Recognizing k-equistable Graphs in FPT Time

A graph G = (V,E) is called equistable if there exist a positive integer t and a weight function w : V → N such that S ⊆ V is a maximal stable set of G if and only if w(S) = t. Such a function w is called an equistable function of G. For a positive integer k, a graph G = (V,E) is said to be k-equistable if it admits an equistable function which is bounded by k. We prove that the problem of recognizing k-equistable graphs is fixed parameter tractable when parameterized by k, affirmatively answering a question of Levit et al. In fact, the problem admits an O(k)-vertex kernel that can be computed in linear time.