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-el ement-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 mini mum order of a j-dominating set in G. In this paper we shall give esti mations of the new domination number mu(j,k(G), and with the help of t hese estimations we prove some new and some known upper bounds for the j-domination number.