Properties of the Global Total k -Domination Number

A nonempty subset D ⊂ 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 ⊆ V is a total k-dominating set if there are at least k vertices in set D adjacent to every vertex v ∈ V , and it is a global total k-dominating set if D is a total k-dominating set of both G and G ¯ . The global total k-domination number of G, denoted by γ k t g ( G ) , is the minimum cardinality of a global total k-dominating set of G, GTkD-set. Here we derive upper and lower bounds of γ k t g ( 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 γ ( k − 1 ) t g ( G ) and γ k t g ( G ) , which, in turn, provides another upper bound on γ k t g ( G ) .