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 $$w : V \rightarrow \mathbb {N}$$ such that $$S \subseteq V$$ is a maximal stable set of G if and only if $$wS = t$$. 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 $$Ok^5$$-vertex kernel that can be computed in linear time.