# Parameterized Algorithms for Recognizing Monopolar and 2-Subcolorable Graphs

## TL;DR

This paper introduces fixed-parameter algorithms for recognizing complex graph classes like monopolar and 2-subcolorable graphs, using a novel inductive recognition technique, and discusses related computational hardness results.

## Contribution

It presents the first fixed-parameter algorithms for NP-hard recognition problems of monopolar and 2-subcolorable graphs, expanding algorithmic tools for these graph classes.

## Key findings

- Developed fixed-parameter algorithms for monopolar and 2-subcolorable recognition.
- Introduced the inductive recognition technique for graph recognition problems.
- Provided hardness results for various $(\

## Abstract

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