A Counterexample Regarding Labelled Well-Quasi-Ordering

Graphs and Combinatorics volume 34, pages1395–1409(2018)Cite this article

Abstract

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.

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Notes

  1. 1.

    For this reason, Proposition 1.1 can be strengthened to state that every 2-wqo class of graphs is defined by finitely many minimal forbidden induced subgraphs.

References

  1. 1.

    Atkinson, M.D.: Restricted permutations. Discret. Math. 195(1–3), 27–38 (1999)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Avis, D.M., Newborn, M.: On pop-stacks in series. Utilitas Math. 19, 129–140 (1981)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Benzaken, C., Hammer, P.L., de Werra, D.: Split graphs of Dilworth number \(2\). Discret. Math. 55(2), 123–127 (1985)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Bose, P., Buss, J.F., Lubiw, A.: Pattern matching for permutations. Inform. Process. Lett. 65(5), 277–283 (1998)

    MathSciNet  Article  Google 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)

    Book  Google Scholar 

  8. 8.

    Brignall, R., Huczynska, S., Vatter, V.R.: Decomposing simple permutations, with enumerative consequences. Combinatorica 28(4), 385–400 (2008)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Daligault, J., Rao, M., Thomassé, S.: Well-quasi-order of relabel functions. Order 27(3), 301–315 (2010)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Damaschke, P.: Induced subgraphs and well-quasi-ordering. J. Graph Theory 14(4), 427–435 (1990)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Dushnik, B., Miller, E.W.: Partially ordered sets. Am. J. Math. 63, 600–610 (1941)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Földes, S., Hammer, P.L.: Split graphs. Congr. Numer. 14, 311–315 (1977)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Földes, S., Hammer, P.L.: Split graphs having Dilworth number two. Can. J. Math. 29(3), 666–672 (1977)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Gallai, T.: Transitiv orientierbare Graphen. Acta Math. Acad. Sci. Hungar. 18, 25–66 (1967)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Higman, G.: Ordering by divisibility in abstract algebras. Proc. Lond. Math. Soc. 3(2), 326–336 (1952)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Homberger, C., Pantone, J.: PermPy. http://permpy.com/ (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)

  18. 18.

    Information System on Graph Classes and their Inclusions (ISGCI). Published electronically at http://www.graphclasses.org/

  19. 19.

    Korpelainen, N., Lozin, V., Mayhill, C.: Split permutation graphs. Graphs Combin. 30(3), 633–646 (2014)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Korpelainen, N., Lozin, V., Razgon, I.: Boundary properties of well-quasi-ordered sets of graphs. Order 30(3), 723–735 (2013)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Murphy, M.M.: Restricted Permutations, Antichains, Atomic Classes, and Stack Sorting. PhD thesis, University of St Andrews (2002). http://hdl.handle.net/10023/11023

  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)

  23. 23.

    Robertson, N., Seymour, P.: Graph minors I–XX. J. Combin. Theory Ser. B (1983–2004)

  24. 24.

    Stankova, Z.E.: Forbidden subsequences. Discret. Math. 132(1–3), 291–316 (1994)

    MathSciNet  Article  Google 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 

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Acknowledgements

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].

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Affiliations

  1. School of Mathematics and Statistics, The Open University, Milton Keynes, England, UK

    Robert Brignall

  2. Department of Mathematics, University of Florida, Gainesville, FL, USA

    Michael Engen & Vincent Vatter

Authors
  1. Robert Brignall
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  2. Michael Engen
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  3. Vincent Vatter
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Corresponding author

Correspondence to Vincent Vatter.

Additional information

V. Vatter: Vatter’s research was sponsored by the National Security Agency under Grant number H98230-16-1-0324. The United States Government is authorized to reproduce and distribute reprints not-withstanding any copyright notation herein.

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Brignall, R., Engen, M. & Vatter, V. A Counterexample Regarding Labelled Well-Quasi-Ordering. Graphs and Combinatorics 34, 1395–1409 (2018). https://doi.org/10.1007/s00373-018-1962-0

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Keywords

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