The Computational Complexity of Disconnected Cut and 2K2-Partition

  • Barnaby Martin
  • Daniël Paulusma
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6876)


For a connected graph G = (V,E), a subset U ⊆ V is called a disconnected cut if U disconnects the graph and the subgraph induced by U is disconnected as well. We show that the problem to test whether a graph has a disconnected cut is NP-complete. This problem is polynomially equivalent to the following problems: testing if a graph has a 2K 2-partition, testing if a graph allows a vertex-surjective homomorphism to the reflexive 4-cycle and testing if a graph has a spanning subgraph that consists of at most two bicliques. Hence, as an immediate consequence, these three decision problems are NP-complete as well. This settles an open problem frequently posed in each of the four settings.


Computational Complexity Constraint Satisfaction Input Graph Surjective Homomorphism Span Subgraph 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Brandstädt, A., Dragan, F.F., Le, V.B., Szymczak, T.: On stable cutsets in graphs. Discrete Appl. Math. 105, 39–50 (2000)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bulatov, A.: Tractable conservative constraint satisfaction problems. In: Proceedings of LICS 2003, pp. 321–330 (2003)Google Scholar
  3. 3.
    Bulatov, A., Krokhin, A., Jeavons, P.G.: Classifying the complexity of constraints using finite algebras. SIAM Journal on Computing 34, 720–742 (2005)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Cameron, K., Eschen, E.M., Hoáng, C.T., Sritharan, R.: The complexity of the list partition problem for graphs. SIAM J. Discrete Math. 21, 900–929 (2007)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Cook, K., Dantas, S., Eschen, E.M., Faria, L., de Figueiredo, C.M.H., Klein, S.: 2K 2 vertex-set partition into nonempty parts. Discrete Math. 310, 1259–1264 (2010)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Chvátal, V.: Recognizing decomposable graphs. J. Graph Theory 8, 51–53 (1984)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Dantas, S., de Figueiredo, C.M.H., Gravier, S., Klein, S.: Finding H-partitions efficiently. RAIRO-Theoret. Inf. Appl. 39, 133–144 (2005)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Dantas, S., Maffray, F., Silva, A.: 2K 2-partition of some classes of graphs. Discrete Applied Mathematics (to appear)Google Scholar
  9. 9.
    Feder, T., Hell, P.: List homomorphisms to reflexive graphs. J. Combin. Theory Ser. B 72, 236–250 (1998)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Feder, T., Hell, P., Klein, S., Motwani, R.: List partitions. SIAM J. Discrete Math. 16, 449–478 (2003)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    de Figueiredo, C.M.H.: The P versus NP-complete dichotomy of some challenging problems in graph theory. Discrete Applied Mathematics (to appear)Google Scholar
  12. 12.
    Fleischner, H., Mujuni, E., Paulusma, D., Szeider, S.: Covering graphs with few complete bipartite subgraphs. Theoret. Comput. Sci. 410, 2045–2053 (2009)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Golovach, P.A., Paulusma, D., Song, J.: Computing vertex-surjective homomorphisms to partially reflexive trees. In: Kulikov, A., Vereshchagin, N. (eds.) CSR 2011. LNCS, vol. 6651, pp. 261–274. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  14. 14.
    Hell, P., Nešetřil, J.: On the complexity of H-colouring. Journal of Combinatorial Theory, Series B 48, 92–110 (1990)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Ito, T., Kamiński, M., Paulusma, D., Thilikos, D.M.: Parameterizing Cut Sets in a Graph by the Number of Their Components. In: Dong, Y., Du, D.-Z., Ibarra, O. (eds.) ISAAC 2009. LNCS, vol. 5878, pp. 605–615. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  16. 16.
    Ito, T., Kaminski, M., Paulusma, D., Thilikos, D.M.: On disconnected cuts and separators. Discrete Applied Mathematics (to appear)Google Scholar
  17. 17.
    Martin, B., Paulusma, D.: The Computational Complexity of Disconnected Cut and 2K2-Partition,
  18. 18.
    Teixeira, R.B., Dantas, S., de Figueiredo, C.M.H.: The external constraint 4 nonempty part sandwich problem. Discrete Applied Mathematics 159, 661–673 (2011)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Vikas, N.: Computational complexity of compaction to reflexive cycles. SIAM Journal on Computing 32, 253–280 (2002)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Vikas, N.: Compaction, Retraction, and Constraint Satisfaction. SIAM Journal on Computing 33, 761–782 (2004)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Vikas, N.: A complete and equal computational complexity classification of compaction and retraction to all graphs with at most four vertices and some general results. J. Comput. Syst. Sci. 71, 406–439 (2005)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Whitesides, S.H.: An algorithm for finding clique cut-sets. Inform. Process. Lett. 12, 31–32 (1981)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Barnaby Martin
    • 1
  • Daniël Paulusma
    • 1
  1. 1.School of Engineering and Computing SciencesDurham University, Science LaboratoriesDurhamUnited Kingdom

Personalised recommendations