Upper bounds for [formula omitted] 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 γ k ( 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 μ j, k of Stracke and Volkmann (1993). Some upper bounds for γ k are also obtained.