Advertisement

Well-Quasi-Orderings and the Robertson–Seymour Theorems

  • Rodney G. Downey
  • Michael R. Fellows
Part of the Texts in Computer Science book series (TCS)

Abstract

As we will see, well-quasi-orderings (WQOs) provide a powerful engine for demonstrating that classes of problems are FPT. In the next section, we will look at the rudiments of the theory of WQOs, and in subsequent sections, we will examine applications to combinatorial problems.

Keywords

Planar Graph Plane Embedding Finite Poset Machine Description 20th Century Mathematics 
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. 4.
    K. Abrahamson, M. Fellows, Cutset regularity beats well-quasi-ordering for bounded treewidth, Preprint (1989) Google Scholar
  2. 5.
    K. Abrahamson, M. Fellows, Finite automata, bounded treewidth, and well-quasiordering, in Proceedings of the AMS Summer Workshop on Graph Minors, Graph Structure Theory, ed. by N. Robertson, P. Seymour. Contemporary Mathematics, vol. 147 (Am. Math. Soc., Providence, 1993), pp. 539–564 Google Scholar
  3. 68.
    J. Blasiok, M. Kaminski, Chain minors are FPT, in Proceedings of IPEC 2013 (2013) Google Scholar
  4. 131.
    K. Cattell, M. Dinneen, A characterization of graphs with vertex cover up to five, in Proceedings of the International Workshop on Orders, Algorithms, and Applications (ORDAL ’94). LNCS, vol. 831 (Springer, London, 1994), pp. 86–99 CrossRefGoogle Scholar
  5. 132.
    K. Cattell, M. Dinneen, R. Downey, M. Fellows, M. Langston, On computing graph minor obstruction sets. Theor. Comput. Sci. 233, 107–127 (2000) MathSciNetCrossRefMATHGoogle Scholar
  6. 133.
    K. Cattell, M. Dinneen, M. Fellows, Obstructions to within a few vertices or edges of acyclic, in Algorithms and Data Structures, Proceedings of 4th International Workshop, WADS 1995, Kingston, Ontario, Canada, August 16–18, 1995, ed. by S. Akl, F. Dehne, J.-R. Sack, N. Santoro. LNCS, vol. 955 (Springer, Berlin, 1995), pp. 415–427 Google Scholar
  7. 176.
    B. Courcelle, R. Downey, M. Fellows, A note on the computability of graph minor obstruction sets for monadic second-order ideals. J. Univers. Comput. Sci. 3, 1194–1198 (1997) MathSciNetMATHGoogle Scholar
  8. 224.
    G. Dirac, S. Schuster, A theorem of Kuratowski. Indag. Math. 16(3), 343–348 (1954) MathSciNetGoogle Scholar
  9. 247.
    R. Downey, M. Fellows, Parameterized Complexity. Monographs in Computer Science (Springer, Berlin, 1999) CrossRefGoogle Scholar
  10. 279.
    L. Euler, Solutio problematis ad geometriam situs pertinentis. Comment. Acad. Sci. Petropolitanae 8, 128–140 (1741) Google Scholar
  11. 297.
    M. Fellows, M. Langston, Nonconstructive proofs of polynomial-time complexity. Inf. Process. Lett. 26, 157–162 (1987/88) MathSciNetCrossRefGoogle Scholar
  12. 299.
    M. Fellows, M. Langston, Nonconstructive tools for proving polynomial-time complexity. J. ACM 35, 727–739 (1988) MathSciNetMATHGoogle Scholar
  13. 301.
    M. Fellows, M. Langston, On search, decision and the efficiency of polynomial time algorithms, in Proceedings of 21st ACM Symposium on Theory of Computing (STOC ’89), Seattle, Washington, USA, May 15–May 17, 1989, ed. by D. Johnson (ACM, New York, 1989), pp. 501–512. http://dl.acm.org/citation.cfm?id=73055 Google Scholar
  14. 359.
    M. Grohe, K. Kawarabayashi, D. Marx, P. Wollan, Finding topological subgraphs is fixed-parameter tractable, in Proceedings of 43rd ACM Symposium on Theory of Computing (STOC ’11), San Jose, California, USA, June 6–June 8, 2011, ed. by L. Fortnow, S. Vadhan (ACM, New York, 2011), pp. 479–488 Google Scholar
