Total k-subdominating function

Total [math]\displaystyle{ k }[/math]-subdominating function --- тотальная [math]\displaystyle{ k }[/math]-субдоминирущая функция.

Let [math]\displaystyle{ G = (V,E) }[/math] be a simple graph. For any real-valued function [math]\displaystyle{ f: \; V \rightarrow R }[/math], the weight of [math]\displaystyle{ f }[/math] is defined as [math]\displaystyle{ f(V) = \sum f(v) }[/math], over all vertices [math]\displaystyle{ v \in V }[/math]. For a positive integer [math]\displaystyle{ k }[/math], a total [math]\displaystyle{ k }[/math]-subdominating function (TkSF) is a function [math]\displaystyle{ f: \; V \rightarrow \{1,-1\} }[/math] such that [math]\displaystyle{ f(N(v)) \geq 1 }[/math] for at least [math]\displaystyle{ k }[/math] vertices [math]\displaystyle{ v }[/math] of [math]\displaystyle{ G }[/math]. The total [math]\displaystyle{ k }[/math]-subdomination number [math]\displaystyle{ \gamma_{ks}^{t}(G) }[/math] of a graph [math]\displaystyle{ G }[/math] equals the minimum weight of a TkSF on [math]\displaystyle{ G }[/math]. In the special case for [math]\displaystyle{ k = |V| }[/math], [math]\displaystyle{ \gamma_{ks}^{t} }[/math] is the signed total domination number.