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 ∑u∈N _G[v]f(u)1 for at least k vertices v of G. The k-subdomination number of G, denoted by γ _ks(G), is the minimum of ∑ _v∈Vf(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 γ _ks(G) in terms of the degree sequence of G. This generalizes some known results on the k-subdomination number γ _ks(G), the signed domination number γ _s(G) and the majority domination number γ _maj(G).