  15. 369.
    J. Gustedt, Well quasi ordering finite posets and formal languages. J. Comb. Theory, Ser. B 65(1), 111–124 (1995) MathSciNetCrossRefMATHGoogle Scholar
  16. 378.
    F. Harary, Graph Theory (Addison-Wesley, Reading, 1969) Google Scholar
  17. 393.
    G. Higman, Ordering by divisibility in abstract algebras. Proc. Lond. Math. Soc. 2, 326–336 (1952) MathSciNetCrossRefMATHGoogle Scholar
  18. 437.
    J. Kennedy, L. Quintas, M. Sysło, The theorem on planar graphs. Hist. Math. 12(4), 356–368 (1985) CrossRefMATHGoogle Scholar
  19. 471.
    J. Kruskal, Well-quasi-ordering, the tree theorem, and Vazsonyi’s conjecture. Trans. Am. Math. Soc. 95(2), 210–225 (1960) MathSciNetMATHGoogle Scholar
  20. 472.
    J. Kruskal, The theory of well-quasi-ordering: a frequently rediscovered concept. J. Comb. Theory, Ser. A 13, 297–305 (1972) MathSciNetCrossRefMATHGoogle Scholar
  21. 473.
    C. Kuratowski, Sur le probleme des courbes gauches en topologie. Fundam. Math. 15, 271–283 (1930) MATHGoogle Scholar
  22. 533.
    B. Mohar, Embedding graphs in an arbitrary surface in linear time, in Proceedings of 28th ACM Symposium on Theory of Computing (STOC ’96), Philadelphia, Pennsylvania, USA, May 22–May 24, 1996, ed. by G. Miller (ACM, New York, 1996), pp. 392–397 Google Scholar
  23. 542.
    C. Nash-Williams, On well-quasi-ordering finite trees. Math. Proc. Camb. Philos. Soc. 59(4), 833–835 (1963) MathSciNetCrossRefMATHGoogle Scholar
  24. 543.
    C. Nash-Williams, On well-quasi-ordering infinite trees. Math. Proc. Camb. Philos. Soc. 61(3), 697–720 (1965) MathSciNetCrossRefMATHGoogle Scholar
  25. 570.
    R. Rado, Partial well-ordering of sets of vectors. Mathematika 1, 88–95 (1954) MathSciNetCrossRefGoogle Scholar
  26. 572.
    F.P. Ramsey, On a problem of formal logic. Proc. Lond. Math. Soc. 30, 264–286 (1930) MathSciNetCrossRefGoogle Scholar
  27. 579.
    H. Rice, Classes of recursively enumerable sets and their decision problems. Trans. Am. Math. Soc. 74, 358–366 (1953) MathSciNetCrossRefMATHGoogle Scholar
  28. 588.
    N. Robertson, P. Seymour, Graph minors. XIII. The disjoint paths problem. J. Comb. Theory, Ser. B 63(1), 65–110 (1995) MathSciNetCrossRefMATHGoogle Scholar
  29. 589.
    N. Robertson, P. Seymour, Graph minors. XX. Wagner’s conjecture. J. Comb. Theory, Ser. B 92(2), 325–357 (2004) MathSciNetCrossRefMATHGoogle Scholar
  30. 612.
    D. Seese, The structure of models of decidable monadic theories of graphs. Ann. Pure Appl. Log. 53(2), 169–195 (1991) MathSciNetCrossRefMATHGoogle Scholar
  31. 642.
    R. Thomas, Well-quasi-ordering infinite graphs with forbidden finite planar minor. Trans. Am. Math. Soc. 312(1), 279–313 (1989) CrossRefMATHGoogle Scholar
  32. 648.
    B. Trakhtenbrot, Impossibility of an algorithm for the decision problem on finite classes. Dokl. Akad. Nauk SSSR 70, 569–572 (1950) Google Scholar
  33. 660.
    K. Wagner, Über einer Eigenschaft der ebener Complexe. Math. Ann. 14, 570–590 (1937) CrossRefGoogle Scholar

Copyright information

© Springer-Verlag London 2013

Authors and Affiliations

  • Rodney G. Downey
    • 1
  • Michael R. Fellows
    • 2
  1. 1.School of Mathematics, Statistics and Operations ResearchVictoria UniversityWellingtonNew Zealand
  2. 2.School of Engineering and Information TechnologyCharles Darwin UniversityDarwinAustralia

Personalised recommendations