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 n n with vertex set V={v1,,vn} V = { v 1 ,  , v n } and suppose that at most ri r i units of some commodity may be placed at any vertex vi v i while at least si s i units must be placed in the closed neighbourhood of vi v i for i=1,,n 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.
? r s ? s -dominating r -function s s s r r r s ? -domination ? r s ? ? r k k -tuple domination k k k k k multiple domination probabilistic method