Split contraction: The untold story

Akanksha Agrawal, Daniel Lokshtanov, Saket Saurabh, Meirav Zehavi

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

4 Scopus citations


The edit operation that contracts edges, which is a fundamental operation in the theory of graph minors, has recently gained substantial scientific attention from the viewpoint of Parameterized Complexity. In this paper, we examine an important family of graphs, namely the family of split graphs, which in the context of edge contractions, is proven to be significantly less obedient than one might expect. Formally, given a graph G and an integer k, SPLIT CONTRACTION asks whether there exists X ⊆ E(G) such that G/X is a split graph and |X| ≤ k. Here, G/X is the graph obtained from G by contracting edges in X. It was previously claimed that SPLIT CONTRACTION is fixed-parameter tractable. However, we show that SPLIT CONTRACTION, despite its deceptive simplicity, is W[1]-hard. Our main result establishes the following conditional lower bound: under the Exponential Time Hypothesis, SPLIT CONTRACTION cannot be solved in time 2o(ℓ2) · nO(1) where ℓ is the vertex cover number of the input graph. We also verify that this lower bound is essentially tight. To the best of our knowledge, this is the first tight lower bound of the form 2o(ℓ2) · nO(1) for problems parameterized by the vertex cover number of the input graph. In particular, our approach to obtain this lower bound borrows the notion of harmonious coloring from Graph Theory, and might be of independent interest.

Original languageEnglish
Title of host publication34th Symposium on Theoretical Aspects of Computer Science, STACS 2017
EditorsBrigitte Vallee, Heribert Vollmer
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959770286
StatePublished - 1 Mar 2017
Externally publishedYes
Event34th Symposium on Theoretical Aspects of Computer Science, STACS 2017 - Hannover, Germany
Duration: 8 Mar 201711 Mar 2017

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
ISSN (Print)1868-8969


Conference34th Symposium on Theoretical Aspects of Computer Science, STACS 2017


  • Edge contraction
  • Parameterized complexity
  • Split graph

ASJC Scopus subject areas

  • Software


Dive into the research topics of 'Split contraction: The untold story'. Together they form a unique fingerprint.

Cite this