On the (r, s) domination number of a graph

Suppose a network facility location problem is modelled by means of an undirected, simple graph G = (V, epsilon) with V = {upsilon(1), ... , upsilon(n)}. Let r = (r(1),..., r(n)) and s = (s(1), ... , s(n)) be vectors of nonnegative integers and consider the combinatorial optimization problem of locating the minimum number, gamma < r, s, G > (say), of commodities on the vertices of G such that at least s(j) commodities are located in the vicinity of (i.e. in the closed neigbourhood of) vertex v(j), with no more than r(j) commodities placed at vertex v(j) itself, for all j = 1, ... , n. In this paper we establish lower and upper bounds on the parameter gamma < r, s, G > for a general graph G. We also determine this parameter exactly for certain classes of graphs, such as paths, cycles, complete graphs, complete bipartite graphs and establish good upper bounds on gamma < r, s, G > for a class of grid graphs in the special case where r(j) = r and s(j) = s for all j = 1, ... , n.