Critical sets, crowns, and local maximum independent sets
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A set S⊆ V(G) is independent (or stable) if no two vertices from S are adjacent, and by Ind(G) we mean the set of all independent sets of G. A set A∈Ind(G) is critical (and we write A∈ CritIndep(G)) if | A| -| N(A)| =max{| I| -| N(I)| :I∈Ind(G)}, where N(I) denotes the neighborhood of I. If S∈Ind(G) and there is a matching from N(S) into S, then S is a crown, and we write S∈ Crown(G). Let Ψ(G) be the family of all local maximum independent sets of graph G, i.e., S∈Ψ(G) if S is a maximum independent set in the subgraph induced by S∪ N(S). In this paper we show that CritIndep(G)⊆ Crown(G) ⊆Ψ(G) are true for every graph. In addition, we present some classes of graphs where these families coincide and form greedoids or even more general set systems that we call augmentoids.
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