Parameterized algorithms for recognizing monopolar and 2-subcolorable graphs

Iyad Kanj,Christian Komusiewicz,Manuel Sorge,Erik Jan Van Leeuwen

doi:10.1016/j.jcss.2017.08.002

Abstract

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.

Highlights

A (ΠA, ΠB)-graph, for graph properties ΠA, ΠB, is a graph G = (V, E) for which V admits a partition into two sets A, B such that G[A] satisfies ΠA and G[B] satisfies ΠB
We show that, unless the Exponential Time Hypothesis fails, no subexponential-time algorithms for the above recognition problems exist, and that, unless P=NP, no generic fixed-parameter algorithm exists for the recognizability of graphs whose vertex set can be bipartitioned such that one part is a disjoint union of k cliques
There is an abundance of (ΠA, ΠB)-graph classes, and important ones include, in addition to bipartite graphs (i.e., 2-colorable graphs), well-known graph classes such as split graphs, and unipolar graphs

Summary

Introduction

This is a stark contrast to the variants of both problems where G[A] is required to consist of a single cluster, which correspond to the recognition of split graphs and unipolar graphs, respectively, and admit polynomial-time algorithms [13, 16, 10, 20] This has left the complexity of Monopolar Recognition and 2-Subcoloring parameterized by the number of clusters in G[A] as intriguing open questions. Observe that Theorems 1.1 and 1.2 give fixed-parameter algorithms for two (ΠA, ΠB)Recognition problems, in both of which ΠA defines the set of all cluster graphs, parameterized by the number of clusters in G[A]. We were inspired by their Observation 2, which we adapt to our setting to help bound the running time of our algorithms

Preliminaries

Foundations for Inductive Recognition

An FPT algorithm for Monopolar Recognition

An FPT algorithm for 2-Subcoloring