Critical properties of bipartite permutation graphs

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