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Acta Informatica

, Volume 51, Issue 7, pp 473–497 | Cite as

Parameterized complexity of three edge contraction problems with degree constraints

  • Rémy Belmonte
  • Petr A. Golovach
  • Pim van ’t HofEmail author
  • Daniël Paulusma
Original Article

Abstract

For any graph class \(\mathcal{H}\), the \(\mathcal{H}\)-Contraction problem takes as input a graph \(G\) and an integer \(k\), and asks whether there exists a graph \(H\in \mathcal{H}\) such that \(G\) can be modified into \(H\) using at most \(k\) edge contractions. We study the parameterized complexity of \(\mathcal{H}\)-Contraction for three different classes \(\mathcal{H}\): the class \(\mathcal{H}_{\le d}\) of graphs with maximum degree at most \(d\), the class \(\mathcal{H}_{=d}\) of \(d\)-regular graphs, and the class of \(d\)-degenerate graphs. We completely classify the parameterized complexity of all three problems with respect to the parameters \(k\), \(d\), and \(d+k\). Moreover, we show that \(\mathcal{H}\)-Contraction admits an \(O(k)\) vertex kernel on connected graphs when \(\mathcal{H}\in \{\mathcal{H}_{\le 2},\mathcal{H}_{=2}\}\), while the problem is \(\mathsf{W}[2]\)-hard when \(\mathcal{H}\) is the class of \(2\)-degenerate graphs and hence is expected not to admit a kernel at all. In particular, our results imply that \(\mathcal{H}\)-Contraction admits a linear vertex kernel when \(\mathcal{H}\) is the class of cycles.

Notes

Acknowledgments

We would like to thank Marcin Kamiński and Dimitrios Thilikos for fruitful discussions on the topic. We also thank the three anonymous referees of the conference version of this paper for insightful comments.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Rémy Belmonte
    • 1
  • Petr A. Golovach
    • 1
  • Pim van ’t Hof
    • 1
    Email author
  • Daniël Paulusma
    • 2
  1. 1.Department of InformaticsUniversity of BergenBergenNorway
  2. 2.School of Engineering and Computing SciencesDurham UniversityDurhamUK

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