Three-Dimensional Orthogonal Graph Drawing with Optimal Volume

  • Therese Biedl
  • Torsten Thiele
  • David R. Wood
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1984)


In this paper, we study three-dimensional orthogonal box-drawings of graphs without loops. We provide lower bounds for three scenarios: (1) drawings where vertices have bounded aspect ratio, (2) drawings where the surface of vertices is proportional to their degree, and (3) drawings without any such restrictions. Then we give constructions that match the lower bounds in all scenarios within an order of magnitude.


  1. 1.
    A. Aggarwal, M. Klawe, and P. Shor. Multilayer grid embeddings for VLSI. Algorithmica, 6(1):129–151, 1991.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    N. Alon and J. Spencer. The Probabilistic Method. John Wiley & Sons, 1992.Google Scholar
  3. 3.
    T. Biedl. Three approaches to 3D-orthogonal box-drawings. In Whitesides [17], pages 30–43.Google Scholar
  4. 4.
    T. Biedl and T. Chan. Cross-coloring: improving the technique by Kolmogorov and Barzdin. Technical Report CS-2000-13. Department of Computer Science, University of Waterloo, Canada, 2000.Google Scholar
  5. 5.
    T. Biedl, T. Shermer, S. Whitesides, and S. Wismath. Bounds for orthogonal 3-D graph drawing. J. Graph Alg. Appl., 3(4):63–79, 1999.MATHMathSciNetGoogle Scholar
  6. 6.
    T. Biedl, T. Thiele and D. R. Wood. Three-Dimensional Orthogonal Graph Drawing with Optimal Volume. Technical Report CS-2000-12. Department of Computer Science, University of Waterloo, Canada, 2000.Google Scholar
  7. 7.
    M. Closson, S. Gartshore, J. Johansen, and S. K. Wismath. Fully dynamic 3-dimensional orthogonal graph drawing. In Kratochvíl [14], pages 49–58.Google Scholar
  8. 8.
    G. Di Battista, M. Patrignani, and F. Vargiu. A split&push approach to 3D orthogonal drawing. In Whitesides [17], pages 87–101.Google Scholar
  9. 9.
    P. Eades, C. Stirk, and S. Whitesides. The techniques of Kolmogorov and Bardzin for three-dimensional orthogonal graph drawings. Information Processing Letters, 60(2):97–103, 1996.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    P. Eades, A. Symvonis, and S. Whitesides. Three dimensional orthogonal graph drawing algorithms. Discrete Applied Math., 103(1–3):55–87, 2000.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    É. K. Fogel. An elementary proof of formulae of de la Vallée Poussin (In Russian). Latvijas PSR Zinatnu Akad. Vestis, 11(40):123–130, 1950.MathSciNetGoogle Scholar
  12. 12.
    K. Hagihara, N. Tokura, and N. Suzuki. Graph embedding on a three-dimensional model. Systems-Comput.-Controls, 14(6):58–66, 1983.MathSciNetGoogle Scholar
  13. 13.
    D. Kleitman and M. Krieger. An optimal bound for two dimensional bin packing. In 16th Annual Symposium on Foundations of Computer Science (FOCS’75), pages 163–168. IEEE, 1975.Google Scholar
  14. 14.
    J. Kratochvíl, editor. Symposium on Graph Drawing 99, volume 1731 of Lecture Notes in Computer Science. Springer-Verlag, 1999.Google Scholar
  15. 15.
    A. Lubotzky, R. Phillips, and P. Sarnak. Ramanujan graphs. Combinatorica, 8:261–277, 1988.MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    A. Papakostas and I. Tollis. Incremental orthogonal graph drawing in three dimensions. J. Graph Alg. Appl., 3(4):81–115, 1999.MATHMathSciNetGoogle Scholar
  17. 17.
    S. Whitesides, editor. Symposium on Graph Drawing 98, volume 1547 of Lecture Notes in Computer Science. Springer-Verlag, 1998.Google Scholar
  18. 18.
    D. R. Wood. An algorithm for three-dimensional orthogonal graph drawing. In Whitesides [17], pages 332–346.Google Scholar
  19. 19.
    D. R. Wood. Multi-dimensional orthogonal graph drawing with small boxes. In Kratochvíl [14], pages 311–322.Google Scholar
  20. 20.
    D. R. Wood. A new algorithm and open problems in three-dimensional orthogonal graph drawing. In R. Raman and J. Simpson, editors, Proc. Australasian Workshop on Combinatorial Algorithms (AWOCA’99), pages 157–167. Curtin University of Technology, Perth, 1999.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Therese Biedl
    • 1
  • Torsten Thiele
    • 2
  • David R. Wood
    • 3
  1. 1.Department of Computer ScienceUniversity of WaterlooWaterlooCanada
  2. 2.Institut für InformatikFreie Universität BerlinBerlinGermany
  3. 3.Basser Department of Computer ScienceThe University of SydneySydneyAustralia

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