Hereditary classes of ordered sets of width at most two

TL;DR

This paper investigates hereditary classes of ordered sets of width two, proving they are well-quasi-ordered and better-quasi-ordered, and extends these results to related structures like bipartite permutation graphs and bichains.

Contribution

It provides new proofs and extends known results on the well-quasi-ordering and better-quasi-ordering of hereditary classes of width-two ordered sets and related structures.

Findings

Hereditary classes of bipartite permutation graphs are wqo and bqo.

Hereditary classes of posets of width at most two are wqo and bqo.

Labelled wqo notions are equivalent for these classes.

Abstract

This paper is a contribution to the study of hereditary classes of relational structures, these classes being quasi-ordered by embeddability. It deals with the specific case of ordered sets of width two and the corresponding bichains and incomparability graphs. Several open problems about hereditary classes of relational structures which have been considered over the years have positive answer in this case. For example, well-quasi-ordered hereditary classes of finite bipartite permutation graphs, respectively finite 321-avoiding permutations, have been characterized by Korpelainen, Lozin and Mayhill, respectively by Albert, Brignall, Ru\v{s}kuc and Vatter. We provide another proof of the results mentioned above. It is based on the existence of a countable universal poset of width two, obtained by the first author in 1978, his notion of multichainability (1978) (a kind of analog to letter-graphs), and metric properties of incomparability graphs. Using Laver's theorem (1971) on better-quasi-ordering (bqo) of countable chains we prove that a wqo hereditary class of finite or countable bipartite permutation graphs is necessarily bqo. This gives a positive answer to a conjecture of Nash-Williams (1965) in this case. We extend a previous result of Albert et al. by proving that if a hereditary class of finite, respectively countable, bipartite permutation graphs is wqo, respectively bqo, then the corresponding hereditary classes of posets of width at most two and bichains are wqo, respectively bqo. Several notions of labelled wqo are also considered. We prove that they are all equivalent in the case of bipartite permutation graphs, posets of width at most two and the corresponding bichains. We characterize hereditary classes of finite bipartite permutation graphs which remain wqo when labels from a wqo are added.