Well quasi-order in combinatorics: embeddings and homomorphisms

Huczynska, Sophie; Ruškuc, Nik

doi:10.1017/cbo9781316106853.009

Cited by 10 publications

(8 citation statements)

References 52 publications

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“…To do so, we must first introduce a few facts about 321‐avoiding antichains. Following Huczynska and Ruškuc , we say that a double‐ended fork is the graph formed from a path by adding four vertices of degree 1, two adjacent to one end of the path, and two adjacent to the other, as shown below.

It is clear that the set of double‐ended forks forms an antichain of graphs in the induced subgraph ordering.…”

Section: Well‐quasi‐ordermentioning

confidence: 99%

Growth rates of permutation classes: from countable to uncountable

Vatter

2019

Proc. London Math. Soc.

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We establish that there is an algebraic number ξ≈2.30522 such that while there are uncountably many growth rates of permutation classes arbitrarily close to ξ, there are only countably many less than ξ. Central to the proof are various structural notions regarding generalized grid classes and a new property of permutation classes called concentration. The classification of growth rates up to ξ is completed in a subsequent paper.

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It is clear that the set of double‐ended forks forms an antichain of graphs in the induced subgraph ordering.…”

Section: Well‐quasi‐ordermentioning

confidence: 99%

Growth rates of permutation classes: from countable to uncountable

Vatter

2019

Proc. London Math. Soc.

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“…As the set of finite graphs under the induced subgraph relation cannot contain an infinite strictly decreasing chain, in this case well-quasi-order is synonymous with the absence of infinite antichains. Well-quasi-ordering has been studied for a wide variety of combinatorial objects, under many different orders; see Huczynska and Ruškuc [17] for a recent survey. Thus while the celebrated Minor Theorem of Robertson and Seymour [23] shows that the family of all graphs is wqo under the minor order, one might instead ask about the induced subgraph order, and in this context the set of graphs is clearly not wqo.…”

Section: Well-quasi-order and Strengtheningsmentioning

confidence: 99%

A Counterexample Regarding Labelled Well-Quasi-Ordering

Brignall

Engen

Vatter

2018

Graphs and Combinatorics

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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-quasiorder 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-wellquasi-ordered. This counterexample is based on the widdershins spiral, which has received some study in the area of permutation patterns.

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“…Three of the most celebrated results in combinatorics-Higman's lemma [51], Kruskal's tree theorem [62], and Robertson and Seymour's graph minor theorem [89]-establish that certain notions of embedding constitute well-quasi-orders. For further background on well-quasi-order in general we refer the reader to the excellent panoramas provided by the recent surveys of Cherlin [31] and Huczynska and Ruškuc [52]. It should be noted that well-quasi-order also has significant applications to algorithmic questions, in particular questions about fixed-parameter tractability, for which the reader is referred to the book of Downey and Fellows [36,Part IV].…”

Section: Introductionmentioning

confidence: 99%

Labelled well-quasi-order for permutation classes

Brignall¹,

Vatter²

2021

Preprint

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While the theory of labeled well-quasi-order has received significant attention in the graph setting, it has not yet been considered in the context of permutation patterns. We initiate this study here, and using labeled well-quasi-order are able to subsume and extend all of the well-quasiorder results in the permutation patterns literature. Connections to the graph setting are emphasized throughout. In particular, we establish that a permutation class is labeled well-quasi-ordered if and only if its corresponding graph class is also labeled well-quasi-ordered. * Vatter's research was partially supported by the Simons Foundation via award number 636113. 1 Our figure of 80 years dates the study of well-quasi-order to Wagner [103]. 2 A well-quasi-ordered partial order is sometimes called a partially well-ordered or well-partially-ordered set, or it is simply called a partial well-order. In particular, these terms are used in some of the early work on well-quasi-order in the permutation patterns context. We tend to agree with Kruskal's sentiment from [63, p. 298], where he wrote that "at the casual level it is easier to work with [partial orders] than [quasi-orders], but in advanced work the reverse is true.

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Well quasi-order in combinatorics: embeddings and homomorphisms

Cited by 10 publications

References 52 publications

Growth rates of permutation classes: from countable to uncountable

Growth rates of permutation classes: from countable to uncountable

A Counterexample Regarding Labelled Well-Quasi-Ordering

Labelled well-quasi-order for permutation classes

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