Obtaining a bipartite graph by contracting few edges

Pinar Heggernes, Pim van 't Hof, Daniel Lokshtanov and Christophe Paul

SIAM Journal on Discrete Mathematics, vol. 27, no. 4, pp. 2143-2156, 2013.
[DOI] [Preprint]

A preliminary version of this paper appeared in the proceedings of FSTTCS 2011, the 31st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (held on December 12-14, 2011 in Mumbai, India), Leibniz International Proceedings in Informatics (LIPIcs), vol. 13, pp. 217-228, 2011


We initiate the study of the Bipartite Contraction problem from the perspective of parameterized complexity. In this problem we are given a graph G on n vertices 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)·nO(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. To obtain our result, we combine several techniques and concepts that are central in parameterized complexity: iterative compression, irrelevant vertex, and important separators. To the best of our knowledge, this is the first time the irrelevant vertex technique and the concept of important separators are applied in unison. Furthermore, our algorithm may serve as a comprehensible example of the usage of the irrelevant vertex technique.