On the total k-domination number of graphs
Adel P. Kazemi · Discussiones Mathematicae Graph Theory · 2012
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 vertexWe know that for any graph G of order n with minimum degree at least k, γ ×k (G) ≤ γ ×k,t (G) ≤ n.Obviously for every k-regular graph, the upper bound n is sharp.Here, we give a sufficient condition for γ ×k,t (G) < n.Then we characterize complete multipartite graphs G with γ ×k (G) = γ ×k,t (G).We also state that the total k-domination number of a graph is the k-transversal number of its open neighborhood hypergraph, and also the domination number of a graph is the transversal number of its closed neighborhood hypergraph.Finally, we give an upper bound for the total k-domination number of the cross product graph G × H of two graphs G and H in terms on the similar numbers of G and H. Also, we show that this upper bound is strict for some graphs, when k = 1.