Treewidth of circle graphs

  • T. Kloks
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 762)


In this paper we show that the treewidth of a circle graph can be computed in polynomial time. A circle graph is a graph that is isomorphic to the intersection graph of a finite collection of chords of a circle. The TREEWIDTH problem can be viewed upon as the problem of finding a chordal embedding of the graph that minimizes the clique number. Our algorithm to determine the treewidth of a circle graph can be implemented to run in O(n3) time, where n is the number of vertices of the graph.


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  1. 1.
    Arnborg, S., D. G. Corneil and A. Proskurowski, Complexity of finding embeddings in a k-tree, SIAM J. Alg. Disc. Meth. 8, (1987), pp. 277–284.Google Scholar
  2. 2.
    Bodlaender, H. L., Achromatic number is NP-complete for cographs and interval graphs, Information Processing Letter 31, (1989), pp. 135–138.Google Scholar
  3. 3.
    Bodlaender, H., A linear time algorithm for finding tree-decompositions of small treewidth, Technical report RUU-CS-92-27, Department of Computer Science, Utrecht University, Utrecht, The Netherlands, (1992).Google Scholar
  4. 4.
    Bodlaender, H., T. Kloks and D. Kratsch, Treewidth and pathwidth of permutation graphs, Technical report RUU-CS-92-30, Department of Computer Science, Utrecht University, Utrecht, The Netherlands, (1992). To appear in Proceedings of the 20 th International colloquium on Automata, Languages and Programming.Google Scholar
  5. 5.
    Bodlaender, H. and R. H. Möhring, The pathwidth and treewidth of cographs, Proceedings 2 nd Scandinavian Workshop on Algorithm Theory, Springer Verlag, Lecture Notes in Computer Science 447, (1990), pp. 301–309.Google Scholar
  6. 6.
    Bouchet, A., A polynomial algorithm for recognizing circle graphs, C. R. Acad. Sci. Paris, Sér. I Math. 300, (1985), pp. 569–572.Google Scholar
  7. 7.
    Bouchet, A., Reducing prime graphs and recognizing circle graphs, Combinatorica 7, (1987), pp. 243–254.Google Scholar
  8. 8.
    Brandstädt, A., Special graph classes — a survey, Schriftenreihe des Fachbereichs Mathematik, SM-DU-199 (1991), Universität Duisburg Gesamthochschule.Google Scholar
  9. 9.
    Dirac, G. A., On rigid circuit graphs, Abh. Math. Sem. Univ. Hamburg 25, (1961), pp. 71–76.Google Scholar
  10. 10.
    Farber, M. and M. Keil, Domination in permutation graphs, J. Algorithms 6, (1985), pp. 309–321.Google Scholar
  11. 11.
    Gabor, C. P., W. L. Hsu and K. J. Supowit, Recognizing circle graphs in polynomial time, 26th Annual IEEE Symposium on Foundations of Computer Science, (1985).Google Scholar
  12. 12.
    Gimbel, J., D. Kratsch and L. Stewart, On cocolourings and cochromatic numbers of graphs. To appear in Disc. Appl. Math. Google Scholar
  13. 13.
    Golumbic, M. C., Algorithmic Graph Theory and Perfect Graphs, Academic Press, New York, 1980.Google Scholar
  14. 14.
    Habib, M. and R. H. Möhring, Treewidth of cocomparability graphs and a new order theoretic parameter, Technical report 336/1992, Technische Universität Berlin, 1992.Google Scholar
  15. 15.
    Johnson, D. S., The NP-completeness column: An ongoing guide, J. Algorithms 6, (1985), pp. 434–451.CrossRefGoogle Scholar
  16. 16.
    Kloks, T., Treewidth, PhD Thesis, Utrecht University, The Netherlands, (1993).Google Scholar
  17. 17.
    Kloks, T. and H. Bodlaender, Approximating treewidth and pathwidth of some classes of perfect graphs, Proceedings Third International Symposium on Algorithms and Computation, ISAAC'92, Springer Verlag, Lecture Notes in Computer Science, 650, (1992), pp. 116–125.Google Scholar
  18. 18.
    Kloks, T., H. Bodlaender, H. Müller and D. Kratsch, Computing treewidth and minimum fill-in: all you need are the minimal separators. To appear in Proceedings of the First Annual European Symposium on Algorithms, (1993).Google Scholar
  19. 19.
    Kloks, T. and D. Kratsch, Treewidth of chordal bipartite graphs, 10th Annual Symposium on Theoretical Aspects of Computer Science, Springer-Verlag, Lecture Notes in Computer Science 665, (1993), pp. 80–89.Google Scholar
  20. 20.
    Kloks, T. and D. Kratsch, Finding all minimal separators of a graph, Computing Science Note, 93/27, Eindhoven University of Technology, Eindhoven, The Netherlands, (1993).Google Scholar
  21. 21.
    Müller, H., Chordal graphs of domino type. Manuscript, (1993).Google Scholar
  22. 22.
    Naji, W., Reconnaissance des graphes de cordes, Discrete Mathematics 54, (1985), pp. 329–337.CrossRefGoogle Scholar
  23. 23.
    Sundaram, R., K. Sher Singh and C. Pandu Rangan, Treewidth of circular arc graphs. To appear in SIAM J. Disc. Math. Google Scholar
  24. 24.
    Supowit, K. J., Finding a maximum planar subset of a set of nets in a channel, IEEE Trans. Computer Aided Design 6, (1987), pp. 93–94.CrossRefGoogle Scholar
  25. 25.
    Supowit, K. J., Decomposing a set of points into chains, with applications to permutation and circle graphs, Information Processing Letters 21, (1985), pp. 249–252.Google Scholar
  26. 26.
    Wagner, K., Monotonic coverings of finite sets, Journal of Information Processing and Cybernetics, EIK, 20, (1984), pp. 633–639.Google Scholar
  27. 27.
    Yannakakis, M., Computing the minimum fill-in is NP-complete, SIAM J. Alg. Disc. Meth. 2, (1981), pp. 77–79.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • T. Kloks
    • 1
  1. 1.Department of Mathematics and Computing ScienceEindhoven University of TechnologyMB EindhovenThe Netherlands

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