Properties of the Global Total k-Domination Number

A nonempty subset D subset of V of vertices of a graph G=(V,E) is a dominating set if every vertex of this graph is adjacent to at least one vertex from this set except the vertices which belong to this set itself. D subset of V is a total k-dominating set if there are at least k vertices in set D adjacent to every vertex v is an element of V, and it is a global total k-dominating set if D is a total k-dominating set of both G and G over bar . The global total k-domination number of G, denoted by gamma(g)(kt)(G), is the minimum cardinality of a global total k-dominating set of G, GTkD-set. Here we derive upper and lower bounds of gamma(g)(kt)(G), and develop a method that generates a GTkD-set from a GT(k-1)D-set for the successively increasing values of k. Based on this method, we establish a relationship between gamma(g)((k-1)t)(G) and gamma(g)(kt)(G), which, in turn, provides another upper bound on gamma(g)(kt)(G).