Critical sets, crowns, and local maximum independent sets
Vadim E. Levit, Eugen Mandrescu

TL;DR
This paper explores the relationships between critical, crown, and local maximum independent sets in graphs, establishing subset relations and identifying classes where these sets coincide and form structured set systems.
Contribution
It proves that critical independent sets are always crowns, which are always local maximum independent sets, and identifies classes where these families coincide and form greedoids or augmentoids.
Findings
Critical independent sets are subsets of crowns.
Crowns are subsets of local maximum independent sets.
Certain graph classes have these families coinciding and forming structured set systems.
Abstract
A set is independent (or stable) if no two vertices from are adjacent, and by we mean the set of all independent sets of . A set is critical (and we write ) if , where denotes the neighborhood of . If and there is a matching from into , then is a crown, and we write . Let be the family of all local maximum independent sets of graph , i.e., if is a maximum independent set in the subgraph induced by . In this paper we show that are true for every graph. In addition, we present some classes of graphs where these…
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