Advanced Search
Article Contents
Article Contents

Total $\{k\}$-domination in special graphs

Abstract Full Text(HTML) Related Papers Cited by
  • For a positive integer $k$ and a graph $G = (V,E)$, a function $f:V \to \{0,1,...,k\}$ is called a total $\{k\}$-dominating function of $G$ if $\sum_{u∈ N_G(v)}f(u)≥ k$ for each $v∈ V$, where $N_G(v)$ is the neighborhood of $v$ in $G$. The total $\{k\}$-domination number of $G$, denoted by $\gamma _t^{\left\{ k \right\}}\left( G \right)$, is the minimum weight of a total $\{k\}$-dominating function $G$, where the weight of $f$ is $\sum_{v∈ V}f(v)$. In this paper, we determine the exact values of the total $\{k\}$-domination number for several commonly-encountered classes of graphs including cycles, paths, wheels, and pans.

    Mathematics Subject Classification: 05C10.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] H. Aram and S. Sheikholeslami, On the total $\{k\}$-domination and total $\{k\}$-domatic number of graphs, Bull. Malays. Math. Sci. Soc., 36 (2013), 39-47. 
    [2] B. Bollobás, Modern Graph Theory, Graduate Texts in Mathematics 184, Springer, Berlin, 1998. doi: 10.1007/978-1-4612-0619-4.
    [3] B. BresarP. DorbecW. GoddardB. HartnellM. HenningS. Klavzar and D. F. Rall, Vizing's conjecture: A survey and recent results, J. Graph Theory, 69 (2012), 46-76.  doi: 10.1002/jgt.20565.
    [4] G. DomkeS. HedetniemiR. Laskar and G. Fricke, Relationships between integer and fractional parameters of graphs, Graph Theory, Combinatorics, and Applications, Proceedings of the Sixth Quadrennial Conference on the Theory and Applications of Graphs (Kalamazoo, MI, 1988), 2 (1991), 371-387. 
    [5] T. Haynes, S. H. ST and P. Slater, Domination in Graphs: Advanced Topics, Marcel Dekker, 1998.
    [6] T. Haynes, S. H. ST and P. Slater, Fundamentals of Domination in Graphs, Marcel Dekker, 1998.
    [7] J. He and H. Liang, Complexity of total $\{k\}$-domination and related problems, Frontiers in Algorithmics and Algorithmic Aspects in Information and Management, (eds. M. Atallah, X. -Y. Li and B. Zhu), vol. 6681 of LNCS, 2011,147–155. doi: 10.1007/978-3-642-21204-8_18.
    [8] M. A. Henning, A short proof of a result on a vizing-like problem for integer total domination, J. Comb. Optim., 20 (2010), 321-323.  doi: 10.1007/s10878-008-9201-x.
    [9] C. Lee, Labelled Domination and Its Variants, PhD thesis, National Chung Cheng University, 2006.
    [10] N. Li and X. Hou, On the total $\{k\}$-domination number of Cartesian products of graphs, J. Comb. Optim., 18 (2009), 173-178.  doi: 10.1007/s10878-008-9144-2.
  • 加载中

Article Metrics

HTML views(1372) PDF downloads(299) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint