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

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Abstract

Let G(V, E) be a connected and undirected graph with n-vertex-set V and m-edge-set E. For each v ∈ V, let N(v) = {u|v ∈ V 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) = f k (v)f k − 1(v) … f 1(v), i.e., f i (v) ∈ {0, 1}, 1 ≤ i ≤ k, such that for any vertex v with f(v) = 0(k) we have ⋈ u ∈ N(v) f(u) = 1(k), for all v ∈ V, where ⋈ u ∈ S f(u) denotes the result of taking bitwise OR operation on f(u), for all u ∈ S. 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 γ k r (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 γ 2r (S(n, m)), γ 2r (S +(n, m)), and γ 2r (S ++(n, m)), where S(n, m), S +(n, m), and S ++(n, m) are Sierpiński graphs and extended Sierpiński graphs.

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References

  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. 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)

    Article  MATH  Google Scholar 

  3. Brešar, B., Kraner Šumenjak, T.: On the 2-rainbow domination in graphs. Discret. Appl. Math. 155, 2394–2400 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Brešar, B., Henning, M.A., Rall, D.F.: Rainbow Domination in Graphs. Taiwan. J. Math. 12(1), 213–225 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chang, G.J., Wu, J., Zhu, X.: Rainbow Domination on Trees. Discret. Appl. Math. 158, 8–12 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen, G.H., Duh, D.R.: Topological properties, communication, and computation on WK-recursive networks. Networks 24, 303–317 (1994)

    Article  MathSciNet  Google Scholar 

  8. Duh, D.R., Chen, G.H.: Topological properties of WK-recursive networks. J. Parallel Distrib. Comput. 23, 468–474 (1994)

    Article  Google Scholar 

  9. Fujita, S., Furuya, M., Magnant, C.: K-Rainbow domatic numbers. Discret. Appl. Math. 160, 1104–1113 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  12. Hinz, A.M., Schief, A.: The average distance on the Sierpiński gasket. Probab. Theory Relat. Fields 87, 129–138 (1990)

    Article  MATH  Google Scholar 

  13. Hinz, A.M.: Pascal’s triangle and the Tower of Hanoi. Am. Math. Mon. 99, 538–544 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hinz, A.M., Parisse, D.: The Average Eccentricity of Sierpiński Graphs. Graphs and Combinatorics 28, 671–686 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hinz, A.M., Klavžar, S., Milutinović, U., Petr, C.: The Tower of Hanoi-Myths and Maths., Birkhuser/Springer Basel AG, Basel (2013)

  17. Jakovac, M., Klavžar, S.: Vertex-, edge- and total-colorings of Sierpiński-like graphs. Discret. Math. 309, 1548–1556 (2009)

    Article  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  20. Klavžar, S., Milutinović, U., Petr, C.: 1-perfect codes in Sierpiński graphs. Bull. Aust. Math. Soc. 66, 369–384 (2002)

    Article  MATH  Google Scholar 

  21. Klavžar, S., Mohar, B.: Crossing numbers of Sierpiński-like graphs. J. Graph Theory 50, 186–198 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  22. Klavžar, S.: Coloring Sierpiński graphs and Sierpiński gasket graphs. Taiwan. J. Math. 12, 513–522 (2008)

    Article  MATH  Google Scholar 

  23. Klix, F., Rautenstrauch-Goede, K.: Struktur-und Komponentenanalyse von Problemlösungsprozessen. Zeitschrift für Psychologie 174, 167–193 (1967)

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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 134–138

  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)

    Article  MathSciNet  MATH  Google Scholar 

  28. Parisse, D.: On some metric properties of the Sierpiński graphs S(n, k). Ars Combinatoria 90, 145–160 (2009)

    MathSciNet  MATH  Google Scholar 

  29. Romik, D.: Shortest paths in the Tower of Hanoi graph and finite automata. SIAM J. Discret. Math. 20, 610–622 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  30. Kraner Šumenjak, T., Rall, D.F., Tepeh, A.: Rainbow domination in the lexicographic product of graphs. Discret. Appl. Math. 161, 2133–2141 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  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. Teguia, A.M., Godbole, A.P.: Sierpiński gasket graphs and some of their properties. Aust. J. Commun. 35, 181–192 (2006)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  34. Vecchia, G.D., Sanges, C.: A recursively scalable network VLSI implementation. Futur. Gener. Comput. Syst. 4, 235–243 (1988)

    Article  Google Scholar 

  35. Wu, Y., Xing, H.: Note on 2-rainbow domination and Raman domination in graphs. Appl. Math. Lett. 23, 706–709 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  36. Xu, G.: 2-rainbow domination of generalized Petersen graphs P(n,3). Discret. Appl. Math. 157, 2570–2573 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Information Management, Shih Hsin University, 1 Lane17 Sec.1, Mu-Cha Rd., Taipei, Taiwan, 10607, Republic of China

    Jia-Jie Liu

  2. Department of Information Management, National Taiwan University of Science and Technology, Taipei, Taiwan, Republic of China

    Shun-Chieh Chang & Chiou-Jiun Lin

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  1. Jia-Jie Liu
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  2. Shun-Chieh Chang
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  3. Chiou-Jiun Lin
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Correspondence to Jia-Jie Liu.

Additional information

This work was supported in part by the National Science Council of Republic of China under contracts MOST 103-2221-E-128-002- and MOST 105-2221-E-128-004 -.

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Liu, JJ., Chang, SC. & Lin, CJ. The 2-Rainbow Domination of Sierpiński Graphs and Extended Sierpiński Graphs. Theory Comput Syst 61, 893–906 (2017). https://doi.org/10.1007/s00224-017-9756-y

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