Weak total domination in graphs - Chellali, Mustapha; Rad, Nader Jafari (Utilitas Mathematica, 2014)
A set D of vertices in a graph G = (V, E) is a total dominating set if every vertex of G is adjacent to some vertex in D. A total dominating set D of G is said to be weak if every vertex ν ∈ V - D is adjacent to a vertex u ∈ 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. In this paper we initiate the study of weak total domination in graphs. We present lower and upper bounds on γwt(G) with some characterizations, and we show that determining the number γwt(G) for an arbitrary graph is NP- complete. Then we provide a constructive characterization of trees with equal weak total domination and total domination numbers.