# American Institute of Mathematical Sciences

August  2018, 1(3): 255-263. doi: 10.3934/mfc.2018011

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

 1 University of Science and Technology of China (USTC), Hefei, China 2 Facebook Seattle, 1101 Dexter Ave N, Seattle, WA 98109, USA

* Corresponding author: Hongyu Liang. Email: hongyuliang86@gmail.com

Received  October 2017 Revised  January 2018 Published  July 2018

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.

Citation: Haisheng Tan, Liuyan Liu, Hongyu Liang. Total $\{k\}$-domination in special graphs. Mathematical Foundations of Computing, 2018, 1 (3) : 255-263. doi: 10.3934/mfc.2018011
##### References:
 [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. Google Scholar [2] B. Bollobás, Modern Graph Theory, Graduate Texts in Mathematics 184, Springer, Berlin, 1998. doi: 10.1007/978-1-4612-0619-4. Google Scholar [3] B. Bresar, P. Dorbec, W. Goddard, B. Hartnell, M. Henning, S. 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. Google Scholar [4] G. Domke, S. Hedetniemi, R. 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. Google Scholar [5] T. Haynes, S. H. ST and P. Slater, Domination in Graphs: Advanced Topics, Marcel Dekker, 1998.Google Scholar [6] T. Haynes, S. H. ST and P. Slater, Fundamentals of Domination in Graphs, Marcel Dekker, 1998. Google Scholar [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. Google Scholar [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. Google Scholar [9] C. Lee, Labelled Domination and Its Variants, PhD thesis, National Chung Cheng University, 2006.Google Scholar [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. Google Scholar

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##### References:
 [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. Google Scholar [2] B. Bollobás, Modern Graph Theory, Graduate Texts in Mathematics 184, Springer, Berlin, 1998. doi: 10.1007/978-1-4612-0619-4. Google Scholar [3] B. Bresar, P. Dorbec, W. Goddard, B. Hartnell, M. Henning, S. 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. Google Scholar [4] G. Domke, S. Hedetniemi, R. 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. Google Scholar [5] T. Haynes, S. H. ST and P. Slater, Domination in Graphs: Advanced Topics, Marcel Dekker, 1998.Google Scholar [6] T. Haynes, S. H. ST and P. Slater, Fundamentals of Domination in Graphs, Marcel Dekker, 1998. Google Scholar [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. Google Scholar [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. Google Scholar [9] C. Lee, Labelled Domination and Its Variants, PhD thesis, National Chung Cheng University, 2006.Google Scholar [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. Google Scholar
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