k-Subdomination in graphs Original Research Article

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∈NG[v]f(u)⩾1 for at least k vertices v of G. The k-subdomination number of G, denoted by γks(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 γks(G) in terms of the degree sequence of G. This generalizes some known results on the k-subdomination number γks(G), the signed domination number γs(G) and the majority domination number γmaj(G).