Upper bounds for f-domination number of graphs

For an integer-valued function f defined on the vertices of a graph G, the f-domination number gamma(f)(G) of G is the smallest cardinality of a subset D subset of or equal to V(G) such that each x is an element of V(G) - D is adjacent to at least f(x) vertices in D. When f(x) = k for all x is an element of V(G), gamma(f)(G) is the k-domination number gamma(k)(G). In this note, we give a tight upper bound for gamma(f) and an improvement of the upper bound for a special f-domination number mu(j, k) of Stracke and Volkmann (1993). Some upper bounds for gamma(k) are also obtained. (C) 1998 Elsevier Science B.V. All rights reserved.