Equistable graph

Equistable graph --- эквиустойчивый граф.

A graph [math]\displaystyle{ G = (V,E) }[/math] is equistable if there is a non-negative weight function [math]\displaystyle{ w }[/math] on [math]\displaystyle{ V }[/math] such that a set [math]\displaystyle{ S \subseteq V }[/math] satisfies [math]\displaystyle{ w(S) = \sum_{v \in S} w(e) = 1 }[/math] if and only if [math]\displaystyle{ S }[/math] is maximal stable. The problem of recognizing equistable graphs in polynomial time is still open.

A graph [math]\displaystyle{ G }[/math] is strongly equistable if for each set [math]\displaystyle{ \emptyset \neq T \subseteq V }[/math] such that [math]\displaystyle{ T }[/math] is not maximal stable, and for each constant [math]\displaystyle{ c \leq 1 }[/math], there is a non-negative weight function [math]\displaystyle{ w }[/math] on [math]\displaystyle{ V }[/math] such that [math]\displaystyle{ w(S) = 1 }[/math] for each maximal stable set [math]\displaystyle{ S }[/math], and [math]\displaystyle{ w(T) \neq c. }[/math]