### 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 - 2001 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Discrete Mathematics and Combinatorics

### Cite this

*Combinatorica*,

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

**Critical facets of the stable set polytope.** / Lipták, László; Lovász, L.

Research output: Contribution to journal › Article

*Combinatorica*, vol. 21, no. 1, pp. 61-88. https://doi.org/10.1007/s004930170005

}

TY - JOUR

T1 - Critical facets of the stable set polytope

AU - Lipták, László

AU - Lovász, L.

PY - 2001

Y1 - 2001

N2 - 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∈Va(υ)cursive Greek chiυ≤b of the stable set polytope of a graph G is defined by δ=∑υ∈Va(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 Kδ+2.

AB - 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∈Va(υ)cursive Greek chiυ≤b of the stable set polytope of a graph G is defined by δ=∑υ∈Va(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 Kδ+2.

UR - http://www.scopus.com/inward/record.url?scp=0035628935&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0035628935&partnerID=8YFLogxK

U2 - 10.1007/s004930170005

DO - 10.1007/s004930170005

M3 - Article

AN - SCOPUS:0035628935

VL - 21

SP - 61

EP - 88

JO - Combinatorica

JF - Combinatorica

SN - 0209-9683

IS - 1

ER -