Partial signed domination number

Partial signed domination number --- частично знаковое число доминирования.

Let [math]\displaystyle{ G = (V,E) }[/math] be a simple graph. For any real valued function [math]\displaystyle{ f : V \rightarrow R }[/math] and [math]\displaystyle{ S \subseteq V }[/math], let [math]\displaystyle{ f(S) = \sum _{v \in S} f(v) }[/math]. Let [math]\displaystyle{ c,d }[/math] be positive integers such that [math]\displaystyle{ gcd(c,d) = 1 }[/math] and [math]\displaystyle{ 0 \lt \frac{c}{d} \leq 1 }[/math]. A [math]\displaystyle{ \frac{c}{d} }[/math]-dominating function (partial signed dominating function) 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{ \frac{c}{d} }[/math] of the vertices [math]\displaystyle{ v \in V }[/math]. The [math]\displaystyle{ \frac{c}{d} }[/math]-domination number (partial signed domination number) of [math]\displaystyle{ G }[/math] is

[math]\displaystyle{ \gamma_{\frac{c}{d}}(G) = \min\{f(V)| f\mbox{ is a }\frac{c}{d}\mbox{-dominating function on }G\}. }[/math]