Algorithms for the treewidth and minimum fill-in of HHD-free graphs

  • H. J. Broersma
  • E. Dahlhaus
  • T. Kloks
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1335)


A graph is HHD-free is it does not contain a house (i.e., the complement of P5), a hole (a cycle of length at least 5) or a domino (the graph obtained from two 4-cycles by identifying an edge in one C4 with an edge in the other C4) as an induced subgraph. The minimum fill-in problem is the problem of finding a chordal supergraph with the smallest possible number of edges. The treewidth problem is the problem of finding a chordal embedding of the graph with the smallest possible clique number. In this note we show that both problems are solvable in polynomial time for HHD-free graphs.


graphs algorithms HHD-free graphs treewidth minimum fill-in 




Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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., T. Kloks, D. Kratsch and H. Müller, Treewidth and minimum fill-in on d-trapezoid graphs, Technical report RUU-CS-1995-34, Utrecht University, The Netherlands, 1995.Google Scholar
  3. 3.
    Bodlaender, H. and R. Möhring, The pathwidth and treewidth of cographs, SIAM Journal on Discrete Mathematics 7 (1993), pp. 181–188.Google Scholar
  4. 4.
    Brandstädt, A., Special graph classes — A survey, Schriftenreihe des Fachbereichs Mathematik, SM-DU-199 (1991), Universität Duisburg Gesamthochschule.Google Scholar
  5. 5.
    Maw-Shang Chang, Algorithms for maximum matching and minimum fill-in on chordal bipartite graphs. ISAAC'96 (T. Asano et al. ed.), LLNCS 1178, pp. 146–155.Google Scholar
  6. 6.
    Dirac, G. A., On rigid circuit graphs, Abh. Math. Sem. Univ. Hamburg 25, (1961), pp. 71–76.Google Scholar
  7. 7.
    Golumbic M. C., Algorithmic graph theory and perfect graphs, Academic Press, New York, 1980.Google Scholar
  8. 8.
    Hammer, P. L. and F. Maffray, Completely separable graphs, Discrete Applied Mathematics 27, (1990), pp. 85–99.Google Scholar
  9. 9.
    Hayward, R., C. T. Hoang and F. Maffray, Optimizing weakly triangulated graphs, Graphs and combinatorics 5, (1989), pp. 339–349.Google Scholar
  10. 10.
    Jamison, B. and S. Olariu, On the semi-perfect elimination, Advances in Applied Mathematics 9, (1988), pp. 364–376.Google Scholar
  11. 11.
    Kloks, T., Treewidth — Computations and Approximations, Springer Verlag, Lecture Notes in Computer Science 842, (1994).Google Scholar
  12. 12.
    Kloks, T., Treewidth of circle graphs, International Journal of Foundations of Computer Science 7, (1996), pp. 111–120.Google Scholar
  13. 13.
    Kloks, T. and D. Kratsch, Treewidth of chordal bipartite graphs, J. of Algorithms 19, (1995), pp. 266–281.Google Scholar
  14. 14.
    Kloks, T., D. Kratsch and H. Müller, Approximating the bandwidth for AT-free graphs, Proceedings of the Third Annual European Symposium on Algorithms (ESA'95), Springer-Verlag, Lecture Notes in Computer Science 979, (1995), pp. 434–447.Google Scholar
  15. 15.
    Kloks, T., D. Kratsch and J. Spinrad, Treewidth and pathwidth of cocomparability graphs of bounded dimension, Computing Science Notes, 93/46, Eindhoven University of Technology, Eindhoven, The Netherlands, (1993), to appear in Order.Google Scholar
  16. 16.
    Kloks, T., D. Kratsch and C. K. Wong, Minimum fill-in of circle and circular arc graphs, Proceedings of the 21 th International Symposium on Automata, Languages and Programming (ICALP'96), Springer-Verlag Lecture Notes in Computer Science 1113, (1996), pp. 256–267.Google Scholar
  17. 17.
    Olariu, S., Results on perfect graphs, PhD thesis, Scool of Computer Science, McGill University, Montreal, 1986.Google Scholar
  18. 18.
    Parra, A., Scheffler, P., How to use minimal separators for its chordal triangulation, ICALP'95, LLNCS 944, pp. 123–134.Google Scholar
  19. 19.
    Spinrad, J., A. Brandstädt and L. Stewart, Bipartite permutation graphs, Discrete Applied mathematics 18, (1987), pp. 279–292.Google Scholar
  20. 20.
    Sundaram, R., K. Sher Singh and C. Pandu Rangan, Treewidth of circular arc graphs. To appear in SIAM J. Disc. Math. Google Scholar
  21. 21.
    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 1997

Authors and Affiliations

  • H. J. Broersma
    • 1
  • E. Dahlhaus
    • 2
  • T. Kloks
    • 1
  1. 1.Faculty of Applied MathematicsUniversity of TwenteAE Enschedethe Netherlands
  2. 2.Department of Computer ScienceUniversity of BonnBonnGermany

Personalised recommendations