The Complexity of the Matching-Cut Problem

  • Maurizio Patrignani
  • Maurizio Pizzonia
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2204)


Finding a cut or finding a matching in a graph are so simple problems that they are hardly considered problems at all. In this paper, by means of a reduction from the NAE3SAT problem, we prove that combining these two problems together, i.e., finding a cut whose split edges are a matching is an NP-complete problem. It remains intractable even if we impose the graph to be simple (no multiple edges allowed) or its maximum degree to be k, with k ≽ 4. On the contrary, we give a linear time algorithm that computes a matching-cut of a series-parallel graph. It’s open whether the problem is tractable or not for planar graphs.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    V. Chvátal. Recogniziong decomposable graphs. J. Graph Theory, 8:51–53, 1984.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    G. Di Battista, P. Eades, R. Tamassia, and I. G. Tollis. Graph Drawing. Prentice Hall, Upper Saddle River, NJ, 1999.zbMATHCrossRefGoogle Scholar
  3. 3.
    G. Di Battista, M. Patrignani, and F. Vargiu. A Split&Push approach to 3D orthogonal drawing. Journ. Graph Alg. Appl., 4:105–133, 2000.zbMATHGoogle Scholar
  4. 4.
    M. R. Garey and D. S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, New York, NY, 1979.zbMATHGoogle Scholar
  5. 5.
    M. X. Goemans and D. P. Williamson. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM, 42:1115–1145, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    M. Patrignani and F. Vargiu. 3DCube: A tool for three dimensional graph drawing. In G. Di Battista, editor, Graph Drawing (Proc. GD’ 97) volume 1353 of Lecture Notes Comput. Sci., pages 284–290. Springer-Verlag, 1997.Google Scholar
  7. 7.
    T. J. Schaefer. The complexity of satisfiability problems. In Proc. 10th Annu. ACM Sympos. Theory Comput., pages 216–226, 1978.Google Scholar
  8. 8.
    J. Valdes, R. E. Tarjan, and E. L. Lawler. The recognition of series-parallel digraphs. SIAM J. Comput., 11(2):298–313, 1982.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Maurizio Patrignani
    • 1
  • Maurizio Pizzonia
    • 1
  1. 1.Dipartimento di Informatica e AutomazioneUniversità di Roma TreRomaItaly

Personalised recommendations