A Counterexample Regarding Labelled Well-Quasi-Ordering

  • Robert Brignall
  • Michael Engen
  • Vincent Vatter
Original Paper


Korpelainen, Lozin, and Razgon conjectured that a hereditary property of graphs which is well-quasi-ordered by the induced subgraph order and defined by only finitely many minimal forbidden induced subgraphs is labelled well-quasi-ordered, a notion stronger than that of n-well-quasi-order introduced by Pouzet in the 1970s. We present a counterexample to this conjecture. In fact, we exhibit a hereditary property of graphs which is well-quasi-ordered by the induced subgraph order and defined by finitely many minimal forbidden induced subgraphs yet is not 2-well-quasi-ordered. This counterexample is based on the widdershins spiral, which has received some study in the area of permutation patterns.


Labelled well-quasi-order Permutation graph Well-quasi-order Widdershins spiral 



We thank Jay Pantone for performing the computer search that established Corollary 3.5. The computation was performed using the PermPy package developed by Homberger and Pantone [16].


  1. 1.
    Atkinson, M.D.: Restricted permutations. Discret. Math. 195(1–3), 27–38 (1999)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Atminas, A., Brignall, R., Lozin, V., Stacho, J.: Minimal classes of graphs of unbounded clique-width defined by finitely many forbidden induced subgraphs. arXiv:1503.01628 [math.CO]
  3. 3.
    Atminas, A., Lozin, V.: Labelled induced subgraphs and well-quasi-ordering. Order 32(3), 313–328 (2015)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Avis, D.M., Newborn, M.: On pop-stacks in series. Utilitas Math. 19, 129–140 (1981)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Benzaken, C., Hammer, P.L., de Werra, D.: Split graphs of Dilworth number \(2\). Discret. Math. 55(2), 123–127 (1985)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bose, P., Buss, J.F., Lubiw, A.: Pattern matching for permutations. Inform. Process. Lett. 65(5), 277–283 (1998)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph Classes: A Survey. SIAM Monographs on Discrete Mathematics and Applications. SIAM, Philadelphia (1999)CrossRefGoogle Scholar
  8. 8.
    Brignall, R., Huczynska, S., Vatter, V.R.: Decomposing simple permutations, with enumerative consequences. Combinatorica 28(4), 385–400 (2008)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Daligault, J., Rao, M., Thomassé, S.: Well-quasi-order of relabel functions. Order 27(3), 301–315 (2010)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Damaschke, P.: Induced subgraphs and well-quasi-ordering. J. Graph Theory 14(4), 427–435 (1990)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Dushnik, B., Miller, E.W.: Partially ordered sets. Am. J. Math. 63, 600–610 (1941)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Földes, S., Hammer, P.L.: Split graphs. Congr. Numer. 14, 311–315 (1977)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Földes, S., Hammer, P.L.: Split graphs having Dilworth number two. Can. J. Math. 29(3), 666–672 (1977)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Gallai, T.: Transitiv orientierbare Graphen. Acta Math. Acad. Sci. Hungar. 18, 25–66 (1967)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Higman, G.: Ordering by divisibility in abstract algebras. Proc. Lond. Math. Soc. 3(2), 326–336 (1952)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Homberger, C., Pantone, J.: PermPy. (2017)
  17. 17.
    Huczynska, S., Ruškuc, N.: Well quasi-order in combinatorics: embeddings and homomorphisms. In: Czumaj, A., Georgakopoulos, A., Král’, D., Lozin, V., Pikhurko, O. (eds.) Surveys in Combinatorics 2015, vol. 424 of London Mathematical Society Lecture Note Series, pp. 261–293. Cambridge University Press, Cambridge (2015)Google Scholar
  18. 18.
    Information System on Graph Classes and their Inclusions (ISGCI). Published electronically at
  19. 19.
    Korpelainen, N., Lozin, V., Mayhill, C.: Split permutation graphs. Graphs Combin. 30(3), 633–646 (2014)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Korpelainen, N., Lozin, V., Razgon, I.: Boundary properties of well-quasi-ordered sets of graphs. Order 30(3), 723–735 (2013)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Murphy, M.M.: Restricted Permutations, Antichains, Atomic Classes, and Stack Sorting. PhD thesis, University of St Andrews (2002).
  22. 22.
    Pouzet, M.: Un bel ordre d’abritement et ses rapports avec les bornes d’une multirelation. C. R. Acad. Sci. Paris Sér. A-B 274, A1677–A1680 (1972)Google Scholar
  23. 23.
    Robertson, N., Seymour, P.: Graph minors I–XX. J. Combin. Theory Ser. B (1983–2004)Google Scholar
  24. 24.
    Stankova, Z.E.: Forbidden subsequences. Discret. Math. 132(1–3), 291–316 (1994)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Vatter, V.R.: Permutation classes. In: Bóna, M. (ed.) Handbook of Enumerative Combinatorics, pp. 754–833. CRC Press, Boca Raton (2015)Google Scholar

Copyright information

© Springer Japan KK, part of Springer Nature 2018

Authors and Affiliations

  • Robert Brignall
    • 1
  • Michael Engen
    • 2
  • Vincent Vatter
    • 2
  1. 1.School of Mathematics and StatisticsThe Open UniversityMilton KeynesUK
  2. 2.Department of MathematicsUniversity of FloridaGainesvilleUSA

Personalised recommendations