Orthogonal Drawings of Cycles in 3D Space

Extended Abstract
  • Giuseppe Di Battista
  • Giuseppe Liotta
  • Anna Lubiw
  • Sue Whitesides
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1984)

Abstract

Let C be a directed cycle, whose edges have each been assigned a desired direction in 3D (East, West, North, South, Up, or Down) but no length. We say that C is a shape cycle. We consider the following problem. Does there exist an orthogonal drawing Γ of C in 3D such that each edge of Γ respects the direction assigned to it and such that Γ does not intersect itself? If the answer is positive, we say that C is simple. This problem arises in the context of extending orthogonal graph drawing techniques and VLSI rectilinear layout techniques from 2D to 3D. We give a combinatorial characterization of simple shape cycles that yields linear time recognition and drawing algorithms.

References

  1. 1.
    T. C. Biedl. Heuristics for 3d-orthogonal graph drawings. In Proc. 4th Twente Workshop on Graphs and Combinatorial Optimization, pp. 41–44, 1995.Google Scholar
  2. 2.
    T. Biedl, T. Shermer, S. Wismath, and S. Whitesides. Orthogonal 3-D graph drawing. J. Graph Algorithms and Applications, 3(4):63–79, 1999.MATHMathSciNetGoogle Scholar
  3. 3.
    R. F. Cohen, P. Eades, T. Lin and F. Ruskey. Three-dimensional graph drawing. Algorithmica, 17(2):199–208, 1997.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    G. Di Battista, P. Eades, R. Tamassia, and I. Tollis. Graph Drawing. Prentice Hall, 1999.Google Scholar
  5. 5.
    G. Di Battista, G. Liotta, A. Lubiw, and S. Whitesides. Embedding problems for paths with direction constrained edges. In D.-Z. Du, P. Eades, V. Estivill-Castro, X. Lin, and A. Sharma, eds., Computing and Combinatorics, 6th Ann. Int. Conf., COCOON 2000, Springer-Verlag LNCS vol. 1858, pp. 64–73, 2000.Google Scholar
  6. 6.
    G. Di Battista and L. Vismara. Angles of planar triangular graphs. SIAM J. Discrete Math., 9(3):349–359, 1996.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    P. Eades, C. Stirk, and S. Whitesides. The techniques of Komolgorov and Bardzin for three dimensional orthogonal graph drawings. Inform. Process. Lett., 60:97–103, 1996.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    P. Eades, A. Symvonis, and S. Whitesides. Three-dimensional orthogonal graph drawing algorithms. Discrete Applied Math., vol. 103, pp. 55–87, 2000.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    A. Garg. New results on drawing angle graphs. Comput. Geom. Theory Appl., 9(1–2):43–82, 1998. Special Issue on Geometric Representations of Graphs, G. Di Battista and R. Tamassia, eds..Google Scholar
  10. 10.
    A. Papakostas and I. Tollis. Algorithms for incremental orthogonal graph drawing in three dimensions. J. Graph Algorithms and Appl., 3(4):81–115, 1999.MATHMathSciNetGoogle Scholar
  11. 11.
    R. Tamassia. On embedding a graph in the grid with the minimum number of bends. SIAM J. Comput., 16(3):421–444, 1987.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    G. Vijayan and A. Wigderson. Rectilinear graphs and their embeddings. SIAM J. Comput., 14:355–372, 1985.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    V. Vijayan. Geometry of planar graphs with angles. In Proc. 2nd Annu. ACM Sympos. Comput. Geom., pp. 116–124, 1986.Google Scholar
  14. 14.
    D. R. Wood. Two-bend three-dimensional orthogonal grid drawing of maximum degree five graphs. TR 98/03, School of Computer Science and Software Engineering, Monash University, 1998.Google Scholar
  15. 15.
    D. R. Wood. An algorithm for three-dimensional orthogonal graph drawing. In S. Whitesides, ed., Graph Drawing (6th Int. Symp., GD’ 98), Springer-Verlag, LNCS vol. 1547, pp. 332–346, 1998.Google Scholar
  16. 16.
    D. R. Wood. Three-Dimensional Orthogonal Graph Drawing. Ph.D. thesis, School of Computer Science and Software Engineering, Monash University, 2000.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Giuseppe Di Battista
    • 1
  • Giuseppe Liotta
    • 2
  • Anna Lubiw
    • 3
  • Sue Whitesides
    • 4
  1. 1.Dipartimento di Informatica ed AutomazioneUniversità di Roma TreRomaItaly
  2. 2.Dipartimento di Ingegneria Elettronica e dell’InformazioneUniversità di PerugiaPerugiaItaly
  3. 3.Department of Computer ScienceUniversity of WaterlooWaterlooCanada
  4. 4.School of Computer ScienceMcGill UniversityMontrealCanada

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