Recognizing k-equistable graphs in FPT time

TL;DR

This paper proves that recognizing k-equistable graphs is fixed parameter tractable, providing an efficient kernelization algorithm with linear time preprocessing for the problem.

Contribution

The paper establishes that recognizing k-equistable graphs is fixed parameter tractable and presents an $O(k^5)$-vertex kernel computable in linear time.

Findings

Recognition problem is fixed parameter tractable

Existence of an $O(k^5)$-vertex kernel

Linear time kernelization algorithm

Abstract

A graph $G = (V,E)$ is called equistable if there exist a positive integer $t$ and a weight function $w : V \to \mathbb{N}$ 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.