Skip to main content
SpringerLink
Log in

An Algorithm for Listing All Minimal 2-Dominating Sets of a Tree

  • Conference paper
Frontiers in Algorithmics and Algorithmic Aspects in Information and Management

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7924))

  • 937 Accesses

Abstract

We provide an algorithm for listing all minimal 2-dominating sets of a tree of order n in time \(\mathcal{O}(1.3248^n)\). This implies that every tree has at most 1.3248n minimal 2-dominating sets. We also show that this bound is tight.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Björklund, A., Husfeldt, T.: Inclusion-exclusion algorithms for counting set partitions. In: Proceedings of FOCS, pp. 575–582 (2006)

    Google Scholar 

  2. Blidia, M., Favaron, O., Lounes, R.: Locating-domination, 2-domination and independence in trees. Australasian Journal of Combinatorics 42, 309–316 (2008)

    MathSciNet  MATH  Google Scholar 

  3. Bród, D., Skupień, Z.: Trees with extremal numbers of dominating sets. Australasian Journal of Combinatorics 35, 273–290 (2006)

    MathSciNet  MATH  Google Scholar 

  4. Bród, D., Włoch, A., Włoch, I.: On the number of minimal dominating sets including the set of leaves in trees. International Journal of Contemporary Mathematical Sciences 4, 1739–1748 (2009)

    MathSciNet  MATH  Google Scholar 

  5. Couturier, J.-F., Heggernes, P., van’t Hof, P., Kratsch, D.: Minimal dominating sets in graph classes: combinatorial bounds and enumeration. In: Bieliková, M., Friedrich, G., Gottlob, G., Katzenbeisser, S., Turán, G. (eds.) SOFSEM 2012. LNCS, vol. 7147, pp. 202–213. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  6. Eppstein, D.: Small maximal independent sets and faster exact graph coloring. Journal of Graph Algorithms and Applications 7, 131–140 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Fink, J., Jacobson, M.: n-domination in graphs. In: Graph Theory with Applications to Algorithms and Computer Science, pp. 282–300. Wiley, New York (1985)

    Google Scholar 

  8. Fomin, F., Grandoni, F., Pyatkin, A., Stepanov, A.: Combinatorial bounds via measure and conquer: bounding minimal dominating sets and applications. ACM Transactions on Algorithms 5, article 9, 17 (2009)

    Google Scholar 

  9. Fomin, F., Kratsch, D.: Exact Exponential Algorithms. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

  10. Fujisawa, J., Hansberg, A., Kubo, T., Saito, A., Sugita, M., Volkmann, L.: Independence and 2-domination in bipartite graphs. Australasian Journal of Combinatorics 40, 265–268 (2008)

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  13. Kanté, M., Limouzy, V., Mary, A., Nourine, L.: Enumeration of minimal dominating sets and variants. In: Owe, O., Steffen, M., Telle, J.A. (eds.) FCT 2011. LNCS, vol. 6914, pp. 298–309. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  14. Koivisto, M.: An \(\mathcal{O}(2^n)\) algorithm for graph coloring and other partitioning problems via inclusion-exclusion. In: Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2006), pp. 583–590. IEEE (2006)

    Google Scholar 

  15. Krzywkowski, M.: Trees having many minimal dominating sets. Information Processing Letters 113, 276–279 (2013)

    Article  MathSciNet  Google Scholar 

  16. Lawler, E.: A note on the complexity of the chromatic number problem. Information Processing Letters 5, 66–67 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  17. Moon, J., Moser, L.: On cliques in graphs. Israel Journal of Mathematics 3, 23–28 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  18. Shaheen, R.: Bounds for the 2-domination number of toroidal grid graphs. International Journal of Computer Mathematics 86, 584–588 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Wilf, H.: The number of maximal independent sets in a tree. SIAM Journal on Algebraic and Discrete Methods 7, 125–130 (1986)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Electronics, Telecommunications and Informatics, Gdańsk University of Technology, Poland

    Marcin Krzywkowski

Authors
  1. Marcin Krzywkowski
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. School of Engineering and Information Technology, Charles Darwin University, 0909, Darwin, NT, Australia

    Michael Fellows

  2. School of High-Technology for Human Welfare, Tokai University, 317 Nishimo, 410-0395, Numazu, Japan

    Xuehou Tan

  3. Department of Computer Science, Montana State University, 59717, Bozeman, MT, USA

    Binhai Zhu

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Krzywkowski, M. (2013). An Algorithm for Listing All Minimal 2-Dominating Sets of a Tree. In: Fellows, M., Tan, X., Zhu, B. (eds) Frontiers in Algorithmics and Algorithmic Aspects in Information and Management. Lecture Notes in Computer Science, vol 7924. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38756-2_4

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-38756-2_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-38755-5

  • Online ISBN: 978-3-642-38756-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation