Computing Well-Covered Vector Spaces of Graphs using Modular Decomposition

TL;DR

This paper introduces a polynomial-time algorithm for computing the well-covered vector space of fork-free graphs using modular decomposition, extending previous results and enabling recognition of well-covered fork-free graphs.

Contribution

The paper presents a novel polynomial-time algorithm for well-covered vector spaces in fork-free graphs, leveraging modular decomposition and Gaussian elimination.

Findings

Polynomial-time algorithm for fork-free graphs

Recognition of well-covered fork-free graphs in polynomial time

Extension of known results on cographs

Abstract

A graph is well-covered if all its maximal independent sets have the same cardinality. This well studied concept was introduced by Plummer in 1970 and naturally generalizes to the weighted case. Given a graph $G$, a real-valued vertex weight function $w$ is said to be a well-covered weighting of $G$ if all its maximal independent sets are of the same weight. The set of all well-covered weightings of a graph $G$ forms a vector space over the field of real numbers, called the well-covered vector space of $G$. Since the problem of recognizing well-covered graphs is $\mathsf{co}$-$\mathsf{NP}$-complete, the problem of computing the well-covered vector space of a given graph is $\mathsf{co}$-$\mathsf{NP}$-hard. Levit and Tankus showed in 2015 that the problem admits a polynomial-time algorithm in the class of claw-free graph. In this paper, we give two general reductions for the problem, one based on anti-neighborhoods and one based on modular decomposition, combined with Gaussian elimination. Building on these results, we develop a polynomial-time algorithm for computing the well-covered vector space of a given fork-free graph, generalizing the result of Levit and Tankus. Our approach implies that well-covered fork-free graphs can be recognized in polynomial time and also generalizes some known results on cographs.