Local Maximum Stable Sets Greedoids Stemmed from Very Well-Covered Graphs

TL;DR

This paper investigates the conditions under which the family of local maximum stable sets in very well-covered graphs forms a greedoid, establishing a precise characterization involving unique perfect matchings.

Contribution

It provides a necessary and sufficient condition for local maximum stable sets to form a greedoid in very well-covered graphs, linking it to the uniqueness of perfect matchings.

Findings

Local maximum stable sets form a greedoid in very well-covered graphs with a unique perfect matching.

The characterization extends known results from forests to a broader class of graphs.

The paper clarifies the structural properties needed for greedoid formation in this context.

Abstract

A maximum stable set in a graph G is a stable set of maximum cardinality. S is called a local maximum stable set of G if S is a maximum stable set of the subgraph induced by the closed neighborhood of S. A greedoid (V,F) is called a local maximum stable set greedoid if there exists a graph G=(V,E) such that its family of local maximum stable sets coinsides with (V,F). It has been shown that the family local maximum stable sets of a forest T forms a greedoid on its vertex set. In this paper we demonstrate that if G is a very well-covered graph, then its family of local maximum stable sets is a greedoid if and only if G has a unique perfect matching.