drawing. In: Wismath S and Wolff A (eds) Lecture Notes in Computer Science, vol. 8242, Springer, pp. 412–423 (2013)

  2. Garg, A., Tamassia, R.: Proceedings of the Second Annual European Symposium on Algorithms (ESA). In: van Leeuwen J (ed) LNCS, vol. 855, Springer, pp. 12–23 (1994)

  3. Eades, P., Gutwenger, C., Hong, S.H., Mutzel, P.: Algorithms and theory of computation handbook: special topics and techniques, pp. 6–6 (2010)

  4. Nishizeki, T., Miura, K., Rahman, M.S.: Algorithms for drawing plane graphs. IEICE Trans. Inf. Syst. 87(2), 281–289 (2004)

    Google Scholar 

  5. Jelínek, V., Jelínková, E., Kratochvíl, J., Lidický, B., Tesař, M., Vyskočil, T.: In: Proceedings of the 17th international conference on Graph Drawing (GD), pp. 304–315 (2010)

  6. Knauer, K.B., Micek, P., Walczak, B.: Proceedings of the 18th Annual International Conference on Computing and Combinatorics (COCOON). In: Gudmundsson J, Mestre J and Viglas T (eds) LNCS, vol. 7434, Springer, pp. 323–334 (2012)

  7. Wade, G.A., Chu, J.H.: Drawability of complete graphs using a minimal slope set. Comput. J. 37, 139–142 (1994)

    Article  MATH  Google Scholar 

  8. Dujmović, V., Eppstein, D., Suderman, M., Wood, D.R.: Drawings of planar graphs with few slopes and segments. Comput. Geom. 38(3), 194–212 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  9. Keszegh, B., Pach, J., Pálvölgyi, D.: Drawing planar graphs of bounded degree with few slopes. SIAM J. Discrete Math. 27(2), 1171–1183 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  10. Jelínek, V., Jelínková, E., Kratochvíl, J., Lidický, B., Tesar, M., Vyskocil, T.: The planar slope number of planar partial \(3\)-trees of bounded degree. Gr. Combinat. 29(4), 981–1005 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  11. Giacomo, E.D., Liotta, G., Montecchiani, F.: Drawing outer 1-planar graphs with few slopes. J. Gr. Algorithms Appl. 19(2), 707–741 (2015). https://doi.org/10.7155/jgaa.00376

    Article  MATH  MathSciNet  Google Scholar 

  12. Chaplick, S., Lozzo, G.D., Giacomo, E.D., Liotta, G., Montecchiani, F.: Algorithms and Data Structures-17th International Symposium, WADS 2021, Virtual Event, August 9–11, 2021, Proceedings. In: Lubiw A, Salavatipour MR (eds) Lecture Notes in Computer Science, vol. 12808, Springer, pp. 271–285. (2021) https://doi.org/10.1007/978-3-030-83508-8_20

  13. Kant, G.: Proceedings of the 18th International Workshop on Graph-Theoretic Concepts in Computer Science (WG). In: LNCS, vol. 657, Springer, pp. 263–276 (1992)

  14. Mondal, D., Nishat, R.I., Biswas, S., Rahman, M.S.: Minimum-segment convex drawings of \(3\)-connected cubic plane graphs. J. Comb. Optim. 25(3), 460–480 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  15. Di Giacomo, E., Liotta, G., Montecchiani, F.: Drawing subcubic planar graphs with four slopes and optimal angular resolution. Theor. Comput. Sci. 714, 51–73 (2018). https://doi.org/10.1016/j.tcs.2017.12.004

    Article  MATH  MathSciNet  Google Scholar 

  16. Formann, M., Hagerup, T., Haralambides, J., Kaufmann, M., Leighton, F.T., Symvonis, A., Welzl, E., Woeginger, G.: Drawing graphs in the plane with high resolution. SIAM J. Comput. 22(5), 1035–1052 (1993). https://doi.org/10.1137/0222063

    Article  MATH  MathSciNet  Google Scholar 

  17. Mukkamala, P., Pálvölgyi, D.: Proceedings of the 19th International Symposium of Graph Drawing (GD). In: van Kreveld MJ and Speckmann B (eds) LNCS, vol. 7034, Springer, pp. 254–265 (2011)

  18. Pach, J., Pálvölgyi, D.: Bounded-degree graphs can have arbitrarily large slope numbers. Electron. J. Combin. 13(1), 555 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  19. Barát, J., Matousek, J., Wood, D.R.: Bounded-degree graphs have arbitrarily large geometric thickness. Electron. J. Comb. 13(1), 554 (2006)

    MATH  MathSciNet  Google Scholar 

  20. Di Battista, G., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing. Prentice- Hall, Englewood Cliffs (1999)

    MATH  Google Scholar 

  21. Papakostas, A., Tollis, I.G.: Proceedings of the DIMACS International Workshop on Graph Drawing (GD). In: Tamassia R and Tollis IG (eds) LNCS, vol. 894, Springer, pp. 40–51 (1994)

