On the Parameterized Complexity of Cutting a Few Vertices from a Graph

  • Fedor V. Fomin
  • Petr A. Golovach
  • Janne H. Korhonen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8087)


We study the parameterized complexity of separating a small set of vertices from a graph by a small vertex-separator. That is, given a graph G and integers k, t, the task is to find a vertex set X with |X| ≤ k and |N(X)| ≤ t. We show that
  • the problem is fixed-parameter tractable (FPT) when parameterized by t but W[1]-hard when parameterized by k, and

  • a terminal variant of the problem, where X must contain a given vertex s, is W[1]-hard when parameterized either by k or by t alone, but is FPT when parameterized by k + t.

We also show that if we consider edge cuts instead of vertex cuts, the terminal variant is NP-hard.


Polynomial Time Parameterized Complexity Polynomial Kernel Terminal Vertex Vertex Input 


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  1. 1.
    Alon, N., Yuster, R., Zwick, U.: Color-coding. Journal of the ACM 42(4), 844–856 (1995)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Armon, A., Zwick, U.: Multicriteria global minimum cuts. Algorithmica 46(1), 15–26 (2006)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bodlaender, H.L., Downey, R.G., Fellows, M.R., Hermelin, D.: On problems without polynomial kernels. Journal of Computer and System Sciences 75(8), 423–434 (2009)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Cai, L., Chan, S.M., Chan, S.O.: Random separation: A new method for solving fixed-cardinality optimization problems. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 239–250. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  5. 5.
    Cao, Y.: A note on small cuts for a terminal (2013), arXiv:1306.2578 [cs.DS]Google Scholar
  6. 6.
    Chen, J., Liu, Y., Lu, S.: An improved parameterized algorithm for the minimum node multiway cut problem. Algorithmica 55(1), 1–13 (2009)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Diestel, R.: Graph theory, 4th edn. Springer (2010)Google Scholar
  8. 8.
    Downey, R.G., Fellows, M.R.: Parameterized complexity. Springer (1999)Google Scholar
  9. 9.
    Feige, U., Krauthgamer, R., Nissim, K.: On cutting a few vertices from a graph. Discrete Applied Mathematics 127(3), 643–649 (2003)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Flum, J., Grohe, M.: Parameterized complexity theory. Springer (2006)Google Scholar
  11. 11.
    Galbiati, G.: Approximating minimum cut with bounded size. In: Pahl, J., Reiners, T., Voß, S. (eds.) INOC 2011. LNCS, vol. 6701, pp. 210–215. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  12. 12.
    Garey, M.R., Johnson, D.S.: Computers and intractability: a guide to the theory of NP-completeness. W. H. Freeman and Co., San Francisco (1979)MATHGoogle Scholar
  13. 13.
    Hayrapetyan, A., Kempe, D., Pál, M., Svitkina, Z.: Unbalanced graph cuts. In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 191–202. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  14. 14.
    Li, A., Zhang, P.: Unbalanced graph partitioning. Theory of Computing Systems 53(3), 454–466 (2013)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Lokshtanov, D., Marx, D.: Clustering with local restrictions. Information and Computation 222, 278–292 (2013)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Marx, D.: Parameterized graph separation problems. Theoretical Computer Science 351(3), 394–406 (2006)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Naor, M., Schulman, L., Srinivasan, A.: Splitters and near-optimal derandomization. In: 36th Annual Symposium on Foundations of Computer Science (FOCS 1995), pp. 182–191. IEEE (1995)Google Scholar
  18. 18.
    Niedermeier, R.: Invitation to fixed-parameter algorithms. Oxford University Press (2006)Google Scholar
  19. 19.
    Watanabe, T., Nakamura, A.: Edge-connectivity augmentation problems. Journal of Computer and System Sciences 35(1), 96–144 (1987)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Fedor V. Fomin
    • 1
  • Petr A. Golovach
    • 1
  • Janne H. Korhonen
    • 2
  1. 1.Department of InformaticsUniversity of BergenNorway
  2. 2.Helsinki Institute for Information Technology HIIT, & Department of Computer ScienceUniversity of HelsinkiFinland

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