Skip to main content

Branchwidth and Branch Decompositions

  • Reference work entry
Encyclopedia of Optimization

Article Outline

Keywords

Introduction

Graph Minors Theorem

Tangles

Constructing Branch Decompositions

Branch-Decomposition-Based Algorithms

Branchwidth of Matroids

Treewidth and Tree Decompositions

References

This research was partially supported by NSF grant DMI-0217265

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 2,500.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 2,499.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Alber J, Niedermeier R (2002) Improved tree decomposition based algorithms for domination-like problems. In: Proceedings of the 5th Latin American Theoretical Informatics (LATIN 2002). Lecture Notes in Computer Science, vol 2286. Springer, Heidelberg, pp 613–627

    Google Scholar 

  2. Alekhnovich M, Razborov A (2002) Satisfiability, branch-width and tseitin tautologies. In: 43rd Annual IEEE Symposium on Foundations of Computer Science. IEEE Computer Society, pp 593–603

    Google Scholar 

  3. Alon N, Seymour PD, Thomas R (1994) Planar separators. SIAM J Discret Math 7:184–193

    Article  MathSciNet  MATH  Google Scholar 

  4. Archdeacon D (1980) A Kuratowski Theorem for the Projective Plane. PhD thesis, Ohio State University

    Google Scholar 

  5. Archdeacon D, Huneke P (1989) A Kuratowski theorem for non-orientable surfaces. J Combin Theory Ser B 46(2):173–231

    Article  MathSciNet  MATH  Google Scholar 

  6. Arnborg S, Lagergren J, Seese D (1991) Easy problems for tree-decomposable graphs. J Algorithms 12:308–340

    Article  MathSciNet  MATH  Google Scholar 

  7. Bodlaender H (1993) A tourist guide through treewidth. Acta Cybernetica 11:1–21

    MathSciNet  MATH  Google Scholar 

  8. Bodlaender H (1996) A linear time algorithm for finding tree-decompositions of small treewidth. SIAM J Comput 25:1305–1317

    Article  MathSciNet  MATH  Google Scholar 

  9. Bodlaender H (1997) Treewidth: Algorithmic techniques and results. In: Privara I, Rvzicka P (eds) Proceedings of the 22nd International Symposium on Mathematical Foundations of Computer Science, MFCS'97. Lecture Notes in Computer Science, vol 1295. Springer, Berlin, pp 29–36

    Google Scholar 

  10. Bodlaender H (1998) A partial k-arboretum of graphs with bounded treewidth. Theoret Comput Sci 209:1–45

    Article  MathSciNet  MATH  Google Scholar 

  11. Bodlaender H, Gilbert J, Hafsteinsson H, Kloks T (1992) Approximation treewidth, pathwidth, and minimum elimination tree height. In: Schmidt G, Berghammer R (eds) Proceedings 17th International Workshop on Graph-Theoretic Concepts in Computer Science WG 1991. Lecture Notes in Computer Science, vol 570. Springer, Berlin, pp 1–12

    Google Scholar 

  12. Bodlaender H, Kloks T (1992) Approximating treewidth and pathwidth of some classes of perfect graphs. In: Proceedings Third International Symposium on Algorithms and Computation, ISAAC 1992. Lecture Notes in Computer Science, vol 650. Springer, Berlin, pp 116–125

    Google Scholar 

  13. Bodlaender H, Kloks T (1996) Efficient and constructive algorithms for the pathwidth and treewidth of graphs. J Algorithms 21:358–402

    Article  MathSciNet  MATH  Google Scholar 

  14. Bodlaender H, Kloks T, Kratsch D, Muller H (1998) Treewidth and minimum fill-in on d-trapezoid graphs. J Graph Algorithms Appl 2(5):1–23

    MathSciNet  Google Scholar 

  15. Bodlaender H, Tan R, Thilikos D, van Leeuwen J (1997) On interval routing schemes and treewidth. Inf Comput 139:92–109

