image-tuple total domination in graphs Original Research Article

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