Lower Bounds for the Number of Bends in Three-Dimensional Orthogonal Graph Drawings

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


In this paper we present the first non-trivial lower bounds for the total number of bends in 3-D orthogonal drawings of maximum degree six graphs. In particular, we prove lower bounds for the number of bends in 3-D orthogonal drawings of complete simple graphs and multigraphs, which are tight in most cases. These result are used as the basis for the construction of infinite classes of c-connected simple graphs and multigraphs (2 ≤c ≤6) of maximum degree Δ (3 ≤Δ ≤6) with lower bounds on the total number of bends for all members of the class. We also present lower bounds for the number of bends in general position 3-D orthogonal graph drawings. These results have significant ramifications for the ‘2-bends’ problem, which is one of the most important open problems in the field.


  1. 1.
    T. C. Biedl. New lower bounds for orthogonal drawings. J. Graph Algorithms Appl., 2(7):1–31, 1998. 260MathSciNetGoogle Scholar
  2. 2.
    M. Closson, S. Gartshore, J. Johansen, and S. K. Wismath. Fully dynamic 3-dimensional orthogonal graph drawing. In J. Kratochvil, editor, Proc. Graph Drawing: 7th International Symp. (GD’99), volume 1731 of Lecture Notes in Comput. Sci., pages 49–58, Springer, 1999. 259, 261Google Scholar
  3. 3.
    G. Di Battista, M. Patrignani, and F. Vargiu. A split&push approach to 3D orthogonal drawing. In Whitesides [10], pages 87–101. 259, 261Google Scholar
  4. 4.
    P. Eades, C. Stirk, and S. Whitesides. The techniques of Komolgorov and Bardzin for three dimensional orthogonal graph drawings. Inform. Proc. Lett., 60(2):97–103, 1996. 259, 260MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    P. Eades, A. Symvonis, and S. Whitesides. Three dimensional orthogonal graph drawing algorithms. Discrete Applied Math., 103:55–87, 2000. 259, 260, 261, 261, 261, 261, 261MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    A. N. Kolmogorov and Ya. M. Barzdin. On the realization of nets in 3-dimensional space. Problems in Cybernetics, 8:261–268, March 1967. 259, 260, 261Google Scholar
  7. 7.
    A. Papakostas and I. G. Tollis. Algorithms for incremental orthogonal graph drawing in three dimensions. J. Graph Algorithms Appl., 3(4):81–115, 1999. 259, 261, 261, 261MATHMathSciNetGoogle Scholar
  8. 8.
    M. Patrignani and F. Vargiu. 3DCube: a tool for three dimensional graph drawing. In G. Di Battista, editor, Proc. Graph Drawing: 5th International Symp. (GD’97), volume 1353 of Lecture Notes in Comput. Sci., pages 284–290, Springer, 1998. 259, 261Google Scholar
  9. 9.
    R. Tamassia, I. G. Tollis, and J. S. Vitter. Lower bounds for planar orthogonal drawings of graphs. Inform. Process. Lett., 39(1):35–40, 1991. 260MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    S. Whitesides, editor. Proc. Graph Drawing: 6th International Symp. (GD’98), volume 1547 of Lecture Notes in Comput. Sci., Springer, 1998. 271, 271Google Scholar
  11. 11.
    D. R. Wood. On higher-dimensional orthogonal graph drawing. In J. Harland, editor, Proc. Computing: the Australasian Theory Symp. (CATS’97), volume 19(2) of Austral. Comput. Sci. Comm., pages 3–8, 1997. 260, 260Google Scholar
  12. 12.
    D. R. Wood. An algorithm for three-dimensional orthogonal graph drawing. In Whitesides [10], pages 332–346. 259, 261, 262, 268, 269Google Scholar
  13. 13.
    D. R. Wood. Lower bounds for the number of bends in three-dimensional orthogonal graph drawings. Technical Report CS-AAG-2000-01, Basser Department of Computer Science, The University of Sydney, 2000. 259, 262Google Scholar
  14. 14.
    D. R. Wood. Three-Dimensional Orthogonal Graph Drawing. PhDthesis, School of Computer Science and Software Engineering, Monash University, Australia, 2000. 259, 260, 261, 261, 261, 268Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • David R. Wood
    • 1
  1. 1.Basser Department of Computer ScienceThe University of SydneySydneyAustralia

Personalised recommendations