Bounds on the weak domination number

Let G = (V, E) a graph. A set D ~ V is a weak dominating set of G if for every vertex y E V D there is a vertex xED with xy E E and d( x, G) ::; d(y, G). The weak domination number rw (G) is defined as the minimum cardinality of a weak dominating set and was introduced by Sampathkumar and Pushpa Latha in [6]. In this paper we present sharp upper bounds on rw(G) for general graphs involving the maximum and minimum degree and characterize all extremal graphs. Furthermore, we give a probabilistic upper bound and a lower bound on rw( G).