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Well-covered graphs and greedoids

Profile image of Eugen MandrescuEugen Mandrescu

2008, … of the fourteenth symposium on Computing …

Abstract

G is a well-covered graph provided all its maximal stable sets are of the same size (Plummer, 1970). S is a local maximum stable set of G, and we denote by S∈ Ψ (G), if S is a maximum stable set of the subgraph induced by S∪ N (S), where N (S) is the neighborhood of S.

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Interval greedoids and families of local maximum stable sets
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A maximum stable set in a graph G is a stable set of maximum cardinality. S is a local maximum stable set of G, and we write S ∈ Ψ(G), if S is a maximum stable set of the subgraph induced by S ∪ N (S), where N (S) is the neighborhood of S. Nemhauser and Trotter Jr. [21], proved that any S ∈ Ψ(G) is a subset of a maximum stable set of G. In [14] we have shown that the family Ψ(T) of a forest T forms a greedoid on its vertex set. The cases where G is bipartite, triangle-free, well-covered, while Ψ(G) is a greedoid, were analyzed in [15], [16], [18], respectively. In this paper we demonstrate that if the family Ψ(G) of the graph G satisfies the accessibility property, then Ψ(G) forms an interval greedoid on its vertex set. We also characterize those graphs whose families of local maximum stable sets are either antimatroids or matroids.

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A new Greedoid: the family of local maximum stable sets of a forest* 1
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Greedoids on vertex sets of unicycle graphs
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A maximum stable set in a graph G is a stable set of maximum size. S is a local maximum stable set of G, and we write S ∈ Ψ(G), if S is a maximum stable set of the subgraph spanned by S ∪ N (S), where N (S) is the neighborhood of S. G is a unicycle graph if it owns only one cycle. In we have shown that the family Ψ(T ) of a forest T forms a greedoid on its vertex set. Bipartite, triangle-free, and well-covered graphs G whose Ψ(G) form greedoids were analyzed in , respectively.

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A new Greedoid: the family of local maximum stable sets of a forest
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A maximum stable set in a graph G is a stable set of maximum cardinality. S is a local maximum stable set if it is a maximum stable set of the subgraph of G spanned by S ∪ N (S), where N (S) is the neighborhood of S. One theorem of Nemhauser and Trotter Jr. [10], working as a useful sufficient local optimality condition for the weighted maximum stable set problem, ensures that any local maximum stable set of G can be enlarged to a maximum stable set of G. In this paper we demonstrate that an inverse assertion is true for forests. Namely, we show that for any non-empty local maximum stable set S of a forest T there exists a local maximum stable set S1 of T , such that S1 ⊂ S and |S1| = |S| − 1. Moreover, as a further strengthening of both the theorem of Nemhauser and Trotter Jr. and its inverse, we prove that the family of all local maximum stable sets of a forest forms a greedoid on its vertex set.

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References (30)

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Related topics

  • Mathematics
  • Computer Science
  • Combinatorics
  • Stable Set
  • Bipartite Graph
  • Induced Subgraph
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