Parameterized algorithms for recognizing monopolar and 2-subcolorable graphs

A graph G is a (ΠA,ΠB)-graph if V(G) can be bipartitioned into A and B such that G[A] satisfies property ΠA and G[B] satisfies property ΠB. The (ΠA,ΠB)-Recognition problem is to recognize whether a given graph is a (ΠA,ΠB)-graph. There are many (ΠA,ΠB)-Recognition problems, including the recognition problems for bipartite, split, and unipolar graphs. We present efficient algorithms for many cases of (ΠA,ΠB)-Recognition based on a technique which we dub inductive recognition. In particular, we give fixed-parameter algorithms for two NP-hard (ΠA,ΠB)-Recognition problems, Monopolar Recognition and 2-Subcoloring, parameterized by the number of maximal cliques in G[A]. We complement our algorithmic results with several hardness results for (ΠA,ΠB)-Recognition.