Very Well-Covered Graphs of Girth at least Four and Local Maximum Stable Set Greedoids

TL;DR

This paper characterizes when the family of local maximum stable sets in very well-covered graphs with girth at least four forms a greedoid, establishing a precise condition related to the uniqueness of perfect matchings.

Contribution

It provides a complete characterization of when a(G) is a greedoid for very well-covered graphs of girth at least four, linking it to the uniqueness of perfect matchings.

Findings

a(G) is a greedoid if and only if G has a unique perfect matching.

The result applies specifically to very well-covered graphs with girth .

The paper extends previous work on well-covered and bipartite graphs to this new class.

Abstract

A \textit{maximum stable set} in a graph $G$ is a stable set of maximum cardinality. $S$ is a \textit{local maximum stable set} of $G$, and we write $S\in\Psi(G)$, if $S$ is a maximum stable set of the subgraph induced by $S\cup N(S)$, where $N(S)$ is the neighborhood of $S$. Nemhauser and Trotter Jr. (1975), proved that any $S\in\Psi(G)$ is a subset of a maximum stable set of $G$. In (Levit & Mandrescu, 2002) we have shown that the family $\Psi(T)$ of a forest $T$ forms a greedoid on its vertex set. The cases where $G$ is bipartite, triangle-free, well-covered, while $\Psi(G)$ is a greedoid, were analyzed in (Levit & Mandrescu, 2002),(Levit & Mandrescu, 2004),(Levit & Mandrescu, 2007), respectively. In this paper we demonstrate that if $G$ is a very well-covered graph of girth $\geq4$, then the family $\Psi(G)$ is a greedoid if and only if $G$ has a unique perfect matching.