  The new graph parameter twin-width, introduced by Bonnet, Kim, Thomass e and
Watrigant in 2020, allows for an FPT algorithm for testing all FO properties of
graphs. This makes classes of efficiently bounded twin-width attractive from
the algorithmic point of view. In particular, classes of efficiently bounded
twin-width include proper interval graphs, and (as digraphs) posets of width k.
Inspired by an existing generalization of interval graphs into so-called k-thin
graphs, we define a new class of proper k-mixed-thin graphs which largely
generalizes proper interval graphs. We prove that proper k-mixed-thin graphs
have twin-width linear in k, and that a slight subclass of k-mixed-thin graphs
is transduction-equivalent to posets of width k' such that there is a
quadratic-polynomial relation between k and k'. In addition to that, we also
give an abstract overview of the so-called red potential method which we use to
prove our twin-width bounds.
