A NEW DOMINATION CONCEPTION

Let f be an integer valued function defined on the vertex set V(G) of a simple graph G. We call a subset Df of V(G) a f-dominating set of G if \N(x, G) and D(f)\ greater-than-or-equal-to f(x) for all x is-an-element-of V(G) - D(f), where N(x, G) is the set of neighbors of x. D(f) is a minimum f-dominating set if G has no f-dominating set D(f)' with \D(f)'\ < \D(f)\. If j,k is-an-element-of N0 = {0, 1, 2,....} with j less-than-or-equal-to k, then we define the integer valued function f(j,k) on V(G) by [GRAPHICS] By mu(j,k)(G) we denote the cardinality of a minimum f(j,k)-dominating set of G. A set D subset-or-equal-to V(G) is j-dominating if every vertex, which is not in D, is adjacent to at least j vertices of D. The j-domination number gamma(j)(G) is the minimum order of a j-dominating set in G. In this paper we shall give estimations of the new domination number mu(j,k(G), and with the help of these estimations we prove some new and some known upper bounds for the j-domination number.