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Dominating Set Counting in Graph Classes

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Computing and Combinatorics (COCOON 2011)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6842))

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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.

The first and second authors are supported by Grant-in-Aid for Scientific Research from Ministry of Education, Science and Culture, Japan, and Japan Society for the Promotion of Science.

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References

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. Haynes, T.W., Hedetniemi, S.T., Slater, P.J. (eds.): Fundamentals of Domination in Graphs. Marcel Dekker, New York (1998)

    MATH  Google Scholar 

  4. Haynes, T.W., Hedetniemi, S.T., Slater, P.J. (eds.): Domination in Graphs: Advanced Topics. Marcel Dekker, New York (1998)

    MATH  Google Scholar 

  5. Kratsch, D.: Personal communication (April 2011)

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  8. Okamoto, Y., Uno, T., Uehara, R.: Counting the number of independent sets in chordal graphs. Journal of Discrete Algorithms 6, 229–242 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Graduate School of Information Science and Electrical Engineering, Kyushu University, Japan

    Shuji Kijima

  2. Center for Graduate Education Initiative, Japan Advanced Institute of Science and Technology, Japan

    Yoshio Okamoto

  3. National Institute of Informatics, Japan

    Takeaki Uno

Authors
  1. Shuji Kijima
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  2. Yoshio Okamoto
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  3. Takeaki Uno
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Editor information

Editors and Affiliations

  1. Department of Computer Science, University of Texas-Pan American, 78539, Edinburg, TX, USA

    Bin Fu

  2. Department of Computer Science, University of Texas at Dallas, 75080, Richardson, TX, USA

    Ding-Zhu Du

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© 2011 Springer-Verlag Berlin Heidelberg

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Kijima, S., Okamoto, Y., Uno, T. (2011). Dominating Set Counting in Graph Classes. In: Fu, B., Du, DZ. (eds) Computing and Combinatorics. COCOON 2011. Lecture Notes in Computer Science, vol 6842. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22685-4_2

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  • DOI: https://doi.org/10.1007/978-3-642-22685-4_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-22684-7

  • Online ISBN: 978-3-642-22685-4

  • eBook Packages: Computer ScienceComputer Science (R0)

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