Geodesic
On the total k-domination number of graphs
Discussiones Mathematicae. Graph Theory, Tome 32 (2012) no. 3, pp. 419-426

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Let k be a positive integer and let G = (V,E) be a simple graph. The k-tuple domination number γ_×k(G) of G is the minimum cardinality of a k-tuple dominating set S, a set that for every vertex v ∈ V, |N_G[v] ∩ S| ≥ k. Also the total k-domination number γ_×k,t(G) of G is the minimum cardinality of a total k -dominating set S, a set that for every vertex v ∈ V, |N_G(v) ∩ S| ≥ k. The k-transversal number τₖ(H) of a hypergraph H is the minimum size of a subset S ⊆ V(H) such that |S ∩e | ≥ k for every edge e ∈ E(H).
Keywords: total k-domination (k-tuple total domination) number, k-tuple domination number, k-transversal number 
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Kazemi, Adel. On the total k-domination number of graphs. Discussiones Mathematicae. Graph Theory, Tome 32 (2012) no. 3, pp. 419-426. http://geodesic.mathdoc.fr/item/DMGT_2012_32_3_a2/