Advertisement

Universal Point Sets for Planar Three-Trees

  • Radoslav Fulek
  • Csaba D. Tóth
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8037)

Abstract

For every n ∈ ℕ, we present a set S n of O(n 5/3) points in the plane such that every planar 3-tree with n vertices has a straight-line embedding in the plane in which the vertices are mapped to a subset of S n . This is the first subquadratic upper bound on the size of universal point sets for planar 3-trees, as well as for the class of 2-trees and serial parallel graphs.

Keywords

planar 3-tree universal point set straight-line embedding 

References

  1. 1.
    Angelini, P., Di Battista, G., Kaufmann, M., Mchedlidze, T., Roselli, V., Squarcella, C.: Small point sets for simply-nested planar graphs. In: Speckmann, B. (ed.) GD 2011. LNCS, vol. 7034, pp. 75–85. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  2. 2.
    Biedl, T.: Small drawings of outerplanar graphs, series-parallel graphs, and other planar graphs. Discrete Computational Geometry 45, 141–160 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Biedl, T., Vatshelle, M.: The point-set embeddability problem for plane graphs, in. In: Proc. Symposuim on Computational Geometry, pp. 41–50. ACM Press (2011)Google Scholar
  4. 4.
    Bose, P.: On embedding an outer-planar graph in a point set. Computational Geometry: Theory and Applications 23(3), 303–312 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Brandenburg, F.-J.: Drawing planar graphs on \(\frac{8}{9}n^2\) area. Electronic Notes in Discrete Mathematics 31, 37–40 (2008)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bukh, B., Matoušek, J., Nivasch, G.: Lower bounds for weak epsilon-nets and stair-convexity. Israel Journal of Mathematics 182, 199–228 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Cabello, S.: Planar embeddability of the vertices of a graph using a fixed point set is NP-hard. Journal of Graph Algorithms and Applications 10(2), 353–363 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Chrobak, M., Karloff, H.J.: A lower bound on the size of universal sets for planar graphs. SIGACT News 20(4), 83–86 (1989)CrossRefGoogle Scholar
  9. 9.
    Chrobak, M., Payne, T.: A linear time algorithm for drawing a planar graph on a grid. Information Processing Letters 54, 241–246 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10(1), 41–51 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Di Battista, G., Frati, F.: Small area drawings of outerplanar graphs. Algorithmica 54(1), 25–53 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Dolev, D., Leighton, F.T., Trickey, H.: Planar embedding of planar graphs. In: Preparata, F. (ed.) Advances in Computing Research, vol. 2. JAI Press Inc., London (1984)Google Scholar
  13. 13.
    Dujmović, V., Evans, W., Lazard, S., Lenhart, W., Liotta, G., Rappaport, D., Wismath, S.: On point-sets that support planar graphs. Computational Geometry: Theory and Applications 46(1), 29–50 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Durocher, S., Mondal, D.: On the hardness of point-set embeddability. In: Rahman, M.S., Nakano, S.-I. (eds.) WALCOM 2012. LNCS, vol. 7157, pp. 148–159. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  15. 15.
    Durocher, S., Mondal, D., Nishat, R.I., Rahman, M.S., Whitesides, S.: Embedding plane 3-trees in ℝ2 and ℝ3. In: Speckmann, B. (ed.) GD 2011. LNCS, vol. 7034, pp. 39–51. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  16. 16.
    Everett, H., Lazard, S., Liotta, G., Wismath, S.: Universal sets of n points for one-bend drawings of planar graphs with n vertices. Discrete and Computational Geometry 43(2), 272–288 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Fáry, I.: On straight lines representation of plane graphs. Acta Scientiarum Mathematicarum (Szeged) 11, 229–233 (1948)zbMATHGoogle Scholar
  18. 18.
    Hossain, M. I., Mondal, D., Rahman, M. S., Salma, S.A.: Universal line-sets for drawing planar 3-trees. In: Rahman, M.S., Nakano, S.-I. (eds.) WALCOM 2012. LNCS, vol. 7157, pp. 136–147. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  19. 19.
    Frati, F.: Lower bounds on the area requirements of series-parallel graphs. Discrete Mathematics and Theoretical Computer Science 12(5), 139–174 (2010)MathSciNetGoogle Scholar
  20. 20.
    Frati, F., Patrignani, M.: A note on minimum-area straight-line drawings of planar graphs. In: Hong, S.-H., Nishizeki, T., Quan, W. (eds.) GD 2007. LNCS, vol. 4875, pp. 339–344. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  21. 21.
    Fulek, R., Tóth, C.D.: Universal point sets for planar three-tree, http://arxiv.org/abs/1212.6148
  22. 22.
    Gritzmann, P., Mohar, B., Pach, J., Pollack, R.: Embedding a planar triangulation with vertices at specified positions. American Mathematic Monthly 98, 165–166 (1991)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Kurowski, M.: A 1.235 lower bound on the number of points needed to draw all n-vertex planar graphs. Information Processing Letters 92, 95–98 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Nishat, R., Mondal, D., Rahman, M.S.: Point-set embeddings of plane 3-trees. Computational Geometry: Theory and Applications 45(3), 88–98 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Schnyder, W.: Embedding planar graphs in the grid, in. In: Proc. 1st Symposium on Discrete Algorithms, pp. 138–147. ACM Press (1990)Google Scholar
  26. 26.
    Zhou, X., Hikino, T., Nishizeki, T.: Small grid drawings of planar graphs with balanced partition. Journal of Combinatorial Optimization 24(2), 99–115 (2012)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Radoslav Fulek
    • 1
  • Csaba D. Tóth
    • 2
    • 3
  1. 1.Charles UniversityPragueCzech Republic
  2. 2.California State UniversityNorthridgeUSA
  3. 3.University of CalgaryCanada

Personalised recommendations