Total dominating function

Total dominating function --- тотально доминирующая функция.

As a fractional generalization of the total dominating set, a total dominating function (TDF) of a graph [math]\displaystyle{ G = (V,E) }[/math] is defined as a function [math]\displaystyle{ f: V \rightarrow [0,1] }[/math] such that [math]\displaystyle{ \sum_{u \in N(v)} f(u) \geq 1 }[/math] for each [math]\displaystyle{ v \in V }[/math]. (Here [math]\displaystyle{ N(v) }[/math] is the open neighborhood of [math]\displaystyle{ v }[/math].) A TDF [math]\displaystyle{ f }[/math] is minimal (MTDF) if no function [math]\displaystyle{ g: V \rightarrow [0,1] }[/math] with [math]\displaystyle{ g \lt f }[/math] is also a TDF of [math]\displaystyle{ G }[/math], where [math]\displaystyle{ g \lt f }[/math] means that [math]\displaystyle{ g(v) \leq f(v) }[/math] for each [math]\displaystyle{ v \in V }[/math] and [math]\displaystyle{ f \neq g }[/math]. Obviously, an integer-valued (minimal) TDF is exactly the characteristic function of a (minimal) total dominating set.