### Abstract

A facet of the stable set polytope of a graph G can be viewed as a generalization of the notion of an α-critical graph. We extend several results from the theory of α-critical graphs to facets. The defect of a nontrivial, full-dimensional facet ∑_{υ∈V}a(υ)cursive Greek chi_{υ}≤b of the stable set polytope of a graph G is defined by δ=∑_{υ∈V}a(υ)-2b. We prove the upper bound a(u)+δ for the degree of any node u in a critical facet-graph, and show that d(u)=2δ can occur only when δ=1. We also give a simple proof of the characterization of critical facet-graphs with defect 2 proved by Sewell [11]. As an application of these techniques we sharpen a result of Surányi [13] by showing that if an α-critical graph has defect δ and contains δ+2 nodes of degree δ+1, then the graph is an odd subdivision of K_{δ+2}.

Original language | English |
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Pages (from-to) | 61-88 |

Number of pages | 28 |

Journal | Combinatorica |

Volume | 21 |

Issue number | 1 |

DOIs | |

Publication status | Published - Dec 1 2001 |

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Computational Mathematics

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## Cite this

*Combinatorica*,

*21*(1), 61-88. https://doi.org/10.1007/s004930170005