Negative k-subdecision numbers in graphs

Let G be a simple connected graph without isolated vertex with vertex set V (G) and edge set E(G). A function f : V (G) → {−1, 1} is said to be a negative k-subdecision function of G if ∑ x∈NG(v) f(x) ≤ 1 for at least k vertices v of G. The value max ∑ x∈V (G) f(x), taking over all negative k-subdecision functions f of G, is called the negative k-subdecision number of G and is denoted by βkD(G). In this paper we initiate the study of the negative k-subdecision numbers in graphs and present some bounds for this parameter.