Discussiones Mathematicae Graph Theory 32(3) (2012) 419-426
doi: 10.7151/dmgt.1616

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On the Total k-domination Number of Graphs

Adel P. Kazemi

Department of Mathematics
University of Mohaghegh Ardabili
P. O. Box 5919911367, Ardabil, Iran

Abstract

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, |NG[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, |NG(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).

We 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.

Keywords: total k-domination (k-tuple total domination) number, k-tuple domination number, k-transversal number

2010 Mathematics Subject Classification: 05C69.

References

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Received 9 March 2011
Revised 23 July 2011
Accepted 25 July 2011