Advertisement

The Algorithmic Complexity of k-Domatic Partition of Graphs

  • Hongyu Liang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7604)

Abstract

Let G = (V,E) be a simple undirected graph, and k be a positive integer. A k-dominating set of G is a set of vertices S ⊆ V satisfying that every vertex in V ∖ S is adjacent to at least k vertices in S. A k-domatic partition of G is a partition of V into k-dominating sets. The k-domatic number of G is the maximum number of k-dominating sets contained in a k-domatic partition of G. In this paper we study the k-domatic number from both algorithmic complexity and graph theoretic points of view. We prove that it is \(\mathcal{NP}\)-complete to decide whether the k-domatic number of a bipartite graph is at least 3, and present a polynomial time algorithm that approximates the k-domatic number of a graph of order n within a factor of \((\frac{1}{k}+o(1))\ln n\), generalizing the (1 + o(1))ln n approximation for the 1-domatic number given in [5]. In addition, we determine the exact values of the k-domatic number of some particular classes of graphs.

Keywords

Sensor Network Bipartite Graph Polynomial Time Algorithm Algorithmic Complexity Domination Number 
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.

References

  1. 1.
    Caro, Y., Roditty, Y.: A note on the k-domination number of a graph. Internat. J. Math. Math. Sci. 13, 205–206 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Cockayne, E.J., Hedetniemi, S.T.: Towards a theory of domination in graphs. Networks 7, 247–261 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Diestel, R.: Graph Theory, 4th edn. Springer (2010)Google Scholar
  4. 4.
    Favaron, O., Hansberg, A., Volkmann, L.: On k-domination and minimum degree in graphs. J. Graph Theory 57, 33–40 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Feige, U., Halldórsson, M.M., Kortsarz, G., Srinivasan, A.: Approximating the domatic number. SIAM J. Comput. 32(1), 172–195 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Fink, J.F., Jacobson, M.S.: n-domination in graphs. In: Graph Theory with Applications to Algorithms and Computer Science, pp. 282–300 (1985)Google Scholar
  7. 7.
    Fink, J.F., Jacobson, M.S.: On n-domination, n-dependence and forbidden subgraphs. In: Graph Theory with Applications to Algorithms and Computer Science, pp. 301–311 (1985)Google Scholar
  8. 8.
    Fujisawa, J., Hansberg, A., Kubo, T., Saito, A., Sugita, M., Volkmann, L.: Independence and 2-domination in bipartite graphs. Australas. J. Combin. 40, 265–268 (2008)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Fujita, S., Yamashita, M., Kameda, T.: A study on r-configurations – a resource assignment problem on graphs. SIAM J. Discrete Math. 13, 227–254 (2000)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman (1979)Google Scholar
  11. 11.
    Hansberg, A., Meierling, D., Volkmann, L.: Distance domination and distance irredundance in graphs. Electron. J. Comb. 14 (2007)Google Scholar
  12. 12.
    Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Domination in Graphs: Advanced Topics. Marcel Dekker (1998)Google Scholar
  13. 13.
    Haynes, T.W., Hedetniemi, S.T.: ST, and P.J. Slater. Fundamentals of Domination in Graphs. Marcel Dekker (1998)Google Scholar
  14. 14.
    Islam, K., Akl, S.G., Meijer, H.: Maximizing the lifetime of wireless sensor networks through domatic partition. In: Proceedings of the 34th IEEE Conference on Local Computer Networks, LCN (2009)Google Scholar
  15. 15.
    Kämmerling, K., Volkmann, L.: The k-domatic number of a graph. Czech. Math. J. 59(2), 539–550 (2009)zbMATHCrossRefGoogle Scholar
  16. 16.
    Meir, A., Moon, J.W.: Relations between packing and covering number of a tree. Pacific J. Math. 61, 225–233 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Misra, R., Mandal, C.: Efficient clusterhead rotation via domatic partition in self-organizing sensor networks. Wireless Communications & Mobile Computing 9(8), 1040–1058 (2009)CrossRefGoogle Scholar
  18. 18.
    Pemmaraju, S.V., Pirwani, I.A.: Energy conservation via domatic partitions. In: Proceedings of the 7th ACM International Symposium on Mobile Ad Hoc Networking and Computing, MobiHoc (2006)Google Scholar
  19. 19.
    Pepper, R.: Implications of some observations about the k-domination number. Congr. Numer. 206, 65–71 (2010)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Rautenbach, D., Volkmann, L.: New bounds on the k-domination number and the k-tuple domination number. Appl. Math. Lett. 20, 98–102 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Telle, J.A.: Complexity of domination-type problems in graphs. Nord. J. Comput. 1(1), 157–171 (1994)MathSciNetGoogle Scholar
  22. 22.
    Zelinka, B.: On k-ply domatic numbers of graphs. Math. Slovaca 34(3), 313–318 (1984)MathSciNetzbMATHGoogle Scholar

Copyright information

© IFIP International Federation for Information Processing 2012

Authors and Affiliations

  • Hongyu Liang
    • 1
  1. 1.Institute for Interdisciplinary Information SciencesTsinghua UniversityBeijingChina

Personalised recommendations