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Exponential Time Algorithms for the Minimum Dominating Set Problem on Some Graph Classes

  • Serge Gaspers
  • Dieter Kratsch
  • Mathieu Liedloff
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4059)

Abstract

The Minimum Dominating Set problem remains NP-hard when restricted to chordal graphs, circle graphs and c-dense graphs (i.e. |E| ≥cn 2 for a constant c, 0<c<1/2). For each of these three graph classes we present an exponential time algorithm solving the Minimum Dominating Set problem. The running times of those algorithms are O(1.4173 n ) for chordal graphs, O(1.4956 n ) for circle graphs, and \(O(1.2303^{(1+\sqrt{1-2c})n})\) for c-dense graphs.

Keywords

Input Graph Tree Decomposition Domination Number Chordal Graph Graph Class 
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.

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References

  1. 1.
    Alber, J., Bodlaender, H.L., Fernau, H., Kloks, T., Niedermeier, R.: Fixed parameter algorithms for dominating set and related problems on planar graphs. Algorithmica 33, 461–493 (2002)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bertossi, A.A.: Dominating sets for split and bipartite graphs. Inform. Process. Lett. 19, 37–40 (1984)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Blair, J.R.S., Peyton, B.W.: An introduction to chordal graphs and clique trees. In: Graph theory and sparse matrix computation. IMA, Math. Appl., vol. 56, pp. 1–29. Springer, Heidelberg (1993)Google Scholar
  4. 4.
    Bodlaender, H.L., Thilikos, D.M.: Graphs with branchwidth at most three. J. Algorithms 32, 167–194 (1999)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Booth, K.S., Johnson, J.H.: Dominating sets in chordal graphs. SIAM J. Comput. 11, 191–199 (1982)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Brandstädt, A., Le, V., Spinrad, J.P.: Graph classes: A survey. SIAM Monogr. Discrete Math. Appl., Philadelphia (1999)MATHCrossRefGoogle Scholar
  7. 7.
    Fomin, F.V., Høie, K.: Pathwidth of cubic graphs and exact algorithms, Technical Report 298, Department of Informatics, University of Bergen, Norway (2005)Google Scholar
  8. 8.
    Fomin, F.V., Grandoni, F., Kratsch, D.: Measure and Conquer: Domination – A Case Study. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 191–203. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  9. 9.
    Fomin, F.V., Grandoni, F., Kratsch, D.: Some new techniques in design and analysis of exact (exponential) algorithms. Bull. EATCS 87, 47–77 (2005)MATHMathSciNetGoogle Scholar
  10. 10.
    Fomin, F.V., Kratsch, D., Woeginger, G.J.: Exact (exponential) algorithms for the dominating set problem. In: Hromkovič, J., Nagl, M., Westfechtel, B. (eds.) WG 2004. LNCS, vol. 3353, pp. 245–256. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  11. 11.
    Golumbic, M.C.: Algorithmic graph theory and perfect graphs. Academic Press, New York (1980)MATHGoogle Scholar
  12. 12.
    Grandoni, F.: A note on the complexity of minimum dominating set. J. Discrete Algorithms (to appear)Google Scholar
  13. 13.
    Keil, J.M.: The complexity of domination problems in circle graphs. Discrete Appl. Math. 42, 51–63 (1993)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Kloks, T.: Treewidth. Computations and approximation. LNCS, vol. 842. Springer, Heidelberg (1994)MATHCrossRefGoogle Scholar
  15. 15.
    Kloks, T.: Treewidth of Circle Graphs. Internat. J. Found. Comput. Sci. 7, 111–120 (1996)MATHCrossRefGoogle Scholar
  16. 16.
    Randerath, B., Schiermeyer, I.: Exact algorithms for Minimum Dominating Set, Technical Report zaik-469, Zentrum fur Angewandte Informatik, Köln, Germany (2004)Google Scholar
  17. 17.
    Robertson, N., Seymour, P.D.: Graph Minors. II. Algorithmic Aspects of Tree-Width. J. Algorithms 7, 309–322 (1986)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Schiermeyer, I.: Problems remaining NP-complete for sparse or dense graphs. Discuss. Math. Graph Theory 15, 33–41 (1995)MATHMathSciNetGoogle Scholar
  19. 19.
    Woeginger, G.J.: Exact algorithms for NP-hard problems: A survey. In: Jünger, M., Reinelt, G., Rinaldi, G. (eds.) Combinatorial Optimization - Eureka, You Shrink! LNCS, vol. 2570, pp. 185–207. Springer, Heidelberg (2003)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Serge Gaspers
    • 1
  • Dieter Kratsch
    • 2
  • Mathieu Liedloff
    • 2
  1. 1.Department of InformaticsUniversity of BergenBergenNorway
  2. 2.LITA, Université de Paul Verlaine – MetzMetzFrance

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