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Branchwidth of Graphs

2003; Fomin, Thilikos

  • Reference work entry
Encyclopedia of Algorithms

Keywords and Synonyms

Tangle Number        

Problem Definition

Branchwidth, along with its better-known counterpart, treewidth, are measures of the “global connectivity” of a graph.

Definition

Let G be a graph on n vertices. A branch decomposition of G is a pair \( { (T,\tau) } \), where T is a tree with vertices of degree 1 or 3 and τ is a bijection from the set of leaves of T to the edges of G. The order, we denote it as α(e), of an edge e in T is the number of vertices v of G such that there are leaves \( { t_{1},t_{2} } \) in T in different components of \( { T(V(T),E(T)-e) } \) with \( { \tau(t_{1}) } \) and \( { \tau(t_{2}) } \) both containing v as an endpoint.

The width of \( { (T,\tau) } \) is equal to \( { \max_{e\in E(T)} \{\alpha(e)\} } \), i. e. is the maximum order over all edges of T. The branchwidth of G is the minimum width over all the branch decompositions of G (in the case where \( { |E(G)|\leq 1 } \), then we define the branchwidth to be 0; if \( { |E(G)|=0 } \), then G...

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  1. Alber, J., Bodlaender, H.L., Fernau, H., Kloks, T., Niedermeier, R.: Fixed parameter algorithms for dominating set and related problems on planar graphs. Algorithmica 33, 461–493 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Arnborg, S.: Efficient algorithms for combinatorial problems on graphs with bounded decomposability – A survey. BIT 25, 2–23 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bodlaender, H.L., Thilikos, D.M.: Constructive linear time algorithms for branchwidth. In: Automata, languages and programming (Bologna, 1997). Lecture Notes in Computer Science, vol. 1256, pp. 627–637. Springer, Berlin (1997)

    Google Scholar 

  4. Bodlaender, H.L., Thilikos, D.M.: Graphs with branchwidth at most three. J. Algorithms 32, 167–194 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cook, W., Seymour, P.D.: Tour merging via branch‑decomposition. Inf. J. Comput. 15, 233–248 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. Demaine, E.D., Fomin, F.V., Hajiaghayi, M., Thilikos, D.M.: Bidimensional parameters and local treewidth. SIAM J. Discret. Math. 18, 501–511 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. Demaine, E.D., Hajiaghayi, M.T., Nishimura, N., Ragde, P., Thilikos, D. M.: Approximation algorithms for classes of graphs excluding single‐crossing graphs as minors. J. Comput. Syst. Sci. 69, 166–195 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dorn, F., Fomin, F.V., Thilikos, D.M.: Subexponential algorithms for non-local problems on H-minor-free graphs. In: Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms (SODA 2008). pp. 631–640. Society for Industrial and Applied Mathematics, Philadelphia (2006)

    Google Scholar 

  9. Dorn, F.: Dynamic programming and fast matrix multiplication. In: Proceedings of the 14th Annual European Symposium on Algorithms (ESA 2006). Lecture Notes in Computer Science, vol. 4168 , pp. 280–291. Springer, Berlin (2006)

    Google Scholar 

  10. Dorn, F., Fomin, F.V., Thilikos, D.M.: Fast subexponential algorithm for non-local problems on graphs of bounded genus. In: Proceedings of the 10th Scandinavian Workshop on Algorithm Theory (SWAT 2006). Lecture Notes in Computer Science. Springer, Berlin (2005)

    Google Scholar 

  11. Dorn, F., Penninkx, E., Bodlaender, H., Fomin, F.V.: Efficient exact algorithms on planar graphs: Exploiting sphere cut branch decompositions. In: Proceedings of the 13th Annual European Symposium on Algorithms (ESA 2005). Lecture Notes in Computer Science, vol. 3669, pp. 95–106. Springer, Berlin (2005)

    Google Scholar 

  12. Downey, R.G., Fellows, M.R.: Parameterized complexity. In: Monographs in Computer Science. Springer, New York (1999)

    Google Scholar 

  13. Feige, U., Hajiaghayi, M., Lee, J.R.: Improved approximation algorithms for minimum‐weight vertex separators. In: Proceedings of the 37th annual ACM Symposium on Theory of computing (STOC 2005), pp. 563–572. ACM Press, New York (2005)

