Upper bounds for f-domination number of graphs,

Abstract For an integer-valued function ƒ defined on the vertices of a graph G, the ƒ- domination number γ ƒ (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), γ ƒ (G) is the k-domination number γk(G). In this note, we give a tight upper bound for γ ƒ 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.