Graph clustering I: Cycles of cliques

Extended abstract
  • F. J. Brandenburg
Clustering and Labelling
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1353)


A graph is a cycle of cliques, if its set of vertices can be partitioned into clusters, such that each cluster is a clique and the cliques form a cycle. Then there is a partition of the set of edges into inner edges of the cliques and interconnection edges between the clusters. Cycles of cliques are a special instance of two-level clustered graphs. Such graphs are drawn by a two phase method: draw the top level graph and then browse into the clusters. In general, it is NP-hard whether or not a graph is a two-level clustered graph of a particular type, e.g. a clique or a planar graph or a triangle of cliques. However, it is efficiently solvable whether or not a graph is a path of cliques or is a large cycle of cliques.


Planar Graph Graph Cluster Graph Grammar Graph Drawing Complement Graph 
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.


  1. 1.
    C.J. Alpert and A.B. Kahng: Recent directions in netlist partitioning: a survey. Integration, the VLSI J. 19 (2995), 1–18.Google Scholar
  2. 2.
    J. Bosik: Decompositions of Graphs. Kluvwer Academic Publishers, Dordrecht (1990).Google Scholar
  3. 3.
    F.J. Brandenburg: Designing graph drawings by layout graph grammars. Graph Drawing 94, LNCS 894 (1995), 416–427.Google Scholar
  4. 4.
    R.C. Brewster, P. Hell and G. MacGillivray: The complexity of restricted graph homomorphisms. Discrete Mathematics 167/168 (1997), 145–154.CrossRefGoogle Scholar
  5. 5.
    U. Dogrusöz, B. Madden and P. Madden: Circular layout in the graph layout toolkit. Graph Drawing 96, LNCS 1190 (1997), 92–100.Google Scholar
  6. 6.
    D. Der and M Tarsi: Graph decomposition is NP-complete: proof of Holyer's conjecture. SIAM J. Comput. 26 (1997), 1166–1187.CrossRefGoogle Scholar
  7. 7.
    P. Eades and Q. W. Feng: Multilevel visualization of clustered graphs.Graph Drawing 96, LNCS 1190 (1997), 101–112.Google Scholar
  8. 8.
    P. Eades, Q.W. Feng and X. Lin: Straight-line drawing algorithms for hierarchical graphs and clustered graphs. Graph Drawing 96, LNCS 1190 (1997), 113–128.Google Scholar
  9. 9.
    P. Eades, J. Marks and S. North: Graph-Drawing Contest Report. Graph Drawing 96, LNCS 1190 (1997), 129–138.Google Scholar
  10. 10.
    M.R. Garey and D.S. Johnson: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Fransisco (1979).Google Scholar
  11. 11.
    D. J. Gschwindt and T. P. Murthagh: A recursive algorithm for drawing hierarchical graphs. Technical Report CS-89-02, Williams College, Williamstown (1989).Google Scholar
  12. 12.
    D. Harel: On visual formalisms. Comm. ACM 31 (1988), 514–530.CrossRefGoogle Scholar
  13. 13.
    P. Hell and J. Nešetřil: On the complexity of H-coloring. J. of Combinatorial Theory, Series B 48 (1990), 92–110.Google Scholar
  14. 14.
    M. Himsolt: Konzeption und Implementierung von Grapheditoren. Shaker Verlag, Aachen (1993).Google Scholar
  15. 15.
    I. Holyer: The NP-completeness of some edge partition problems. SIAM J. Comput., 10 (1981), 713–717.CrossRefGoogle Scholar
  16. 16.
    J. Kratochvil: String Graphs. II. Recognizing string graphs is NP-Hard. J. of Combinatorial Theory, Series B 52 (1991), 67–78.Google Scholar
  17. 17.
    J. Kratochvil, M. Goljan and P. Kučera: String Graphs. Academia, Prague (1986).Google Scholar
  18. 18.
    T. Lengauer: Combinatorial Algorithms for Integrated Circuit Layout. Wiley-Teubner Series (1990).Google Scholar
  19. 19.
    U. Manber: Introduction to Algorithms a Creative Approach. Addison Wesley, Reading (1989).Google Scholar
  20. 20.
    E.B. Messinger, L.A. Rowe and R.R. Henry: A divide-and conquer algorithm for the automatic layout of large directed graphs. IEEE Trans. Systems Man Cybernetics 21 (1991), 1–12.Google Scholar
  21. 21.
    R. Sablowski and A. Frick: Automatic graph clustering. Graph Drawing 96, LNCS 1190 (1997), 396–400.Google Scholar
  22. 22.
    F.-S. Shieh and C.L. McCreary: Directed graphs drawing by clan-based decomposition. Graph Drawing 95, LNCS 1027 (1996), 472–482.Google Scholar
  23. 23.
    K. Sugiyama and K. Misue: Visualization of structural information: automatic drawing of compound graphs. IEEE Trans. Systems Man Cybernetics 21 (1991), 876–892.CrossRefGoogle Scholar
  24. 24.
    R.E. Tarjan: Decomposition by clique separators. Discrete Math. 55 (1985), 221–232.CrossRefGoogle Scholar
  25. 25.
    S.H. Whitesides: An algorithm for finding clique cut-sets. Inf. Proc. Letters 12 (1981), 31–32.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • F. J. Brandenburg
    • 1
  1. 1.Lehrstuhl für InformatikUniversität PassauPassauGermany

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