An algorithm for the (r,s)-domination number of a tree

Abstract

Suppose that at most r units of some commodity may be positioned at any vertex of a graph G = (V, E) while at least s (≥ r) units must be present in the vicinity (i.e. closed neighbourhood) of each vertex. Suppose that the function f : V → {0, . . . , r}, whose values are the numbers of units stationed at vertices, satisfies the above requirement. Then f is called an s-dominating r-function. We present an algorithm which finds the minimum number of units required in such a function and a function which attains this minimum, for any tree.