Inequalities involving independence domination, f-domination, connected and total f-domination numbers

Abstract

Let f be an integer-valued function defined on the vertex set V(G) of a graph G. A subset D of V(G) is an f-dominating set if each vertex x outside D is adjacent to at least f(x) vertices in D. The minimum number of vertices in an f-dominating set is defined to be the f-domination number, denoted by γ f (G). In a similar way one can define the connected and total f-domination numbers γ c,f (G) and γ t,f (G). If f(x) = 1 for all vertices x, then these are the ordinary domination number, connected domination number and total domination number of G, respectively. In this paper we prove some inequalities involving γ f (G), γ c,f (G), γ t,f (G) and the independence domination number i(G). In particular, several known results are generalized.