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Abstract

It is known to be NP-hard to decide whether a graph can be made chordal by the deletion of k vertices. Here we present a uniformly polynomial-time algorithm for the problem: the running time is f(k) Open image in new window n α for some constant α not depending on k and some f depending only on k. For large values of n, such an algorithm is much better than trying all the O(n k ) possibilities. Therefore, the chordal deletion problem parameterized by the number k of vertices to be deleted is fixed-parameter tractable. This answers an open question of Cai [2].

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References

  1. 1.
    Cai, L.: Fixed-parameter tractability of graph modification problems for hereditary properties. Inform. Process. Lett. 58(4), 171–176 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Cai, L.: Parameterized complexity of vertex colouring. Discrete Appl. Math. 127, 415–429 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Downey, R.G., Fellows, M.R.: Parameterized complexity. Springer, Heidelberg (1999)Google Scholar
  4. 4.
    Golumbic, M.C.: Algorithmic graph theory and perfect graphs. Academic Press, New York (1980)zbMATHGoogle Scholar
  5. 5.
    Kaplan, H., Shamir, R., Tarjan, R.E.: Tractability of parameterized completion problems on chordal, strongly chordal, and proper interval graphs. SIAM J. Comput. 28(5), 1906–1922 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Lewis, J.M., Yannakakis, M.: The node-deletion problem for hereditary properties is NP-complete. J. Comput. System Sci. 20(2), 219–230 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Natanzon, A., Shamir, R., Sharan, R.: Complexity classification of some edge modification problems. Discrete Appl. Math. 113(1), 109–128 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Reed, B., Smith, K., Vetta, A.: Finding odd cycle transversals. Operations Research Letters 32(4), 299–301 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Rose, D.J., Tarjan, R.E., Lueker, G.S.: Algorithmic aspects of vertex elimination on graphs. SIAM J. Comput. 5(2), 266–283 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Yannakakis, M.: Computing the minimum fill-in is NP-complete. SIAM J. Algebraic Discrete Methods 2(1), 77–79 (1981)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Dániel Marx
    • 1
  1. 1.Institut für InformatikHumboldt-Universität zu BerlinBerlinGermany

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