A generalization of global dominating function

Abstract

Let G be a graph. A function f : V (G) −→ {0, 1}, satisfying the condition that every vertex u with f(u) = 0 is adjacent with at least one vertex v such that f(v) = 1, is called a dominating function (DF ). The weight of f is defined as wet(f) = Σv∈V (G)f(v). The minimum weight of a dominating function of G is denoted by γ(G), and is called the domination number of G. A dominating function f is called a global dominating function (GDF ) if f is also a DF of G. The minimum weight of a global dominating function is denoted by γg(G) and is called global domination number of G. In this paper we introduce a generalization of global dominating function. Suppose G is a graph and s ≥ 2 and Kn is the complete graph on V (G). A function f : V (G) −→ {0, 1} on G is s-dominating function (s−DF ), if there exists some factorization {G1, . . . , Gs} of Kn, such that G1 = G and f is dominating function of each Gi.