Buchsbaumness of the second powers of edge ideals

Hoang, Do Trong; Trung, Trân Nam

doi:10.48550/arxiv.1606.02815

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“…Thus Theorems 4.1,5.2 provide us with a complete description of well-covered graphs of girth ≥ 6 containing a pair of disjoint maximum independent sets. The graph G is locally triangle-free if G−N [v] is triangle-free for any vertex v ∈ V (G) [18]. If α(G) ≤ 2, the structure of locally triangle-free graphs belonging to W 2 is as follows.…”

Section: Discussionmentioning

confidence: 99%

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1-well-covered graphs revisited

Levit

Mândrescu

2019

European Journal of Combinatorics

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A graph is well-covered if all its maximal independent sets are of the same size (M. D. Plummer, 1970). A well-covered graph is 1-well-covered if the deletion of every vertex leaves a graph which is well-covered as well (J. W. Staples, 1975).A graph G belongs to class Wn if every n pairwise disjoint independent sets in G are included in n pairwise disjoint maximum independent sets (J. W. Staples, 1975). Clearly, W1 is the family of all well-covered graphs. It turns out that G ∈ W2 if and only if it is a 1-well-covered graph without isolated vertices.We show that deleting a shedding vertex does not change the maximum size of a maximal independent set including a given A ∈ Ind (G) in a graph G, where Ind(G) is the family of all the independent sets. Specifically, for well-covered graphs it means that the vertex v is shedding if and only if G−v is well-covered. In addition, we provide new characterizations of 1-well-covered graphs, which we further use in building 1-well-covered graphs by corona, join, and concatenation operations.

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Section: Discussionmentioning

confidence: 99%

“…A characterization of triangle-dominating graphs (i.e., graphs where every triangle is also a dominating set) from W 2 in terms of forbidden configurations is presented in [18]. 1-well-covered circulant graphs are considered in [10].…”

Section: Introductionmentioning

confidence: 99%

1-well-covered graphs revisited

Levit

Mândrescu

2019

European Journal of Combinatorics

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Buchsbaumness of the second powers of edge ideals

Cited by 1 publication

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1-well-covered graphs revisited

1-well-covered graphs revisited

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