Abstract
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.
This work is supported by EPSRC (EP/G020604/1 and EP/G043434/1).
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References
Brandstädt, A., Dragan, F.F., Le, V.B., Szymczak, T.: On stable cutsets in graphs. Discrete Appl. Math. 105, 39–50 (2000)
Bulatov, A.: Tractable conservative constraint satisfaction problems. In: Proceedings of LICS 2003, pp. 321–330 (2003)
Bulatov, A., Krokhin, A., Jeavons, P.G.: Classifying the complexity of constraints using finite algebras. SIAM Journal on Computing 34, 720–742 (2005)
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)
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)
Chvátal, V.: Recognizing decomposable graphs. J. Graph Theory 8, 51–53 (1984)
Dantas, S., de Figueiredo, C.M.H., Gravier, S., Klein, S.: Finding H-partitions efficiently. RAIRO-Theoret. Inf. Appl. 39, 133–144 (2005)
Dantas, S., Maffray, F., Silva, A.: 2K 2-partition of some classes of graphs. Discrete Applied Mathematics (to appear)
Feder, T., Hell, P.: List homomorphisms to reflexive graphs. J. Combin. Theory Ser. B 72, 236–250 (1998)
Feder, T., Hell, P., Klein, S., Motwani, R.: List partitions. SIAM J. Discrete Math. 16, 449–478 (2003)
de Figueiredo, C.M.H.: The P versus NP-complete dichotomy of some challenging problems in graph theory. Discrete Applied Mathematics (to appear)
Fleischner, H., Mujuni, E., Paulusma, D., Szeider, S.: Covering graphs with few complete bipartite subgraphs. Theoret. Comput. Sci. 410, 2045–2053 (2009)
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)
Hell, P., Nešetřil, J.: On the complexity of H-colouring. Journal of Combinatorial Theory, Series B 48, 92–110 (1990)
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)
Ito, T., Kaminski, M., Paulusma, D., Thilikos, D.M.: On disconnected cuts and separators. Discrete Applied Mathematics (to appear)
Martin, B., Paulusma, D.: The Computational Complexity of Disconnected Cut and 2K2-Partition, http://arxiv.org/abs/1104.4779
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)
Vikas, N.: Computational complexity of compaction to reflexive cycles. SIAM Journal on Computing 32, 253–280 (2002)
Vikas, N.: Compaction, Retraction, and Constraint Satisfaction. SIAM Journal on Computing 33, 761–782 (2004)
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)
Whitesides, S.H.: An algorithm for finding clique cut-sets. Inform. Process. Lett. 12, 31–32 (1981)
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Martin, B., Paulusma, D. (2011). The Computational Complexity of Disconnected Cut and 2K 2-Partition. In: Lee, J. (eds) Principles and Practice of Constraint Programming – CP 2011. CP 2011. Lecture Notes in Computer Science, vol 6876. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23786-7_43
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