Advertisement

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)

Abstract

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).

Keywords

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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Asano, T., Hirata, T.: Edge-contraction problems. J. Comput. Syst. Sci. 26(2), 197–208 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Brouwer, A.E., Veldman, H.J.: Contractibility and NP-completeness. J. Graph Theory 11, 71–79 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cai, L.: Parameterized complexity of cardinality constrained optimization problems. The Computer Journal 51(1), 102–121 (2008)CrossRefGoogle Scholar
  4. 4.
    Diestel, R.: Graph Theory, Electronic Edition. Springer-Verlag (2005)Google Scholar
  5. 5.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer (1999)Google Scholar
  6. 6.
    Fellows, M.R., Hermelin, D., Rosamond, F., Vialette, S.: On the parameterized complexity of multiple-interval problems. Theor. Comp. Sci. 410, 53–61 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Golovach, P.A., van ’t Hof, P., Paulusma, D.: Obtaining planarity by contracting few edges. Theor. Comp. Sci. 476, 38–46 (2013)CrossRefzbMATHGoogle Scholar
  8. 8.
    Golovach, P.A., Kamiński, M., Paulusma, D., Thilikos, D.M.: Increasing the minimum degree of a graph by contractions. Theor. Comp. Sci. 481, 74–84 (2013)CrossRefzbMATHGoogle Scholar
  9. 9.
    Heggernes, P., van ’t Hof, P., Lévêque, B., Lokshtanov, D., Paul, C.: Contracting graphs to paths and trees. Algorithmica (to appear) doi:10.1007/s00453-012-9670-2Google Scholar
  10. 10.
    Heggernes, P., van ’t Hof, P., Lokshtanov, D., Paul, C.: Obtaining a bipartite graph by contracting few edges. In: FSTTCS 2011, LIPIcs, vol. 13, pp. 217–228 (2011)Google Scholar
  11. 11.
    Lewis, J.M., Yannakakis, M.: The node-deletion problem for hereditary properties is NP-complete. J. Comp. System Sci. 20, 219–230 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Marx, D., O’Sullivan, B., Razgon, I.: Finding small separators in linear time via treewidth reduction. In: ACM Trans. Algorithms (to appear), Manuscript available at http://www.cs.bme.hu/~dmarx/papers/marx-tw-reduction-talg.pdf
  13. 13.
    Mathieson, L., Szeider, S.: Editing graphs to satisfy degree constraints: A parameterized approach. J. Comput. Syst. Sci. 78, 179–191 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Moser, H., Thilikos, D.M.: Parameterized complexity of finding regular induced subgraphs. J. Discr. Algorithms 7, 181–190 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Yannakakis, M.: Edge-deletion problems. SIAM J. Comput. 10(2), 297–309 (1981)MathSciNetCrossRefzbMATHGoogle Scholar

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

Personalised recommendations