Geodesic
On the Total k-Domination in Graphs
Discussiones Mathematicae. Graph Theory, Tome 38 (2018) no. 1, pp. 301-317

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Let G=(V, E) be a graph; a set S ⊆ V is a total k-dominating set if every vertex v ∈ V has at least k neighbors in S. The total k-domination number γkt(G) is the minimum cardinality among all total k-dominating sets. In this paper we obtain several tight bounds for the total k-domination number of a graph. In particular, we investigate the relationship between the total k-domination number of a graph and the order, the size, the girth, the minimum and maximum degree, the diameter, and other domination parameters of the graph.
Keywords: k -domination, total k -domination, k -tuple domination, k -tuple total domination 
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Bermudo, Sergio; Hernández-Gómez, Juan C.; Sigarreta, José M. On the Total k-Domination in Graphs. Discussiones Mathematicae. Graph Theory, Tome 38 (2018) no. 1, pp. 301-317. http://geodesic.mathdoc.fr/item/DMGT_2018_38_1_a23/