A Study of Graph Value Functions in Domination Theory

The Y -domination number of a graph for a given number set Y was introduced by D.W. Bange, A. E. Barkauskas, L. H. Host and P. J. Slater as a generalization of the domination number of a graph. It is defined using the concept of a Y -dominating function. The total dominating graph Dt(G) of G is the graph with the vertex set V∪S in which two vertices u and v are adjacent if u ∈ V and v is a minimal total dominating set of G containing u. In this paper, some properties of this new graph are obtained. Also characterizations are given for graphs.