A counterexample regarding labelled well-quasi-ordering

TL;DR

This paper provides a counterexample to a conjecture in graph theory, showing that certain hereditary graph properties are not necessarily labelled well-quasi-ordered despite being well-quasi-ordered and finitely characterized.

Contribution

It introduces a specific hereditary graph property based on the widdershins spiral that disproves a previous conjecture about labelled well-quasi-ordering.

Findings

Counterexample based on the widdershins spiral disproves the conjecture.

Hereditary property is well-quasi-ordered but not 2-well-quasi-ordered.

Challenges assumptions about the relationship between minimal forbidden subgraphs and labelled well-quasi-ordering.

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.