Obtaining a Bipartite Graph by Contracting Few Edges

TL;DR

This paper introduces a fixed-parameter tractable algorithm for the Bipartite Contraction problem, which determines if a graph can be made bipartite through a limited number of edge contractions, using novel combinations of advanced techniques.

Contribution

It presents the first fixed-parameter algorithm for Bipartite Contraction, combining irrelevant vertex and important separator techniques in a novel way.

Findings

Developed an $f(k) n^{O(1)}$ time algorithm for Bipartite Contraction.

Demonstrated the applicability of combining irrelevant vertex and important separator techniques.

Provided insights into the complexity of transforming graphs into bipartite forms.

Abstract

We initiate the study of the Bipartite Contraction problem from the perspective of parameterized complexity. In this problem we are given a graph $G$ and an integer $k$, and the task is to determine whether we can obtain a bipartite graph from $G$ by a sequence of at most $k$ edge contractions. Our main result is an $f(k) n^{O(1)}$ time algorithm for Bipartite Contraction. Despite a strong resemblance between Bipartite Contraction and the classical Odd Cycle Transversal (OCT) problem, the methods developed to tackle OCT do not seem to be directly applicable to Bipartite Contraction. Our algorithm is based on a novel combination of the irrelevant vertex technique, introduced by Robertson and Seymour, and the concept of important separators. Both techniques have previously been used as key components of algorithms for fundamental problems in parameterized complexity. However, to the best of our knowledge, this is the first time the two techniques are applied in unison.