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Dominating Set Counting in Graph Classes

  • Shuji Kijima
  • Yoshio Okamoto
  • Takeaki Uno
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6842)

Abstract

We make an attempt to understand the dominating set counting problem in graph classes from the viewpoint of polynomial-time computability. We give polynomial-time algorithms to count the number of dominating sets (and minimum dominating sets) in interval graphs and trapezoid graphs. They are based on dynamic programming. With the help of dynamic update on a binary tree, we further reduce the time complexity. On the other hand, we prove that counting the number of dominating sets (and minimum dominating sets) in split graphs and chordal bipartite graphs is #P-complete. These results are in vivid contrast with the recent results on counting the independent sets and the matchings in chordal graphs and chordal bipartite graphs.

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References

  1. 1.
    Booth, K.S., Lueker, G.S.: Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms. Journal of Computer and System Sciences 13, 335–379 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Brouwer, A.E., Csorba, P., Schrijver, A.: The number of dominating sets of a finite graph is odd. Preprint (2009), http://www.win.tue.nl/~aeb/preprints/domin4a.pdf
  3. 3.
    Haynes, T.W., Hedetniemi, S.T., Slater, P.J. (eds.): Fundamentals of Domination in Graphs. Marcel Dekker, New York (1998)zbMATHGoogle Scholar
  4. 4.
    Haynes, T.W., Hedetniemi, S.T., Slater, P.J. (eds.): Domination in Graphs: Advanced Topics. Marcel Dekker, New York (1998)zbMATHGoogle Scholar
  5. 5.
    Kratsch, D.: Personal communication (April 2011)Google Scholar
  6. 6.
    Lin, M.-S., Chen, Y.-J.: Linear time algorithms for counting the number of minimal vertex covers with minimum/maximum size in an interval graph. Information Processing Letters 107, 257–264 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Okamoto, Y., Uehara, R., Uno, T.: Counting the number of matchings in chordal and chordal bipartite graph classes. In: Paul, C., Habib, M. (eds.) WG 2009. LNCS, vol. 5911, pp. 296–307. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  8. 8.
    Okamoto, Y., Uno, T., Uehara, R.: Counting the number of independent sets in chordal graphs. Journal of Discrete Algorithms 6, 229–242 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Provan, J.S., Ball, M.O.: The complexity of counting cuts and of computing the probability that a graph is connected. SIAM Journal on Computing 12, 777–788 (1983)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Shuji Kijima
    • 1
  • Yoshio Okamoto
    • 2
  • Takeaki Uno
    • 3
  1. 1.Graduate School of Information Science and Electrical EngineeringKyushu UniversityJapan
  2. 2.Center for Graduate Education InitiativeJapan Advanced Institute of Science and TechnologyJapan
  3. 3.National Institute of InformaticsJapan

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