### Abstract

A graph is called very well-covered if it is unmixed without isolated vertices such that the cardinality of each minimal vertex cover is half the number of vertices. We first prove that a very well-covered graph is Cohen-Macaulay if and only if it is vertex decomposable. Next, we show that the Castelnuovo-Mumford regularity of the quotient ring of the edge ideal of a very well-covered graph is equal to the maximum number of pairwise 3-disjoint edges.

Original language | English |
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Pages (from-to) | 2473-2480 |

Number of pages | 8 |

Journal | Journal of Pure and Applied Algebra |

Volume | 215 |

Issue number | 10 |

DOIs | |

Publication status | Published - Oct 2011 |

Externally published | Yes |

### ASJC Scopus subject areas

- Algebra and Number Theory

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## Cite this

Mahmoudi, M., Mousivand, A., Crupi, M., Rinaldo, G., Terai, N., & Yassemi, S. (2011). Vertex decomposability and regularity of very well-covered graphs.

*Journal of Pure and Applied Algebra*,*215*(10), 2473-2480. https://doi.org/10.1016/j.jpaa.2011.02.005