Interval greedoids and families of local maximum stable sets of graphs

TL;DR

This paper explores the structure of local maximum stable sets in graphs, showing they form interval greedoids under certain conditions and characterizing graphs with matroid or antimatroid families of such sets.

Contribution

It introduces the concept of interval greedoids in the context of local maximum stable sets and characterizes graphs with matroid or antimatroid families of these sets.

Findings

Local maximum stable sets form interval greedoids under the accessibility property.

Characterization of graphs with families of local maximum stable sets as matroids or antimatroids.

Extension of previous work on greedoids in specific graph classes.

Abstract

A maximum stable set in a graph G is a stable set of maximum cardinality. S is a local maximum stable set of G, if S is a maximum stable set of the subgraph induced by its closed neighborhood. Nemhauser and Trotter Jr. proved in 1975 that any local maximum stable set is a subset of a maximum stable set of G. In 2002 we showed that the family of all local maximum stable sets of a forest forms a greedoid on its vertex set. The cases where G is bipartite, triangle-free, well-covered, while the family of all local maximum stable sets is a greedoid, were analyzed in 2004, 2007, and 2008, respectively. In this paper we demonstrate that if the family of all local maximum stable sets of the graph satisfies the accessibility property, then it is an interval greedoid. We also characterize those graphs whose families of local maximum stable sets are either antimatroids or matroids.