Planarity-Preserving Clustering and Embedding for Large Planar Graphs

  • Christian A. Duncan
  • Michael T. Goodrich
  • Stephen G. Kobourov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1731)


In this paper we present a novel approach for cluster-based drawing of large planar graphs that maintains planarity. Our technique works for arbitrary planar graphs and produces a clustering which satisfies the conditions for compound-planarity (c-planarity). Using the clustering, we obtain a representation of the graph as a collection of O(log n) layers, where each succeeding layer represents the graph in an increasing level of detail. At the same time, the difference between two graphs on neighboring layers of the hierarchy is small, thus preserving the viewer’s mental map. The overall running time of the algorithm is O(n log n), where n is the number of vertices of graph G.


  1. 1.
    C. A. Duncan, M. T. Goodrich, and S. G. Kobourov. Balanced aspect ratio trees and their use for drawing very large graphs. Proc. of 6th Symposium on Graph Drawing (GD’98), LNCS 1190:101–112, 1998.Google Scholar
  2. 2.
    P. Eades and Q. W. Feng. Multilevel visualization of clustered graphs. Proc. of 4th Symposium on Graph Drawing (GD’96), LNCS 1190:101–112, 1996.Google Scholar
  3. 3.
    P. Eades, Q. W. Feng, and X. Lin. Straight-line drawing algorithms for hierarchical graphs and clustered graphs. Proc. of the 4th Symposium on Graph Drawing (GD’96), LNCS 1190:113–128, 1997.Google Scholar
  4. 4.
    I. Fary. On straight lines representation of planar graphs. Acta Sci. Math. Szeged, 11:229–233, 1948.MathSciNetGoogle Scholar
  5. 5.
    Q.-W. Feng, R. F. Cohen, and P. Eades. Planarity for clustered graphs. ESA’95, LNCS 979:213–226, 1995.Google Scholar
  6. 6.
    R. J. Lipton and R. E. Tarjan. A separator theorem for planar graphs. SIAM J. Appl. Math., 36:177–189, 1979.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    F. J. Newbery. Edge concentration: A method for clustering directed graphs. In Proceedings of the 2nd International Workshop on Software Configuration Management, pages 76–85, Princeton, New Jersey, October 1989.Google Scholar
  8. 8.
    S. C. North. Drawing ranked digraphs with recursive clusters. ALCOM International Workshop PARIS 1993 on Graph Drawing and Topological Graph Algorithms (GD’93), September 1993.Google Scholar
  9. 9.
    Sablowski and Frick. Automatic graph clustering. Proc. of 4th Symposium on Graph Drawing (GD’96), LNCS 1190:395–400, 1996.Google Scholar
  10. 10.
    W. Schnyder. Embedding planar graphs on the grid. In Proc. 1st ACM-SIAM Sympos. Discrete Algorithms, pages 138–148, 1990.Google Scholar
  11. 11.
    K. Sugiyama and K. Misue. Visualization of structural information: Automatic drawing of compound digraphs. IEEE Trans. Softw. Eng., 21(4):876–892, 1991.MathSciNetGoogle Scholar
  12. 12.
    K. Wagner. Bemerkungen zum vierfarbenproblem. Jahresbericht der Deutschen Mathematiker-Vereinigung, 46:26–32, 1936.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Christian A. Duncan
    • 1
  • Michael T. Goodrich
    • 1
  • Stephen G. Kobourov
    • 1
  1. 1.Center for Geometric ComputingThe Johns Hopkins UniversityBaltimore

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