    Google Scholar 

  14. Fomin, F.V., Mazoit, F., Todinca, I.: Computing branchwidth via efficient triangulations and blocks. In: Proceedings of the 31st Workshop on Graph Theoretic Concepts in Computer Science (WG 2005). Lecture Notes Computer Science, vol. 3787, pp. 374–384. Springer, Berlin (2005)

    Google Scholar 

  15. Fomin, F.V., Thilikos, D.M.: Fast parameterized algorithms for graphs on surfaces: Linear kernel and exponential speed-up. In: Proceedings of the 31st International Colloquium on Automata, Languages and Programming (ICALP 2004). Lecture Notes Computer Science, vol. 3142, pp. 581–592. Springer, Berlin (2004)

    Google Scholar 

  16. Fomin, F.V., Thilikos, D.M.: Dominating sets in planar graphs: Branch-width and exponential speed-up. SIAM J. Comput. 36, 281–309 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. Fomin, F.V., Thilikos, D.M.: New upper bounds on the decomposability of planar graphs. J. Graph Theor. 51, 53–81 (2006)

    Article  MathSciNet  Google Scholar 

  18. Gu, Q.P., Tamaki, H.: Optimal branch‐decomposition of planar graphs in O(n 3) time. In: Proceedings of the 32nd International Colloquium on Automata, Languages and Programming (ICALP 2005). Lecture Notes Computer Science, vol. 3580, pp. 373–384. Springer, Berlin (2005)

    Google Scholar 

  19. Gu, Q.P., Tamaki, H.: Branch-width, parse trees, and monadic second-order logic for matroids. J. Combin. Theor. Ser. B 96, 325–351 (2006)

    Article  Google Scholar 

  20. Kloks, T., Kratochvíl, J., Müller, H.: Computing the branchwidth of interval graphs. Discret. Appl. Math. 145, 266–275 (2005)

    Article  MATH  Google Scholar 

  21. Mazoit, F.: The branch-width of circular-arc graphs. In: 7th Latin American Symposium on Theoretical Informatics (LATIN 2006), 2006, pp. 727–736

    Google Scholar 

  22. Oum, S.I., Seymour, P.: Approximating clique-width and branch-width. J. Combin. Theor. Ser. B 96, 514–528 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  23. Paul, C., Proskurowski, A., Telle, J.A.: Generating graphs of bounded branchwidth. In: Proceedings of the 32nd Workshop on Graph Theoretic Concepts in Computer Science (WG 2006). Lecture Notes Computer Science, vol. 4271, pp. 205–216. Springer, Berlin (2006)

    Google Scholar 

  24. Paul, C., Telle, J.A.: New tools and simpler algorithms for branchwidth. In: Proceedings of the 13th Annual European Symposium on Algorithms (ESA 2005), 2005 pp. 379–390

    Google Scholar 

  25. Robertson, N. Seymour, P.D.: Graph minors. X. Obstructions to tree‐decomposition J. Combin. Theor. Ser. B 52, 153–190 (1991)

    Google Scholar 

  26. Robertson, N. Seymour, P.D.: Graph minors. XII. Distance on a surface. J. Combin. Theor. Ser. B 64, 240–272 (1995)

    Google Scholar 

  27. Robertson, N. Seymour, P.D.: Graph minors. XIII. The disjoint paths problem. J. Combin. Theor. Ser. B 63, 65–110 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  28. Robertson, N., Seymour, P.D., Thomas, R.: Quickly excluding a planar graph. J. Combin. Theor. Ser. B 62, 323–348 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  29. Seymour, P.D., Thomas, R.: Call routing and the ratcatcher. Combinatorica 14, 217–241 (1994)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Informatics, University of Bergen, Bergen, Norway

    Fedor Fomin

  2. Department of Mathematics, National and Kapodistrian University of Athens, Athens, Greece

    Dimitrios Thilikos

Authors
  1. Fedor Fomin
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  2. Dimitrios Thilikos
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Editor information

Editors and Affiliations

  1. Department of Electrical Engineering and Computer ScienceMcCormick School of Engineering and Applied Science, Northwestern University, Evanston, IL, 60208, USA

    Ming-Yang Kao Professor of Computer Science

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© 2008 Springer-Verlag

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Fomin, F., Thilikos, D. (2008). Branchwidth of Graphs. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-30162-4_55

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