Recognizing k-equistable Graphs in FPT Time

Abstract

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 $w(S) = t$. 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 $O(k^5)$-vertex kernel that can be computed in linear time.

This work is supported in part by the Slovenian Research Agency (I0-0035, research program P1-0285 and research projects N1-0032, J1-5433, J1-6720, and J1-6743) and by the bilateral project BI-FR/15–16–PROTEUS–003.