Drawing stressed planar graphs in three dimensions

  • Peter Eades
  • Patrick Garvan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1027)

Abstract

There is much current interest among researchers to find algorithms that will draw graphs in three dimensions. It is well known that every 3-connected planar graph can be represented as a strictly convex polyhedron. However, no practical algorithms exist to draw a general 3-connected planar graph as a convex polyhedron. In this paper we review the concept of a stressed graph and how it relates to convex polyhedra; we present a practical algorithm that uses stressed graphs to draw 3-connected planar graphs as strictly convex polyhedra; and show some examples.

Key words

graph stressed graph convex polyhedron reciprocal polyhedron 

References

  1. 1.
    H. S. M. Coxeter. Regular Polytopes. Dover, NY, 1973.Google Scholar
  2. 2.
    I. Cahit. Drawing the Complete Graph in 3-D with Straight Lines and Without Crossings. Bulletin of the ICA, Vol 12, Sep 94.Google Scholar
  3. 3.
    R. F. Cohen, P. Eades, T. Lin, and F. Ruskey. Three-Dimensional Graph Drawing. Proc. Graph Drawing 94. Lecture Notes in Computer Science. Volume 8941–11. Springer-Verlag, Berlin, 1995.Google Scholar
  4. 4.
    G. Das, M. T. Goodrich. On the Complexity of Approximating and Illuminating Three-Dimensional Convex Polyhedra. To appear, WADS 1995.Google Scholar
  5. 5.
    R. Davidson and D. Harel. Drawing Graphs Nicely Using Simulated Annealing, Technical Report CS89-13, Department of Applied Mathematics and Computer Science, The Weizmann Institute of Science, Rehovot, Israel, July 1989.Google Scholar
  6. 6.
    T. M. J. Fruchterman and E. M. Reingold. Graph Drawing by Force-directed Placement. Software — Practice and Experience, vol. 21(11), 1129–1164 (Nov 1991).Google Scholar
  7. 7.
    B. Grünbaum. Convex Polytopes. Wiley, NY, 1967.Google Scholar
  8. 8.
    B. Grünbaum and G. C. Shephard. Duality of Polyhedra. In Shaping Space: A Polyhedral Approach. Birkhauser, Boston, 1988.Google Scholar
  9. 9.
    J. E. Hopcroft and P. J. Kahn. A Paradigm for Robust Geometric Algorithms. Algorithmica (1992) 7:339–380.CrossRefMathSciNetGoogle Scholar
  10. 10.
    C. L. Liu. Introduction to Combinatorial Mathematics. McGraw-Hill, NY, 1968.Google Scholar
  11. 11.
    J. C. Maxwell. On Reciprocal Figures and Diagrams of Forces. Phil. Mag. S. 4. vol. xxvii. (1864) pp.250–261Google Scholar
  12. 12.
    T. Nishizeki and N. Chiba. Planar Graphs: Theory and Algorithms. Annals of Discrete Mathematics, Volume 32. North-Holland, Netherlands, 1988.Google Scholar
  13. 13.
    W. T. Tutte. How to Draw a Graph. Proc. London. Math. Soc. (3), 13 (1963), 743–768.Google Scholar

Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • Peter Eades
    • 1
  • Patrick Garvan
    • 1
  1. 1.Dept. of Computer ScienceUniversity of NewcastleNewcastleAustralia

Personalised recommendations