  22. Malitz, S., Papakostas, A.: On the angular resolution of planar graphs. SIAM J. Discret. Math. 7(2), 172–183 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  23. Kindermann, P., Montecchiani, F., Schlipf, L., Schulz, A.: Drawing subcubic 1-planar graphs with few bends, few slopes, and large angles. J. Gr. Algorithms Appl. 25(1), 1–28 (2021). https://doi.org/10.7155/jgaa.00547

    Article  MATH  MathSciNet  Google Scholar 

  24. Kindermann, P., Montecchiani, F., Schlipf, L., Schulz, A.: Graph Drawing and Network Visualization - 26th International Symposium, GD 2018, Barcelona, Spain, September 26-28, 2018, Proceedings. In: Biedl TC and Kerren A (eds) Lecture Notes in Computer Science, vol. 11282, Springer, pp. 152–166. (2018) https://doi.org/10.1007/978-3-030-04414-5_11

  25. Brandes, U., Shubina, G.: R. Tamassia. In: de Leeuw, W.C., van Liere, R. (eds.) Data Visualization 2000, pp. 23–32. Springer Vienna, Vienna (2000)

    Chapter  Google Scholar 

  26. Igamberdiev, A., Meulemans, W., Schulz, A.: Drawing planar cubic 3-connected graphs with few segments: Algorithms & experiments. J. Gr. Algorithms Appl. 21(4), 561–588 (2017). https://doi.org/10.7155/jgaa.00430

    Article  MATH  MathSciNet  Google Scholar 

  27. Gutwenger, C., Mutzel, P.: Proceedings of the 8th International Symposium on Graph Drawing (GD). In: Marks J (ed) LNCS, 1984, Springer, pp. 77–90 (2001)

  28. Knauer, K.B., Micek, P., Walczak, B.: Outerplanar graph drawings with few slopes. Comput. Geom. 47(5), 614–624 (2014). https://doi.org/10.1016/j.comgeo.2014.01.003

    Article  MATH  MathSciNet  Google Scholar 

  29. Giacomo, E.D.: Graph Drawing. In: Liotta G (ed) Lecture Notes in Computer Science, vol. 2912, Springer, pp. 238–246 (2003)

  30. Giacomo, E.D., Liotta, G., Wismath, S.K.: Proceedings of the 14th Canadian Conference on Computational Geometry, University of Lethbridge, Alberta, Canada, August 12–14, 2002, pp. 149–153 (2002)

  31. Giacomo, E.D., Didimo, W., Liotta, G., Montecchiani, F.: Area requirement of graph drawings with few crossings per edge. Comput. Geom. 46(8), 909–916 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  32. Giacomo, E.D., Didimo, W., Liotta, G., Montecchiani, F.: WG. In: Golumbic MC, Stern M, Levy A and Morgenstern G (eds) Lecture Notes in Computer Science, vol. 7551, Springer, pp. 91–102 (2012)

  33. Chung, F.R.K., Leighton, F.T., Rosenberg, A.L.: Embedding graphs in books: a layout problem with applications to VLSI design. SIAM J. Algebr. Discrete Methods 8(1), 33–58 (1987)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

Part of this research took place during the 12th INRIA-McGill-Victoria Workshop on Computational Geometry, held in February 2–8, 2013, at the Bellairs Research Institute of McGill University. We thank the organizers and the participants for the opportunity and the useful discussions. Special thanks go to Zahed Rahmati for early thoughtful insights on the problems of this paper. The research of Giuseppe Liotta was partially supported by MUR Project “AHeAD”under PRIN 20174LF3T8. The research of Debajyoti Mondal is supported by NSERC.

Author information

Author notes

  1. William Lenhart, Giuseppe Liotta, Debajyoti Mondal and Rahnuma Islam Nishat these authors contributed equally to this work.

Authors and Affiliations

  1. Department of Computer Science, Williams College, Williamstown, MA, USA

    William Lenhart

  2. Dipartimento di Ingegneria, Universita’ degli Studi di Perugia, Perugia, Italy

    Giuseppe Liotta

  3. Department of Computer Science, University of Saskatchewan, Saskatoon, SK, Canada

    Debajyoti Mondal

  4. Department of Computer Science, Mathematics, Physics and Statistics, University of British Columbia, Kelowna, BC, Canada

    Rahnuma Islam Nishat

Authors
  1. William Lenhart
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Giuseppe Liotta
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Debajyoti Mondal
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Rahnuma Islam Nishat
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rahnuma Islam Nishat.

Ethics declarations

Conflict of interest

The authors have no conflicts of interest to declare that are relevant to the content of this article.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

A preliminary version of this work appeared in Proceedings of GD 2013 [1].

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lenhart, W., Liotta, G., Mondal, D. et al. Drawing Partial 2-Trees with Few Slopes. Algorithmica 85, 1156–1175 (2023). https://doi.org/10.1007/s00453-022-01065-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-022-01065-0

Keywords

Search

Navigation