    Google Scholar 

  16. Bodlaender H, Thilikos D (1997) Constructive linear time algorithms for branchwidth. In: Degano P, Gorrieri R, Marchetti-Spaccamela A (eds) Lecture Notes in Computer Science: Proceedings of the 24th International Colloquium on Automata, Languages, and Programming. Springer, Berlin, pp 627–637

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  18. Borie RB, Parker RG, Tovey CA (1992) Automatic generation of linear-time algorithms from predicate calculus descriptions of problems on recursively constructed graph families. Algorithmica 7:555–581

    Article  MathSciNet  MATH  Google Scholar 

  19. Christian WA (2003) Linear-Time Algorithms for Graphs with Bounded Branchwidth. PhD thesis, Rice University

    Google Scholar 

  20. Cook W, Seymour PD (1994) An algorithm for the ring-router problem. Technical report, Bellcore

    Google Scholar 

  21. Cook W, Seymour PD (2003) Tour merging via branch-decomposition. INFORMS J Comput 15(3):233–248

    Article  MathSciNet  Google Scholar 

  22. Courcelle B (1990) The monadic second-order logic of graphs I: Recognizable sets of finite graphs. Inf Comput 85:12–75

    Article  MathSciNet  MATH  Google Scholar 

  23. Dharmatilake JS (1996) A min-max theorem using matroid separations. Contemp Math 197:333–342

    MathSciNet  Google Scholar 

  24. Fomin F, Thilikos D (2003) Dominating sets in planar graphs: Branch-width and exponential speed-up. In: Proceedings of the Fourthteenth Annual ACM-SIAM Symposium on Discrete Algorithms (Baltimore, MD 2003). ACM, New York, pp 168–177

    Google Scholar 

  25. Geelen JF, Gerards AMH, Robertson N, Whittle GP (2003) On the excluded minors for the matroids of branch-width k. J Combin Theory Ser B 88:261–265

    Article  MathSciNet  MATH  Google Scholar 

  26. Geelen JF, Gerards AMH, Whittle G (2002) Branch width and well-quasi-ordering in matroids and graphs. J Combin Theory Ser B 84:270–290

    Article  MathSciNet  MATH  Google Scholar 

  27. Glover H, Huneke P, Wang CS (1979) 103 graphs that are irreducible for the projective plane. J Combin Theory Ser B 27:332–370

    Article  MathSciNet  MATH  Google Scholar 

  28. Gu QP, Tamaki H (2005) Optimal branch-decomposition of planar graphs in \( o(n^3) \) time. In: Proceedings of the 31st International Colloquium on Automata, Languages and Programming. LNCS, vol 3580, pp 373–384

    Google Scholar 

  29. Hall R, Oxley J, Semple C, Whittle G (2002) On matroids of branch-width three. J Combin Theory Ser B 86:148–171

    Article  MathSciNet  MATH  Google Scholar 

  30. Hicks IV (2000) Branch Decompositions and their Applications. PhD thesis, Rice University

    Google Scholar 

  31. Hicks IV (2002) Branchwidth heuristics. Congressus Numerantium 159:31–50

    MathSciNet  MATH  Google Scholar 

  32. Hicks IV (2004) Branch decompositions and minor containment. Networks 43(1):1–9

    Article  MathSciNet  MATH  Google Scholar 

  33. Hicks IV (2005) Graphs, branchwidth, and tangles! oh my! Networks 45:55–60

    Article  MathSciNet  MATH  Google Scholar 

  34. Hicks IV (2005) Planar branch decompositions I: The ratcatcher. INFORMS J Comput 17(4):402–412

    Article  MathSciNet  Google Scholar 

  35. Hicks IV (2005) Planar branch decompositions II: The cycle method. INFORMS J Comput 17(4):413–421

    Article  MathSciNet  Google Scholar 

  36. Hicks IV, Koster AMCA, Kolotoğlu E (2005) Branch and tree decomposition techniques for discrete optimization. In: Cole Smith J (ed) Tutorials in Operations Research 2005. INFORMS, Hanover, MD, pp 1–29

