Generation of Graphs with Bounded Branchwidth

  • Christophe Paul
  • Andrzej Proskurowski
  • Jan Arne Telle
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4271)


Branchwidth is a connectivity parameter of graphs closely related to treewidth. Graphs of treewidth at most k can be generated algorithmically as the subgraphs of k-trees. n this paper, we investigate the family of edge-maximal graphs of branchwidth k, that we call k-branches. The k-branches are, just as the k-trees, a subclass of the chordal graphs where all minimal separators have size k. However, a striking difference arises when considering subgraph-minimal members of the family. Whereas K k + 1 is the only subgraph-minimal k-tree, we show that for any k ≥7 a minimal k-branch having q maximal cliques exists for any value of \(q \not\in \{3,5\}\), except for k=8,q=2. We characterize subgraph-minimal k-branches for all values of k. Our investigation leads to a generation algorithm, that adds one or two new maximal cliques in each step, producing exactly the k-branches.


Planar Graph Maximal Clique Chordal Graph Minimal Separator Mathematical Society Lecture Note Series 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bodlaender, H.L., Kloks, T., Kratsch, D.: Treewidth and pathwidth of permutation graphs. SIAM J. Computing 25, 1305–1317 (1996)MATHCrossRefGoogle Scholar
  2. 2.
    Bodlaender, H.L.: Treewidth: Algorithmic techniques and results. In: Privara, I., Ružička, P. (eds.) MFCS 1997. LNCS, vol. 1295, pp. 19–36. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  3. 3.
    Bodlaender, H.L., Thilikos, D.M.: Graphs with branchwidth at most three. Journal of Algorithms 32, 167–194 (1999)MATHCrossRefMathSciNetGoogle 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.
    Dorn, F., Penninkx, E., Bodlaender, H.L., Fomin, F.V.: Efficient Exact Algorithms on Planar Graphs: Exploiting Sphere Cut Branch Decompositions. In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 95–106. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  7. 7.
    Fomin, F., Thilikos, D.M.: 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
  8. 8.
    Fomin, F., Thilikos, D.M.: 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
  9. 9.
    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
  10. 10.
    Fomin, F., Mazoit, F., Todinca, I.: Computing branchwidth via efficient triangulations and blocks. In: Kratsch, D. (ed.) WG 2005. LNCS, vol. 3787, pp. 374–384. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  11. 11.
    Kloks, T., Kratochvil, J., Müller, H.: New branchwidth territories. Discrete Applied Mathematics 145, 266–275 (2005)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Kleinberg, J., Tardos, E.: Algorithm design. Addison-Wesley, Reading (2005)Google Scholar
  13. 13.
    Paul, C., Telle, J.A.: Edge-maximal graphs of branchwidth k. In: International Conference on Graph Theory - ICGT. Electronic Notes in Discrete Mathematics, vol. 23, pp. 363–368 (2005)Google Scholar
  14. 14.
    Paul, C., Proskurowski, A., Telle, J.A.: Algorithm generation of graphs of branchwidht ≤ k. LIRMM Technical report number RR-05047 (2005)Google Scholar
  15. 15.
    Paul, C., Telle, J.A.: New tools and simpler algorithms for branchwidth. In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 379–390. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  16. 16.
    Reed, B.: Treewidth and tangles, a new measure of connectivity and some applications. In: Surveys in Combinatorics. London Mathematical Society Lecture Note Series, vol. 241. Cambridge University Press, Cambridge (1997)Google Scholar
  17. 17.
    Robertson, N., Seymour, P.D.: Graph minors X: Obstructions to tree-decomposition. Journal on Combinatorial Theory Series B 52, 153–190 (1991)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Rose, D.: On simple characterization of k-trees. Discrete Mathematics 7, 317–322 (1974)MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Christophe Paul
    • 1
  • Andrzej Proskurowski
    • 2
  • Jan Arne Telle
    • 3
  1. 1.CNRS – LIRMMMontpellierFrance
  2. 2.Department of Computer and Information ScienceUniversity of OregonUSA
  3. 3.Department of InformaticsUniversity of BergenNorway

Personalised recommendations