Bounds on weak and strong total domination number in graphs

A set D of vertices in a graph G = (V;E) is a total dominating set if every vertex of G is adjacentto some vertex in D. A total dominating set D of G is said to be weak if every vertex v (element of) V-D is adjacent to a vertex u (element of) D such that dG(v) ≥ dG(u). The weak total domination number γwt(G) of G is the minimum cardinality of a weak total dominating set of G. A total dominating set D of G is said to be strong if every vertex v (element of) V- D is adjacent to a vertex u (element of) D such that dG(v) ≤ dG(u). The strong total domination number γst(G) of G is the minimum cardinality of a strong total dominating set of G. We present some bounds on weak and strong total domination number of a graph.