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A Counterexample Regarding Labelled Well-Quasi-Ordering

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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. 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.

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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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Authors and 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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  • DOI: https://doi.org/10.1007/s00373-018-1962-0

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