W2-graphs and shedding vertices

Vadim E. Levit, Eugen Mandrescu

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

Abstract

A graph is well-covered if all its maximal independent sets are of the same size (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 (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 (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.

Original languageEnglish
Pages (from-to)797-803
Number of pages7
JournalElectronic Notes in Discrete Mathematics
Volume61
DOIs
StatePublished - Aug 2017
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2017 Elsevier B.V.

Keywords

  • 1-well-covered graph
  • differential of a set
  • matching
  • maximum independent set
  • shedding vertex
  • well-covered graph

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