Research Article
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Year 2020, , 128 - 135, 31.12.2020
https://doi.org/10.47000/tjmcs.691030

Abstract

References

  • Amjadi, J., Dehgardi, N., Mohammadi, N., Sheikholeslami, S.M., Volkmann, L., \emph{Independent $2$-rainbow domination numbers in trees}, Asian-Eur. J. Math., {\bf08}(2015) 02, 1550035.
  • Bresar, B., Henning, M., Rall, D., \emph{Rainbow domination in graphs}, Taiwanese J. Math. {\bf12}(2008), 23--225.
  • Bresar, B., Sumnjak, T., \emph{On 2-rainbow domination in graphs}, Discrete Appl. Math. {\bf 155}(2007), 2394--2400.
  • Chang, G., Wu, J., Zhu, X., \emph{Rainbow domination on trees}, Discrete Appl. Math. {\bf 158}(2010), 8--12.
  • Hao, G., Mojdeh, D.A., Wei, S., Xie, Z., \emph{Rainbow domination in the Cartesian product of directed paths}, Australas. J. Combin. {\bf 70}(3)(2018), 349--361.
  • Mansouri, Z., Mojdeh, D.A., \emph{Outer independent rainbow dominating functions in graphs}, Opuscula Math. \textbf{40}(5)(2020), 599-615 https://doi.org/10.7494/OpMath.2020.40.5.599.
  • Mojdeh, D.A., Kazemi, A.P., \emph{Domination in Harary graphs}, Bulletin of the Institute of Combinatorics and its Application {\bf 49}(2007), 61--78.
  • Pai, K., Chiu, W., \emph{$3$-rainbow domination number in graphs}, In Proceedings of the Institute of Industrial Engineers Asian Conference 2013, 713--720. Springer, Science+ Business Media Singapore, 2013.
  • West, D.B., Introduction to Graph Theory (Second Edition) (Prentice Hall, USA, 2001).
  • Wu, Y., Jafari Rad, N., \emph{Bounds on the $2$-rainbow domination number of graphs}, Graphs and Combin. {\bf 29}(2013), 1125--1133.

(Independent) $k$-Rainbow Domination of a Graph

Year 2020, , 128 - 135, 31.12.2020
https://doi.org/10.47000/tjmcs.691030

Abstract

Let $G=(V,E)$ be a graph with the vertex set $V=V(G)$ and the edge set $E=E(G)$. Let $k$ be a positive integer and $\gamma_{rk}(G)$ ($\gamma_{i_{rk}}(G)$) be $k$-rainbow domination (independent $k$-rainbow domination) number of a graph $G$. In this paper, we study the $k$-rainbow domination and independent $k$-rainbow domination numbers of graphs. We obtain bounds for $\gamma_{rk}(G-e)$ ($\gamma_{i_{rk}}(G-e)$) in terms of $\gamma_{rk}(G)$ ($\gamma_{i_{rk}}(G)$). Finally, the relation between weak $3$-domination and $3$-rainbow domination number of graphs will be investigated.

References

  • Amjadi, J., Dehgardi, N., Mohammadi, N., Sheikholeslami, S.M., Volkmann, L., \emph{Independent $2$-rainbow domination numbers in trees}, Asian-Eur. J. Math., {\bf08}(2015) 02, 1550035.
  • Bresar, B., Henning, M., Rall, D., \emph{Rainbow domination in graphs}, Taiwanese J. Math. {\bf12}(2008), 23--225.
  • Bresar, B., Sumnjak, T., \emph{On 2-rainbow domination in graphs}, Discrete Appl. Math. {\bf 155}(2007), 2394--2400.
  • Chang, G., Wu, J., Zhu, X., \emph{Rainbow domination on trees}, Discrete Appl. Math. {\bf 158}(2010), 8--12.
  • Hao, G., Mojdeh, D.A., Wei, S., Xie, Z., \emph{Rainbow domination in the Cartesian product of directed paths}, Australas. J. Combin. {\bf 70}(3)(2018), 349--361.
  • Mansouri, Z., Mojdeh, D.A., \emph{Outer independent rainbow dominating functions in graphs}, Opuscula Math. \textbf{40}(5)(2020), 599-615 https://doi.org/10.7494/OpMath.2020.40.5.599.
  • Mojdeh, D.A., Kazemi, A.P., \emph{Domination in Harary graphs}, Bulletin of the Institute of Combinatorics and its Application {\bf 49}(2007), 61--78.
  • Pai, K., Chiu, W., \emph{$3$-rainbow domination number in graphs}, In Proceedings of the Institute of Industrial Engineers Asian Conference 2013, 713--720. Springer, Science+ Business Media Singapore, 2013.
  • West, D.B., Introduction to Graph Theory (Second Edition) (Prentice Hall, USA, 2001).
  • Wu, Y., Jafari Rad, N., \emph{Bounds on the $2$-rainbow domination number of graphs}, Graphs and Combin. {\bf 29}(2013), 1125--1133.
There are 10 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Zhila Mansouri This is me 0000-0002-2918-9615

Doost Ali Mojdeh 0000-0001-9373-3390

Publication Date December 31, 2020
Published in Issue Year 2020

Cite

APA Mansouri, Z., & Mojdeh, D. A. (2020). (Independent) $k$-Rainbow Domination of a Graph. Turkish Journal of Mathematics and Computer Science, 12(2), 128-135. https://doi.org/10.47000/tjmcs.691030
AMA Mansouri Z, Mojdeh DA. (Independent) $k$-Rainbow Domination of a Graph. TJMCS. December 2020;12(2):128-135. doi:10.47000/tjmcs.691030
Chicago Mansouri, Zhila, and Doost Ali Mojdeh. “(Independent) $k$-Rainbow Domination of a Graph”. Turkish Journal of Mathematics and Computer Science 12, no. 2 (December 2020): 128-35. https://doi.org/10.47000/tjmcs.691030.
EndNote Mansouri Z, Mojdeh DA (December 1, 2020) (Independent) $k$-Rainbow Domination of a Graph. Turkish Journal of Mathematics and Computer Science 12 2 128–135.
IEEE Z. Mansouri and D. A. Mojdeh, “(Independent) $k$-Rainbow Domination of a Graph”, TJMCS, vol. 12, no. 2, pp. 128–135, 2020, doi: 10.47000/tjmcs.691030.
ISNAD Mansouri, Zhila - Mojdeh, Doost Ali. “(Independent) $k$-Rainbow Domination of a Graph”. Turkish Journal of Mathematics and Computer Science 12/2 (December 2020), 128-135. https://doi.org/10.47000/tjmcs.691030.
JAMA Mansouri Z, Mojdeh DA. (Independent) $k$-Rainbow Domination of a Graph. TJMCS. 2020;12:128–135.
MLA Mansouri, Zhila and Doost Ali Mojdeh. “(Independent) $k$-Rainbow Domination of a Graph”. Turkish Journal of Mathematics and Computer Science, vol. 12, no. 2, 2020, pp. 128-35, doi:10.47000/tjmcs.691030.
Vancouver Mansouri Z, Mojdeh DA. (Independent) $k$-Rainbow Domination of a Graph. TJMCS. 2020;12(2):128-35.

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