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Unifying Duality Theorems for Width Parameters in Graphs and Matroids (Extended Abstract)

  • Reinhard DiestelEmail author
  • Sang-il Oum
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8747)

Abstract

We prove a general duality theorem for width parameters in combinatorial structures such as graphs and matroids. It implies the classical such theorems for path-width, tree-width, branch-width and rank-width, and gives rise to new width parameters with associated duality theorems. The dense substructures witnessing large width are presented in a unified way akin to tangles, as orientations of separation systems satisfying certain consistency axioms.

Keywords

Duality Theorem Separation System Width Parameter Order Function Connectivity Function 
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.

References

  1. 1.
    Amini, O., Mazoit, F., Nisse, N., Thomassé, S.: Submodular partition functions. Discrete Appl. Math. 309(20), 6000–6008 (2009)CrossRefzbMATHGoogle Scholar
  2. 2.
    Bienstock, D., Robertson, N., Seymour, P., Thomas, R.: Quickly excluding a forest. J. Combin. Theor. Ser. B 52(2), 274–283 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Diestel, R.: Graph Theory, 4th edn. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  4. 4.
    Geelen, J., Gerards, B., Robertson, N., Whittle, G.: Obstructions to branch-decomposition of matroids. J. Combin. Theor. Ser. B 96, 560–570 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Hliněný, P., Whittle, G.: Matroid tree-width. European J. Combin. 27(7), 1117–1128 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Hliněný, P., Whittle, G.: Addendum to matroid tree-width. European J. Combin. 30, 1036–1044 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Mazoit, F.: A simple proof of the tree-width duality theorem. arXiv:1309.2266 (2013)
  8. 8.
    Oum, S., Seymour, P.: Approximating clique-width and branch-width. J. Combin. Theor. Ser. B 96(4), 514–528 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Oxley, J.: Matroid Theory. Oxford University Press, Oxford (1992)zbMATHGoogle Scholar
  10. 10.
    Robertson, N., Seymour, P.: Graph minors. X. Obstructions to tree-decomposition. J. Combin. Theor. Ser. B 52, 153–190 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Seymour, P., Thomas, R.: Graph searching and a min-max theorem for tree-width. J. Combin. Theor. Ser. B 58(1), 22–33 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Seymour, P., Thomas, R.: Call routing and the ratcatcher. Combinatorica 14(2), 217–241 (1994)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Mathematisches SeminarUniversität HamburgHamburgGermany
  2. 2.Department of Mathematical SciencesKAISTDaejeonSouth Korea

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