Treewidth and pathwidth of permutation graphs

  • Hans Bodlaender
  • Ton Kloks
  • Dieter Kratsch
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 700)


In this paper we show that the treewidth and pathwidth of a permutation graph can be computed in polynomial time. In fact we show that, for permutation graphs, the treewidth and pathwidth are equal. These results make permutation graphs one of the few non-trivial graph classes for which at the moment, treewidth is known to be computable in polynomial time. Our algorithm to decide whether the treewidth (pathwidth) is at most some given integer k, can be implemented to run in O(nk2) time, when the matching diagram is given. We show that this algorithm can easily be adapted to compute the pathwidth of a permutation graph in O(nk2) time, where k is the pathwidth.


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  1. 1.
    S. Arnborg, Efficient algorithms for combinatorial problems on graphs with bounded decomposability-A survey. BIT 25, 2–23, 1985.Google Scholar
  2. 2.
    S. Arnborg, D.G. Cornell and A. Proskurowski, Complexity of finding embeddings in a k-tree, SIAM J. Alg. Disc. Meth. 8, 277–284, 1987.Google Scholar
  3. 3.
    S. Arnborg and A. Proskurowski, Linear time algorithms for NP-hard problems restricted to partial k-trees. Disc. Appl. Math. 23, 11–24, 1989.Google Scholar
  4. 4.
    C. Berge and C. Chvatal, Topics on Perfect Graphs, Annals of Discrete Math. 21, 1984.Google Scholar
  5. 5.
    H.L. Bodlaender, A tourist guide through treewidth, Technical report RUU-CS-92-12, Department of Computer Science, Utrecht University, Utrecht, The Netherlands, 1992. To appear in: Proceedings 7th Int. Meeting of Young Computer Scientists.Google Scholar
  6. 6.
    H. Bodlaender and R.H. Möhring, The pathwidth and treewidth of cographs, In Proceedings 2nd Scandinavian Workshop on Algorithm Theory, 301–309, Springer Verlag, Lect. Notes in Comp. Sc., vol. 447, 1990. To appear in: SIAM J. Discr. Math.Google Scholar
  7. 7.
    H. Bodlaender and T. Kloks, Better algorithms for the pathwidth and treewidth of graphs, Proceedings of the 18th Int. Coll. on Automata, Languages and Programming, 544–555, Springer Verlag, Lect. Notes in Comp. Sc., vol. 510, 1991.Google Scholar
  8. 8.
    A. Brandstädt, Special graph classes — a survey, Schriftenreihe des Fachbereichs Mathematik, SM-DU-199 (1991) Universität Duisburg Gesamthochschule.Google Scholar
  9. 9.
    S. Even, A. Pnueli and A. Lempel, Permutation graphs and transitive graphs, J. Assoc. Comput. Mach. 19, 400–410, 1972.Google Scholar
  10. 10.
    M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs, Academic Press, New York, 1980.Google Scholar
  11. 11.
    J. Gustedt, Pathwidth for chordal graphs is NP-complete. To appear in: Discr. Appl. Math. Google Scholar
  12. 12.
    T. Kloks and H. Bodlaender, Approximating treewidth and pathwidth of some classes of perfect graphs. 3th Ann. Int. Symp. on Algorithms and Computation (ISAAC'92), 116–125, Springer Verlag, Lect. Notes in Comp. Sc., vol. 650, 1993.Google Scholar
  13. 13.
    T. Kloks and D. Kratsch, Treewidth of chordal bipartite graphs. In: Proc. 10th Ann. Symp. on Theoretical Aspects of Computer Science, 80–89, Springer Verlag, Lect. Notes in Comp. Sc., vol. 665, 1993.Google Scholar
  14. 14.
    J. Lagergren and S. Arnborg, Finding minimal forbidden minors using a finite congruence, Proceedings of the 18th Int. Coll. on Automata, Languages and Programming, 532–543, Springer Verlag, Lect. Notes in Comp. Sc., vol. 510, 1991.Google Scholar
  15. 15.
    J. van Leeuwen, Graph algorithms. In Handbook of Theoretical Computer Science, A: Algorithms an Complexity Theory, 527–631, Amsterdam, 1990. North Holland Publ. Comp.Google Scholar
  16. 16.
    C.G. Lekkerkerker and J.Ch. Boland, Representation of a finite graph by a set of intervals on the real line, Fund. Math. 51, 45–64, 1962.Google Scholar
  17. 17.
    B. Reed, Finding approximate separators and computing treewidth quickly, Proc. 24th Ann. ACM Symp. on Theory of Computing, 221–228, 1992.Google Scholar
  18. 18.
    J. Spinrad, On comparability and permutation graphs, SIAM J. Comp. 14, 658–670, 1985.Google Scholar
  19. 19.
    R. Sundaram, K. Sher Singh and C. Pandu Rangan, Treewidth of circular arc graphs, Manuscript, 1991, to appear in SIAM J. Disc. Math.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • Hans Bodlaender
    • 1
  • Ton Kloks
    • 1
  • Dieter Kratsch
    • 2
  1. 1.Department of Computer ScienceUtrecht UniversityTB Utrechtthe Netherlands
  2. 2.Fakultät MathematikFriedrich-Schiller-Universität, UniversitätshochhausJenaGermany

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