kk-tuple total domination in graphs

A set SS of vertices in a graph GG is a kk-tuple total dominating set, abbreviated kTDS, of GG if every vertex of GG is adjacent to least kk vertices in SS. The minimum cardinality of a kTDS of GG is the kk-tuple total domination number of GG. For a graph to have a kTDS, its minimum degree is at least kk. When k=1k=1, a kk-tuple total domination number is the well-studied total domination number. When k=2k=2, a kTDS is called a double total dominating set and the kk-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 kk-transversals in hypergraphs. We investigate the kk-tuple total domination number for complete multipartite graphs. Upper bounds on the kk-tuple total domination number of general graphs are presented.