Upper bounds for the k-subdomination number of graphs

For a positive integer k, a k-subdominating function of G=(V,E) is a function f : V→{−1,1} such that the sum of the function values, taken over closed neighborhoods of vertices, is at least one for at least k vertices of G. The sum of the function values taken over all vertices is called the aggregate of f and the minimum aggregate among all k-subdominating functions of G is the k-subdomination number γks(G). In this paper, we solve a conjecture proposed in (Ars. Combin 43 (1996) 235), which determines a sharp upper bound on γks(G) for trees if k>|V|/2 and give an upper bound on γks for connected graphs.