Parameterized Algorithms for Recognizing Monopolar and 2-Subcolorable Graphs

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