A linear time algorithm to recognize clustered planar graphs and its parallelization

  • Elias Dahlhaus
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1380)


We develop a linear time algorithm for the following problem: Given a graph G and a hierarchical clustering of the vertices such that all clusters induce connected subgraphs, determine whether G may be embedded into the plane such that no cluster has a hole.

This is an improvement to the O(n 2)-algorithm of Q.W. Feng et al. [6] and the algorithm of Lengauer [12] that operates in linear time on a replacement system. The size of the input of Lengauer's algorithm is not necessarily linear with respect to the number of vertices.


Planar Graph Maximum Weight Dual Graph Connected Subgraph Linear Time Algorithm 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    K. Abrahamson, N. Dadoun, D. Kirkpatrick, T. Przyticka, A Simple Parallel Tree Contraction Algorithm, Journal of Algorithms 10 (1988), pp. 287–302.CrossRefGoogle Scholar
  2. 2.
    T. Biedl, G. Kant, A better heuristic for Orthogonal Graph Drawing, ESA 94, LLNCS 855, pp. 24–35.Google Scholar
  3. 3.
    K. Booth, G. Lueker, Testing for the Consecutive Ones Property, Interval Graphs, and Graph Planarity Using PQ-Tree Algorithms, Journal of Computer and Systems Sciences 13(1976), pp. 335–379.MathSciNetGoogle Scholar
  4. 4.
    R. Cole, Paralle J Merge Sort, 27. IEEE-FOCS (1986), pp. 511–516.Google Scholar
  5. 5.
    P. Eades, Q.W. Feng, X. Lin, Straight Line Drawing Algorithms for Hierarchical Graphs and Clustered Graphs, Graph Drawing, GD'96, LLNCS 1190, pp. 113–128.Google Scholar
  6. 6.
    Q.W. Feng, R. Cohen, P. Eades, Planarity for Clustered Graphs, ESA'95, LLNCS 979, pp. 213–226.Google Scholar
  7. 7.
    A. Gibbons, W. Rytter, Efficient Parallel Algorithms, Cambridge University Press, Cambridge, 1989.Google Scholar
  8. 8.
    D. Harel, On Visual Formalisms, Communications of the ACM 21 (1988), pp. 549–568.MathSciNetGoogle Scholar
  9. 9.
    T. Kameda, Visualizing Abstract Objects and Relations, World Scientific Series in Computer Science, 1989.Google Scholar
  10. 10.
    G. Kant, Drawing Planar Graphs using the lmc-Ordering, 33rd FOCS (1991), pp. 793–801.Google Scholar
  11. 11.
    T. Lengauer, Combinatorial Algorithms for Integrated Circuit Lyout, Applicable Theory in Computer Science, Teubner/Wiley, Stuttgart/New York, 1990.Google Scholar
  12. 12.
    T. Lengauer, Hierarchical Planarity Testing Algorithm, Journal of the ACM 36 (1989), pp. 474–509.zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Y. Maon, B. Schieber, U. Vishkin, Parallel Ear Decomposition Search (EDS) and st-Numberings in Graphs, Theoretical Computer Science 47 (1986), pp. 277–296.CrossRefMathSciNetGoogle Scholar
  14. 14.
    R. Möhring, Algorithmic Aspects of the Substitution Decomposition in Optimization over Relations, Set Systems and Boolean Functions, Ann. Oper. Res., 4 (1985), pp. 195–225.CrossRefMathSciNetGoogle Scholar
  15. 15.
    T. Nishizeki, N. Chiba, Planar Graphs: Theory and Algorithms, Annals of Discrete Mathematics 32, North Holland, 1988.Google Scholar
  16. 16.
    V. Ramachandran, J. Reif, Planarity Testing in Parallel, Journal of Computer and Systems Sciences 49 (1994), pp. 517–561.CrossRefMathSciNetGoogle Scholar
  17. 17.
    Y. Shiloach, U. Vishkin, An O(log n) Parallel Connectivity Algorithm, Journal of Algorithms 3 (1982), pp. 57–67.CrossRefMathSciNetGoogle Scholar
  18. 18.
    W.T. Tutte, How to Draw a Graph, Proceedings London Mathematical Society 3, pp. 743–768.Google Scholar
  19. 19.
    C. Williams, J. Rasure, C. Hansen, The State of the Art of Visual Languages for Visualization, Visualization 92 (1992), pp. 202–209.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Elias Dahlhaus
    • 1
    • 2
  1. 1.Department of Mathematics and Department of Computer ScienceUniversity of CologneGermany
  2. 2.Dept. of Computer ScienceUniversity of BonnGermany

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