  The class of bipartite permutation graphs enjoys many nice and important
properties. In particular, this class is critically important in the study of
clique- and rank-width of graphs, because it is one of the minimal hereditary
classes of graphs of unbounded clique- and rank-width. It also contains a
number of important subclasses, which are critical with respect to other
parameters, such as graph lettericity or shrub-depth, and with respect to other
notions, such as well-quasi-ordering or complexity of algorithmic problems. In
the present paper we identify critical subclasses of bipartite permutation
graphs of various types.
