Upper bounds on the balanced 〈r, s〉-domination number of a graph

Let G = ( V , E ) be a simple graph of order n with vertex set V = { v 1 , ? , v n } and suppose that at most r i units of some commodity may be placed at any vertex v i while at least s i units must be placed in the closed neighbourhood of v i for i = 1 , ? , n . The smallest number of units that may be placed on the vertices of the graph satisfying the above requirements is called the { r , s } -domination number of the graph. The case where r = r , ? , r and s = s , ? , s is called the balanced case of { r , s } -domination. We establish three upper bounds on the { r , s } -domination number of a graph for the balanced case in this paper.