Theory of Computing Systems

, Volume 61, Issue 3, pp 893–906 | Cite as

The 2-Rainbow Domination of Sierpiński Graphs and Extended Sierpiński Graphs

Article

Abstract

Let G(V, E) be a connected and undirected graph with n-vertex-set V and m-edge-set E. For each vV, let N(v) = {u|vV and(u, v) ∈ E}. For a positive integer k, a k-rainbow dominating function of a graph G is a function f from V(G) to a k-bit Boolean string f(v) = fk(v)fk − 1(v) … f1(v), i.e., fi(v) ∈ {0, 1}, 1 ≤ ik, such that for any vertex v with f(v) = 0(k) we have ⋈uN(v)f(u) = 1(k), for all vV, where ⋈uSf(u) denotes the result of taking bitwise OR operation on f(u), for all uS. The weight of f is defined as \(w(f) = {\sum }_{v\in V}{\sum }^{k}_{i=1} f_{i}(v)\). The k-rainbow domination number γkr(G) is the minimum weight of a k-rainbow dominating function over all k-rainbow dominating functions of G. The 1-rainbow domination is the same as the ordinary domination. The k-rainbow domination problem is to determine the k-rainbow domination number of a graph G. In this paper, we determine γ2r(S(n, m)), γ2r(S+(n, m)), and γ2r(S++(n, m)), where S(n, m), S+(n, m), and S++(n, m) are Sierpiński graphs and extended Sierpiński graphs.

Keywords

k-rainbow domination function Dominating set Sierpiński graphs Extended Sierpiński graphs 

