k-Subdomination in graphs

For a positive integer k, a k-subdominating function of a graph G =(V,E) is a function f:V →{−1; 1} such that Σu∈NG[v] f(u)≧1 for at least k vertices v of G. The k-subdomination number of G, denoted by νks(G), is the minimum of Σv∈V f(v) taken over all k-subdominating functions f of G. In this article, we prove a conjecture for k-subdomination on trees proposed by Cockayne and Mynhardt. We also give a lower bound for νks(G) in terms of the degree sequence of G. This generalizes some known results on the k-subdomination number νks(G), the signed domination number
νs(G) & the majority domination number
νmaj(G).