  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.