References

  1. 1.
    Ali, M., Rahim, M.T., Zeb, M., Ali, G.: On 2-rainbow domination of some families of graphs. International Journal of Mathematics and Soft Computing 1, 47–53 (2011)Google Scholar
  2. 2.
    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)CrossRefMATHGoogle Scholar
  3. 3.
    Brešar, B., Kraner Šumenjak, T.: On the 2-rainbow domination in graphs. Discret. Appl. Math. 155, 2394–2400 (2007)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Brešar, B., Henning, M.A., Rall, D.F.: Rainbow Domination in Graphs. Taiwan. J. Math. 12(1), 213–225 (2008)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Chang, G.J., Wu, J., Zhu, X.: Rainbow Domination on Trees. Discret. Appl. Math. 158, 8–12 (2010)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Chang, S.C., Liu, J.J., Wang, Y.L.: The Outer-connected Domination Number of Sierpiński-like Graphs. Theory of Computing Systems 58, 345–356 (2016)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Chen, G.H., Duh, D.R.: Topological properties, communication, and computation on WK-recursive networks. Networks 24, 303–317 (1994)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Duh, D.R., Chen, G.H.: Topological properties of WK-recursive networks. J. Parallel Distrib. Comput. 23, 468–474 (1994)CrossRefGoogle Scholar
  9. 9.
    Fujita, S., Furuya, M., Magnant, C.: K-Rainbow domatic numbers. Discret. Appl. Math. 160, 1104–1113 (2012)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Fundamentals of Domination in Graphs. Marcel Dekker, Inc., New York (1998)MATHGoogle Scholar
  11. 11.
    Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Domination in Graphs: Advanced Topics. Marcel Dekker, Inc., New York (1998)MATHGoogle Scholar
  12. 12.
    Hinz, A.M., Schief, A.: The average distance on the Sierpiński gasket. Probab. Theory Relat. Fields 87, 129–138 (1990)CrossRefMATHGoogle Scholar
  13. 13.
    Hinz, A.M.: Pascal’s triangle and the Tower of Hanoi. Am. Math. Mon. 99, 538–544 (1992)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Hinz, A.M., Klavžar, S., Milutinović, U., Parisse, D., Petr, C.: Metric properties of the Tower of Hanoi graphs and Stern’s diatomic sequence. Eur. J. Comb. 26, 693–708 (2005)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Hinz, A.M., Parisse, D.: The Average Eccentricity of Sierpiński Graphs. Graphs and Combinatorics 28, 671–686 (2012)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Hinz, A.M., Klavžar, S., Milutinović, U., Petr, C.: The Tower of Hanoi-Myths and Maths., Birkhuser/Springer Basel AG, Basel (2013)Google Scholar
  17. 17.
    Jakovac, M., Klavžar, S.: Vertex-, edge- and total-colorings of Sierpiński-like graphs. Discret. Math. 309, 1548–1556 (2009)CrossRefMATHGoogle Scholar
  18. 18.
    Kaimanovich, V.A.: Random walks on Sierpiński graphs: hyperbolicity and stochastic homogenization. In: Grabner, P., Birkhaüser, W.W. (eds.) Fractals in Graz 2001, pp 145–183 (2003)CrossRefGoogle Scholar
  19. 19.
    Klavžar, S., Milutinović, U.: Graphs S(n, k) and a variant of the Tower of Hanoi problem. Czechoslov. Math. J. 47, 95–104 (1997)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Klavžar, S., Milutinović, U., Petr, C.: 1-perfect codes in Sierpiński graphs. Bull. Aust. Math. Soc. 66, 369–384 (2002)CrossRefMATHGoogle Scholar
  21. 21.
    Klavžar, S., Mohar, B.: Crossing numbers of Sierpiński-like graphs. J. Graph Theory 50, 186–198 (2005)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Klavžar, S.: Coloring Sierpiński graphs and Sierpiński gasket graphs. Taiwan. J. Math. 12, 513–522 (2008)CrossRefMATHGoogle Scholar
  23. 23.
    Klix, F., Rautenstrauch-Goede, K.: Struktur-und Komponentenanalyse von Problemlösungsprozessen. Zeitschrift für Psychologie 174, 167–193 (1967)Google Scholar
  24. 24.
    Lin, C.H., Liu, J.J., Wang, Y.L., Yen, W.C.K.: The hub number of Sierpiński-like graphs. Theory of Computing Systems 49(3), 588–600 (2011)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Lin, C.H., Liu, J.J., Wang, Y.L.: Global strong defensive alliances of Sierpiński-like graphs. Theory of Computing Systems 53(3), 365–385 (2013)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Liu, T.W., Pai, K.J., Wu, R.Y.: Upper bounds on 2 and 3-rainbow domination number of Sierpiński graphs, The 31st Workshop on Combinatorial Mathematics and Computation Theory 134–138Google Scholar
  27. 27.
    Meierling, D., Sheikholeslami, S.M., Volkmann, L.: Nordhaus-Gaddum Bounds on the k-Rainbow Domatic Number of a Graph. Appl. Math. Lett. 24, 1758–1761 (2011)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Parisse, D.: On some metric properties of the Sierpiński graphs S(n, k). Ars Combinatoria 90, 145–160 (2009)MathSciNetMATHGoogle Scholar
  29. 29.
    Romik, D.: Shortest paths in the Tower of Hanoi graph and finite automata. SIAM J. Discret. Math. 20, 610–622 (2006)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Kraner Šumenjak, T., Rall, D.F., Tepeh, A.: Rainbow domination in the lexicographic product of graphs. Discret. Appl. Math. 161, 2133–2141 (2013)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Sydow, H.: Zur metrischen Erfasung von subjektiven Problemzuständen und zu deren Veränderung im Denkprozes. Zeitschrift für Psychologie 177, 145–198 (1970)Google Scholar
  32. 32.
    Teguia, A.M., Godbole, A.P.: Sierpiński gasket graphs and some of their properties. Aust. J. Commun. 35, 181–192 (2006)MATHGoogle Scholar
  33. 33.
    Tong, C.L., Lin, X.H., Yang, Y.S., Lou, M.Q.: 2-rainbow domination of generalized Petersen graphs P(n,2). Discret. Appl. Math. 157, 1932–1937 (2009)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Vecchia, G.D., Sanges, C.: A recursively scalable network VLSI implementation. Futur. Gener. Comput. Syst. 4, 235–243 (1988)CrossRefGoogle Scholar
  35. 35.
    Wu, Y., Xing, H.: Note on 2-rainbow domination and Raman domination in graphs. Appl. Math. Lett. 23, 706–709 (2010)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Xu, G.: 2-rainbow domination of generalized Petersen graphs P(n,3). Discret. Appl. Math. 157, 2570–2573 (2009)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of Information ManagementShih Hsin UniversityTaipeiRepublic of China
  2. 2.Department of Information ManagementNational Taiwan University of Science and TechnologyTaipeiRepublic of China

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