New Tools and Simpler Algorithms for Branchwidth

  • Christophe Paul
  • Jan Arne Telle
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3669)


We provide new tools, such as k-troikas and good subtree-representations, that allow us to give fast and simple algorithms computing branchwidth. We show that a graph G has branchwidth at most k if and only if it is a subgraph of a chordal graph in which every maximal clique has a k-troika respecting its minimal separators. Moreover, if G itself is chordal with clique tree T then such a chordal supergraph exists having clique tree a minor of T. We use these tools to give a straightforward O(m+n+q 2) algorithm computing branchwidth for an interval graph on m edges, n vertices and q maximal cliques. We also prove a conjecture of F. Mazoit [13] by showing that branchwidth is polynomial on a chordal graph given with a clique tree having a polynomial number of subtrees.


Maximal Clique Interval Graph Chordal Graph Minimal Separator Polynomial Number 
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.
    Bodlaender, H.L., Thilikos, D.M.: Constructive linear time algorithms for branchwidth. In: Degano, P., Gorrieri, R., Marchetti-Spaccamela, A. (eds.) ICALP 1997. LNCS, vol. 1256, pp. 627–637. Springer, Heidelberg (1997)Google Scholar
  2. 2.
    Bodlaender, H.L., Thilikos, D.M.: Graphs with branchwidth at most three. Journal of Algorithms 32, 167–194 (1999)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Booth, K., Lueker, G.: Testing of the consecutive ones property, interval graphs, and graph planarity testing using PQ-tree algorithms. Journal of Computer and System Sciences 13, 335–379 (1976)MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Cook, W., Seymour, P.D.: Tour merging via branch-decompositions. Journal on Computing 15, 233–248 (2003)MathSciNetGoogle Scholar
  5. 5.
    Demaine, E., Fomin, F., Hajiaghayi, M., Thilikos, D.M.: Fixed-parameter algorithms for (k,r)-center in planar graphs and map graphs. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 829–844. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  6. 6.
    Fomin, F., Thilikos, D.: Dominating sets in planar graphs: Branch-width and exponential speedup. In: 14th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 168–177 (2003)Google Scholar
  7. 7.
    Fomin, F., Thilikos, D.: A simple and fast approach for solving problems on planar graphs. In: Diekert, V., Habib, M. (eds.) STACS 2004. LNCS, vol. 2996, pp. 56–67. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  8. 8.
    Fomin, F., Thilikos, D.: Fast parameterized algorithms for graphs on surfaces: Linear kernel and exponential speedup. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 581–592. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  9. 9.
    Gavril, F.: The intersection graphs of subtrees in trees are exactly the chordal graphs. Journal of Combinatorial Theory Series B 16, 47–56 (1974)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Acad. Press, New York (1980)MATHGoogle Scholar
  11. 11.
    Kloks, T., Kratochvil, J., Müller, H.: New branchwidth territories. In: Meinel, C., Tison, S. (eds.) STACS 1999. LNCS, vol. 1563, pp. 173–183. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  12. 12.
    Kloks, T., Kratochvil, J., Müller, H.: Computing the branchwidth of interval graphs. Discrete Applied Mathematics 145, 145–266 (2005)CrossRefGoogle Scholar
  13. 13.
    Mazoit, F.: A general scheme for deciding the branchwidth. Technical Report RR2004-34, LIP - École Normale Supérieure de Lyon (2004),
  14. 14.
    Robertson, N., Seymour, P.D.: Graph minors X: Obstructions to tree-decomposition. Journal on Combinatorial Theory Series B 52, 153–190 (1991)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Seymour, P.D., Thomas, R.: Call routing and the ratcatcher. Combinatorica 14(2), 217–241 (1994)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Christophe Paul
    • 1
  • Jan Arne Telle
    • 2
  1. 1.CNRS – LIRMMMontpellierFrance
  2. 2.Department of InformaticsUniversity of BergenNorway

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