A set S of vertices in a graph G is a k-tuple total dominating set, abbreviated kTDS, of G if every vertex of G is adjacent to least k vertices in S. The minimum cardinality of a kTDS of G is the k-tuple total domination number of G. For a graph to have a kTDS, its minimum degree is at least k. When k = 1, a k-tuple total domination number is the well-studied total domination number. When k = 2, a kTDS is called a double total dominating set and the k-tuple total domination number is called the double total domination number. We present properties of minimal kTDS and show that the problem of finding kTDSs in graphs can be translated to the problem of finding k-transversals in hypergraphs. We investigate the k-tuple total domination number for complete multipartite graphs. Upper bounds on the k-tuple total domination number of general graphs are presented.
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