    Google Scholar 

  37. Hicks IV, McMurray N (2007) The branchwidth of graphs and their cycle matroids. J Combin Theory Ser B 97:681–692

    Google Scholar 

  38. Hliněný P (2002) On the exclued minors for matroids of branch-width three. preprint

    Google Scholar 

  39. Kloks T, Kratochvil J, Müller H (1999) New branchwidth territories. In: Meinel C, Tison S (eds) STAC'99, 16th Annual Symposium on Theoretical Aspects of Computer Science, Trier, Germany, March 1999 Proceedings. Springer, Berlin, pp 173–183

    Google Scholar 

  40. Koster A, van Hoesel S, Kolen A (2002) Solving partial constraint satisfaction problems with tree-decompositions. Networks 40:170–180

    Article  MathSciNet  MATH  Google Scholar 

  41. Koster AMCA, Bodlaender HL, van Hoesel SPM (2001) Treewidth: Computational experiments. Electr Notes Discret Math 8:54–57

    Article  Google Scholar 

  42. Kuratowski K (1930) Sur le probleme des courbes gauches en topologie. Fundamenta Mathematicae 15:271–283

    MATH  Google Scholar 

  43. Oxley JG (1992) Matroid Theory. Oxford University Press, Oxford

    MATH  Google Scholar 

  44. Ramachandramurthi S (1997) The structure and number of obstructions to treewidth. SIAM J Discret Math 10:146–157

    Article  MathSciNet  MATH  Google Scholar 

  45. Reed B (1992) Finding approximate separators and computing tree width quickly. In: Proceeding of the 24th Annual Association for Computing Machinery Symposium on Theory of Computing. ACM Press, New York, pp 221–228

    Google Scholar 

  46. Reed B (1997) Tree width and tangles: A new connectivity measure and some applications. In: Bailey RA (ed) Survey in Combinatorics. Cambridge University Press, Cambridge, pp 87–162

    Google Scholar 

  47. Reinelt G (1991) TSPLIB – a traveling salesman library. ORSA J Comput 3:376–384

    MATH  Google Scholar 

  48. Robertson N, Seymour PD (1985) Graph minors: A survey. In: Surveys in Combinatorics, London Math Society Lecture Note Series, edition 103. Cambridge University Press, Cambridge, pp 153–171

    Google Scholar 

  49. Robertson N, Seymour PD (1991) Graph minors X: Obstructions to tree-decompositions. J Combin Theory Ser B 52:153–190

    Article  MathSciNet  MATH  Google Scholar 

  50. Robertson N, Seymour PD (1994) Graph minors XI: Circuits on a surface. J Combin Theory Ser B 60:72–106

    Article  MathSciNet  MATH  Google Scholar 

  51. Robertson N, Seymour PD (1995) Graph minors XIII: The disjoint paths problem. J Combin Theory Ser B 63:65–110

    Article  MathSciNet  MATH  Google Scholar 

  52. Seymour P, Thomas R (1993) Graph searching and a min-max theorem for tree-width. J Combin Theory Ser B 58:22–33

    Google Scholar 

  53. Seymour PD, Thomas R (1994) Call routing and the ratcatcher. Combinatorica 14(2):217–241

    Article  MathSciNet  MATH  Google Scholar 

  54. Tamaki H (2003) A linear time heuristic for the branch-decomposition of planar graphs. Technical Report MPI-I-2003-1-010, Max-Planck-Institut fuer Informatik

    Google Scholar 

  55. Telle JA, Proskurowski A (1997) Algorithms for vertex partitioning problems on partial k-trees. SIAM J Discret Math 10(4):529–550

    Article  MathSciNet  MATH  Google Scholar 

  56. Thomas R (1990) A Menger-like property of tree-width: The finite case. J Combin Theory Ser B 48:67–76

    Article  MathSciNet  MATH  Google Scholar 

  57. Wagner K (1937) Uber eine eigenschaft der ebenen komplexe. Math Annal 115:570–590

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag

About this entry

Cite this entry

Hicks, I.V. (2008). Branchwidth and Branch Decompositions . In: Floudas, C., Pardalos, P. (eds) Encyclopedia of Optimization. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-74759-0_59

Download citation

Publish with us

Policies and ethics