Parameterized Complexity of Two Edge Contraction Problems with Degree Constraints

  • Rémy Belmonte
  • Petr A. Golovach
  • Pim van ’t Hof
  • Daniël Paulusma
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8246)


Motivated by recent results of Mathieson and Szeider (J. Comput. Syst. Sci. 78(1): 179–191, 2012), we study two graph modification problems where the goal is to obtain a graph whose vertices satisfy certain degree constraints. The Regular Contraction problem takes as input a graph G and two integers d and k, and the task is to decide whether G can be modified into a d-regular graph using at most k edge contractions. The Bounded Degree Contraction problem is defined similarly, but here the objective is to modify G into a graph with maximum degree at most d. We observe that both problems are fixed-parameter tractable when parameterized jointly by k and d. We show that when only k is chosen as the parameter, Regular Contraction becomes W[1]-hard, while Bounded Degree Contraction becomes W[2]-hard even when restricted to split graphs. We also prove both problems to be NP-complete for any fixed d ≥ 2. On the positive side, we show that the problem of deciding whether a graph can be modified into a cycle using at most k edge contractions, which is equivalent to Regular Contraction when d = 2, admits an O(k) vertex kernel. This complements recent results stating that the same holds when the target is a path, but that the problem admits no polynomial kernel when the target is a tree, unless NP ⊆ coNP/poly (Heggernes et al., IPEC 2011).


Regular Graph Parameterized Complexity Polynomial Kernel Reduction Rule Optimal Cycle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

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

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