Straight-Line Drawings on Restricted Integer Grids in Two and Three Dimensions

(Extended Abstract)
  • Stefan Felsner
  • Giuseppe Liotta
  • Stephen Wismath
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2265)

Abstract

This paper investigates the following question: Given an integer grid ϕ, where ϕ is a proper subset of the integer plane or a proper subset of the integer 3d space, which graphs admit straight-line crossingfree drawings with vertices located at the grid points of ϕ? We characterize the trees that can be drawn on a two dimensional c · n × к grid, where к and c are given integer constants, and on a two dimensional grid consisting of к parallel horizontal lines of infinite length. Motivated by the results on the plane we investigate restrictions of the integer grid in 3 dimensions and show that every outerplanar graph with n vertices can be drawn crossing-free with straight lines in linear volume on a grid called a prism. This prism consists of 3n integer grid points and is universal — it supports all outerplanar graphs of n vertices. This is the first algorithm that computes crossing-free straight line 3d drawings in linear volume for a non-trivial family of planar graphs. We also show that there exist planar graphs that cannot be drawn on the prism and that the extension to a n × 2 × 2 integer grid, called a box, does not admit the entire class of planar graphs.

References

  1. 1.
    T. Calamoneri and A. Sterbini. Drawing 2-, 3-, and 4-colorable graphs in o(n 2) volume. In S. North, editor, Graph Drawing (Proc. GD’ 96), volume 1190 of Lecture Notes Comput. Sci., pages 53–62. Springer-Verlag, 1997.Google Scholar
  2. 2.
    T.M. Chan. A near-linear area bound for drawing binary trees. In Proc. 10th Annu. ACM-SIAM Sympos. on Discrete Algorithms., pages 161–168, 1999.Google Scholar
  3. 3.
    M. Chrobak and G. Kant. Convex grid drawings of 3-connected planar graphs. Internat. J. Comput. Geom. Appl., 7(3):211–223, 1997.CrossRefMathSciNetGoogle Scholar
  4. 4.
    Marek Chrobak, Michael T. Goodrich, and Roberto Tamassia. Convex drawings of graphs in two and three dimensions. In Proc. 12th Annu. ACM Sympos. Comput. Geom., pages 319–328, 1996.Google Scholar
  5. 5.
    Marek Chrobak and Shin ichi Nakano. Minimum-width grid drawings of plane graphs. Comput. Geom. Theory Appl., 11:29–54, 1998.MATHMathSciNetGoogle Scholar
  6. 6.
    R. F. Cohen, P. Eades, T. Lin, and F. Ruskey. Three-dimensional graph drawing. Algorithmica, 17:199–208, 1997.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    H. de Fraysseix, J. Pach, and R. Pollack. Small sets supporting Fary embeddings of planar graphs. In Proc. 20th ACMSymp os. Theory Comput., pages 426–433, 1988.Google Scholar
  8. 8.
    H. de Fraysseix, J. Pach, and R. Pollack. How to draw a planar graph on a grid. Combinatorica, 10(1):41–51, 1990.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    G. Di Battista, P. Eades, R. Tamassia, and I. G. Tollis. Graph Drawing. Prentice Hall, Upper Saddle River, NJ, 1999.MATHCrossRefGoogle Scholar
  10. 10.
    Reinhard Diestel. Graph theory. Graduate Texts in Mathematics. 173. Springer, 2000. Transl. from the German. 2nd ed.Google Scholar
  11. 11.
    V. Dujmovic, M. Fellows, M. Hallett, M. Kitching, G. Liotta, C. McCartin, N. Nishimura, P. Ragde, F. Rosamond, M. Suderman, S. Whitesides, D. R. Wood. On the Parameterized Complexity of Layered Graph Drawing. ESA, 1–12, 2001.Google Scholar
  12. 12.
    Stefan Felsner. Convex drawings of planar graphs and the order dimension of 3-polytopes. Order-accepted to appear.Google Scholar
  13. 13.
    A. Garg, M. T. Goodrich, and R. Tamassia. Planar upward tree drawings with optimal area. Internat. J. Comput. Geom. Appl., 6:333–356, 1996.MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    A. Garg, R. Tamassia, and P. Vocca. Drawing with colors. In Proc. 4th Annu. European Sympos. Algorithms, volume 1136 of Lecture Notes Comput. Sci., pages 12–26. Springer-Verlag, 1996.Google Scholar
  15. 15.
    Michael Juenger and Sebastian Leipert. Level planar embedding in linear time. In J. Kratochvil, editor, Graph Drawing (Proc. GD’ 99), volume 1731 of Lecture Notes Comput. Sci., pages 72–81. Springer-Verlag, 1999.Google Scholar
  16. 16.
    G. Kant. A new method for planar graph drawings on a grid. In Proc. 33rd Annu. IEEE Sympos. Found. Comput. Sci., pages 101–110, 1992.Google Scholar
  17. 17.
    G. Kant. Drawing planar graphs using the canonical ordering. Algorithmica, 16:4–32, 1996.MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    János Pach, Torsten Thiele, and Géza Tóth. Three-dimensional grid drawings of graphs. In G. Di Battista, editor, Graph Drawing (Proc. GD’ 97), volume 1353 of Lecture Notes Comput. Sci., pages 47–51. Springer-Verlag, 1997.Google Scholar
  19. 19.
    F. P. Preparata and M. I. Shamos. Computational Geometry: An Introduction. Springer-Verlag, 3rd edition, October 1990.Google Scholar
  20. 20.
    W. Schnyder. Embedding planar graphs on the grid. In Proc. 1st ACM-SIAM Sympos. Discrete Algorithms, pages 138–148, 1990.Google Scholar
  21. 21.
    W. Schnyder and W. T. Trotter. Convex embeddings of 3-connected plane graphs. Abstracts of the AMS, 13(5):502, 1992.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Stefan Felsner
    • 1
  • Giuseppe Liotta
    • 2
  • Stephen Wismath
    • 3
  1. 1.Fachbereich Mathematik und InformatikFreie Universität BerlinBerlinGermany
  2. 2.Dipartimento di Ingegneria Elettronica e dell’InformazioneUniversità degli Studi di PerugiaVia DurantiItaly
  3. 3.Dept. of Mathematics and Computer ScienceU. of LethbridgeAlbertaCanada

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