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

Let G=(V,E)G=(V,E) be a simple graph of order nn with vertex set V={v1,…,vn}V={v1,…,vn} and suppose that at most riri units of some commodity may be placed at any vertex vivi while at least sisi units must be placed in the closed neighbourhood of vivi for i=1,…,ni=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.