Parameterized Algorithms for Recognizing Monopolar and 2-Subcolorable Graphs

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