## 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 any one vertex leaves a graph, which is well-covered as well (Staples, 1975). A graph G belongs to class W_{n} if every n pairwise disjoint independent sets in G are included in n pairwise disjoint maximum independent sets (Staples, 1975). Clearly, W_{1} is the family of all well-covered graphs. It turns out that G∈W_{2} 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 independent set. 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.

Original language | English |
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Pages (from-to) | 261-272 |

Number of pages | 12 |

Journal | European Journal of Combinatorics |

Volume | 80 |

DOIs | |

State | Published - Aug 2019 |

Externally published | Yes |

### Bibliographical note

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