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Independent Rainbow Domination of Graphs

  • Zehui Shao
  • Zepeng Li
  • Aljoša Peperko
  • Jiafu Wan
  • Janez ŽerovnikEmail author
Article

Abstract

Given a positive integer t and a graph F, the goal is to assign a subset of the color set \(\{1,2,\ldots ,t\}\) to every vertex of F such that every vertex with the empty set assigned has all t colors in its neighborhood. Such an assignment is called the t-rainbow dominating function (\(t\mathrm{RDF}\)) of the graph F. A \(t\mathrm{RDF}\) is independent (\(It\mathrm{RDF}\)) if vertices assigned with non-empty sets are pairwise non-adjacent. The weight of a \(t\mathrm{RDF}\)g of a graph F is the value \(w(g) =\sum _{v \in V(F)}|g(v)|\). The independent t-rainbow domination number \(i_{rt}(F)\) is the minimum weight over all \(It\mathrm{RDF}\)s of F. In this article, it is proved that the independent t-rainbow domination problem is NP-complete even if the input graph is restricted to a bipartite graph or a planar graph, and the results of the study provide some bounds for the independent t-rainbow domination number of any graph for a positive integer t. Moreover, the exact values and bounds of the independent t-rainbow domination numbers of some Petersen graphs and torus graphs are given.

Keywords

Rainbow domination Domination number Independent rainbow domination NP-complete 

Mathematics Subject Classification

05C69 05C15 05C76 05C85 

References

  1. 1.
    Brešar, B., Henning, M.A., Rall, D.F.: Paired-domination of Cartesian products of graphs and rainbow domination. Electron. Notes Discrete Math. 22, 233–237 (2005)CrossRefzbMATHGoogle Scholar
  2. 2.
    Brešar, B., Henning, M.A., Rall, D.F.: Rainbow domination in graphs. Taiwan. J. Math. 12(1), 213–225 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Brešar, B., Šumenjak, T.K.: On the 2-rainbow domination in graphs. Discrete Appl. Math. 155(1), 2394–2400 (2007)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Chang, G.J., Wu, J., Zhu, X.: Rainbow domination on trees. Discrete Appl. Math. 158, 8–12 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chen, W., Lu, Z., Wu, W.: Dominating problems in swapped networks. Inf. Sci. 269, 286–299 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ebrahimi, B.J., Jahanbakht, N., Mahmoodianc, E.S.: Vertex domination of generalized Petersen graphs. Discrete Math. 309, 4355–4361 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman and Co, San Francisco (1979)zbMATHGoogle Scholar
  8. 8.
    Goddard, W., Henning, M.A.: Independent domination in graphs: a survey and recent results. Discrete Math. 313(7), 839–854 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Fundamentals of Domination in Graphs. Marcel Dekker, New York (1998)zbMATHGoogle Scholar
  10. 10.
    Kelleher, L.L., Cozzens, M.B.: Dominating sets in social network graphs. Math. Soc. Sci. 16(3), 267–279 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Klavžar, S., Žerovnik, J.: Algebraic approach to fasciagraphs and rotagraphs. Discrete Appl. Math. 68, 93–100 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Šumenjak, T.K., Rall, D.F., Tepeh, A.: Rainbow domination in the lexicographic product of graphs. arXiv:1210.0514v2. 13 Mar 2013
  13. 13.
    Ore, O.: Theory of Graphs. American Mathematical Society, Providence (1967)zbMATHGoogle Scholar
  14. 14.
    Pang, C., Zhang, R., Zhang, Q., Wang, J.: Dominating sets in directed graphs. Inf. Sci. 180(19), 3647–3652 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Pavlič, P., Žerovnik, J.: Roman domination number of the Cartesian products of paths and cycles. Electron. J. Comb. 16, P19 (2012)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Pavlič, P., Žerovnik, J.: A note on the domination number of the cartesian products of paths and cycles. Kragujev. J. Math. 37, 275–285 (2013)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Pavlič, P., Žerovnik, J.: Formulas for various domination numbers of products of paths and cycles. Ars combinatoria, accepted for publication. preprint 1180 (2012). http://preprinti.imfm.si
  18. 18.
    Shao, Z., Zhu, E., Lang, F.: On the domination number of Cartesian product of two directed cycles. J. Appl. Math. (2013), Article ID 619695Google Scholar
  19. 19.
    Shao, Z., Liang, M., Yin, C., Xu, X., Pavlič, P., Žerovnik, J.: On rainbow domination numbers of graphs. Inf. Sci. 254, 225–234 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Steimle, A., Staton, W.: The isomorphism classes of the generalized Petersen graphs. Discrete Math. 309, 231–237 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Tong, C., Lin, X., Yang, Y., Luo, M.: 2-rainbow domination of generalized Petersen graphs \(P(n, 2)\). Discrete Appl. Math. 157, 1932–1937 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Tsai, Y., Lin, Y., Hsu, F.R.: Efficient algorithms for the minimum connected domination on trapezoid graphs. Inf. Sci. 177(12), 2405–2417 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Watkins, M.: A theorem on Tait colorings with an application to the generalized Petersen graph. J. Comb. Theory 6, 152–164 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Wu, L., Shan, E., Liu, Z.: On the \(k\)-tuple domination of generalized de Brujin and Kautz digraphs. Inf. Sci. 180(22), 4430–4435 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Xu, G.: 2-rainbow domination in generalized Petersen graphs \(P(n, 3)\). Discrete Appl. Math. 157, 2570–2573 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Yen, W.: The bottleneck independent domination on the classes of bipartite graphs and block graphs. Inf. Sci. 157, 199–215 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Žerovnik, J.: Deriving formulas for domination numbers of fasciagraphs and rotagraphs. Lect. Notes Comput. Sci. 1684, 559–568 (1999)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2017

Authors and Affiliations

  1. 1.School of Computer Science and Educational SoftwareGuangzhou UniversityGuangzhouChina
  2. 2.School of Information Science and EngineeringChengdu UniversityChengduChina
  3. 3.School of Electronic Engineering and Computer SciencePeking UniversityBeijingChina
  4. 4.FME, University of LjubljanaLjubljanaSlovenia
  5. 5.Institute of Mathematics, Physics and MechanicsLjubljanaSlovenia
  6. 6.School of Mechanical and Automotive EngineeringSouth China University of TechnologyGuangzhouChina

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