Graph Operations that are Good for Greedoids

TL;DR

This paper investigates conditions under which the family of local maximum stable sets in various graph constructions forms a greedoid, extending previous results on forests, bipartite, and triangle-free graphs.

Contribution

It provides necessary and sufficient conditions for local maximum stable sets to form greedoids in disjoint unions, Zykov sums, and corona graphs.

Findings

Characterizes when local maximum stable sets form greedoids in specific graph operations.

Extends previous results from forests and special graph classes to more complex constructions.

Provides a unifying framework for understanding greedoid structures in graph theory.

Abstract

S is a local maximum stable set of a graph G, if the set S is a maximum stable set of the subgraph induced by its closed neighborhood. In (Levit, Mandrescu, 2002) we have proved that the family of all local maximum stable sets is a greedoid for every forest. The cases of bipartite graphs and triangle-free graphs were analyzed in (Levit, Mandrescu, 2004) and (Levit, Mandrescu, 2007), respectively. In this paper we give necessary and sufficient conditions for the family of all local maximum stable sets of a graph G to form a greedoid, where G is: (a) the disjoint union of a family of graphs; (b) the Zykov sum of a family of graphs, or (c) the corona X*{H_1,H_2,...,H_n} obtained by joining each vertex k of a graph X to all the vertices of a graph H_k.