Discrete & Computational Geometry

, Volume 1, Issue 4, pp 343–353 | Cite as

Rectilinear planar layouts and bipolar orientations of planar graphs

  • Pierre Rosenstiehl
  • Robert E. Tarjan
Article

Abstract

We propose a linear-time algorithm for generating a planar layout of a planar graph. Each vertex is represented by a horizontal line segment and each edge by a vertical line segment. All endpoints of the segments have integer coordinates. The total space occupied by the layout is at mostn by at most 2n–4. Our algorithm, a variant of one by Otten and van Wijk, generally produces a more compact layout than theirs and allows the dual of the graph to be laid out in an interlocking way. The algorithm is based on the concept of abipolar orientation. We discuss relationships among the bipolar orientations of a planar graph.

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References

  1. 1.
    K. S. Booth and G. S. Lueker, Testing for the consecutive ones property, interval graphs, and graph planarity usingPQ-tree algorithms,J. Comput. System Sci. 13 (1976), 335–379.MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    M. H. Brown and R. Sedgewick, A system for algorithm animation, Technical Report No. CS-84-01, Department of Computer Science, Brown University, Providence, RI, 1984.Google Scholar
  3. 3.
    N. Chiba, T. Yamanouchi, and T. Nishizeki, Linear algorithms for convex drawings of planar graphs,Proceedings of the Silver Jubilee Conference on Combinatorics, 1982, University of Waterloo, Waterloo, Ontario, to appear.Google Scholar
  4. 4.
    R. Cori and J. Hardouin-Duparc, Manipulation des cartes planaires à partir de leur codage, Département de Mathématiques, Université de Bordeaux, Talence, France, 1975.Google Scholar
  5. 5.
    P. Duchet, Y. Hamidoune, M. Las Vergnas, and H. Meyniel, Representing a planar graph by vertical lines joining different intervals,Discrete Math. 46 (1983), 319–321.MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    J. Ebert,st-ordering the vertices of biconnected graphs,Computing 30 (1983), 19–33.MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    S. Even and R. E. Tarjan, Computing anst-numbering,Theoret. Comput. Sci. 2 (1976), 339–344.MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    I. Fáry, On straight line representing of planar graphs,Acta. Sci. Math. (Szeged)11 (1948), 229–233.MATHGoogle Scholar
  9. 9.
    H. de Fraysseix, Drawing planar and non-planar graphs from the half-edge code, to appear.Google Scholar
  10. 10.
    H. de Fraysseix and P. Rosenstiehl, Structures combinatoires pour des traćes automatiques de réseaux,Proceedings of the Third European Conference on CAD/CAM and Computer Graphics, 332–337, 1984.Google Scholar
  11. 11.
    H. de Fraysseix and P. Rosenstiehl, L'algorithme gauche-droite pour le plongement des graphes dans le plan, to appear.Google Scholar
  12. 12.
    D. H. Greene, Efficient coding and drawing of planar graphs, Xerox Palo Alto Research Center, Palo Alto, CA, 1983.Google Scholar
  13. 13.
    J. Hopcroft and R. Tarjan, Efficient planarity testing,J. Comput. Mach. 21 (1974), 549–568.MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    D. E. Knuth,The Art of Computer Programming, Vol. 1, 2nd ed., 258–265, Addison-Wesley, Reading, MA, 1973.Google Scholar
  15. 15.
    A. Lempel, S. Even, and I. Cederbaum, An algorithm for planarity testing of graphs,Theory of Graphs (International Symposium, Rome, July 1966) (P. Rosenstiehl, ed.), 215–232, Gordon and Breach, New York, 1967.Google Scholar
  16. 16.
    C. Mead and L. Conway,Introduction to VLSI Systems, Addison-Wesley, Reading, MA, 1980.Google Scholar
  17. 17.
    R. H. J. M. Otten and J. G. van Wijk, Graph representations in interactive layout design,Proceedings of the IEEE International Symposium on Circuits and Systems 914–918, 1978.Google Scholar
  18. 18.
    P. Rosenstiehl, Embedding in the plane with orientation constraints,Ann. N.Y. Acad. Sci., to appear.Google Scholar
  19. 19.
    Y. Shiloach, Arrangement of planar graphs on the planar lattice, Department of Applied Mathematics, The Weizmann Institute of Science, Rehovot, Israel, 1976.Google Scholar
  20. 20.
    R. Tamassia and I. G. Tollis, Algorithms for visibility representations of planar graphs, Coordinated Science Laboratory, University of Illinois, Urbana, IL, 1985.Google Scholar
  21. 21.
    R. E. Tarjan, Finding dominators in directed graphs,SIAM J. Comput. 3 (1974), 62–69.MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    R. E. Tarjan, Two streamlined depth-first search algorithms,Fund. Inform.,IX (1986), 85–94.MathSciNetGoogle Scholar
  23. 23.
    W. T. Tutte, How to draw a graph,Proc. London Math. Soc. 13 (1963), 743–768.MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    D. R. Woods, Drawing planar graphs, Report No. STAN-CS-82-943, Computer Science Department, Stanford University, Stanford, CA, 1981.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1986

Authors and Affiliations

  • Pierre Rosenstiehl
    • 1
  • Robert E. Tarjan
    • 2
    • 3
  1. 1.Centre de Mathematique SocialeParisFrance
  2. 2.Computer Science DepartmentPrinceton UniversityPrincetonUSA
  3. 3.AT&T Bell LaboratoriesMurray HillUSA

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