Upper bounds for ƒ-domination number of graphs

For an integer-valued function ƒ defined on the vertices of a graph G, the ƒ-domination number γƒ(G) of G is the smallest cardinality of a subset D ⊆ V(G) such that each x ∈ V(G) − D is adjacent to at least ƒ(x) vertices in D. When ƒ(x) = k for all x ∈ V(G), γƒ(G) is the k-domination number γk(G). In this note, we give a tight upper bound for γƒ and an improvement of the upper bound for a special ƒ-domination number μj,k of Stracke and Volkmann (1993). Some upper bounds for γk are also obtained.