Let G = (V, E) be a graph and u, v ~ V. Then, u strongly dominates v and v weakly dominates u if (i) uv ~ E and (ii) deg u >/deg v. A set D c V is a strong-dominating set (sd-set) of G if every vertex in V-D is strongly dominated by at least one vertex in D. Similarly, a weak-dominating set (wd-set) is defined. The strong (weak) domination number 7s (7w) of G is the minimum cardinality of an sd-set (wd-set). Besides investigating some relationship of ?s and ?w with other known parameters of G, some bounds are obtained. A graph G is domination balanced if there exists an sd-set D1 and a wd-set/)2 such that D1 c~D2 = 0. A study of domination balanced graphs is initiated.

Let G = (V, E ) be a connected graph. A set D C V is a set-dominating set (sd-set) if for every set T C V -D, there exists a nonempty set S C Dsuch that the subgraph (S U T ) induced by S U Tis connected. The set-domination number ys(G) of G is the minimum cardinality of a sd-set. In this paper we develop properties of this new parameter and relate it to some other known domination parameters. A set D C V is a set-dominating set (sd-set) of G if for every set T C V -D , there exists a nonempty subset S C D such that the subgraph (S U T) is connected. The set-domination number y,(G) of G is the minimum cardinality of a sd-set.Besides obtaining some bounds for y,(G), we relate it to some other known parameters of G.