Bichain graphs: Geometric model and universal graphs

Bichain graphs form a bipartite analog of split permutation graphs, also known as split graphs of Dilworth number 2. Unlike graphs of Dilworth number 1 that enjoy many nice properties, split permutation graphs are substantially more complex. To better understand the global structure of split permutation graphs, in the present paper we study their bipartite analog. We show that bichain graphs admit a simple geometric representation and have a universal element of quadratic order, i.e. an n-universal bichain graph with n vertices. The latter result improves a recent cubic construction of universal split permutation graphs.