A Counterexample Regarding Labelled Well-Quasi-Ordering

Brignall, Robert; Engen, Michael and Vatter, Vincent (2018). A Counterexample Regarding Labelled Well-Quasi-Ordering. Graphs and Combinatorics, 34(6) pp. 1395–1409.

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

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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