Critical facets of the stable set polytope

László Lipták, L. Lovász

Research output: Contribution to journalArticle

12 Citations (Scopus)

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∈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.

Original languageEnglish
Pages (from-to)61-88
Number of pages28
JournalCombinatorica
Volume21
Issue number1
DOIs
Publication statusPublished - 2001

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Stable Set Polytope
Facet
Critical Graph
Defects
Graph in graph theory
Vertex of a graph
Subdivision
Odd
Upper bound

ASJC Scopus subject areas

  • Mathematics(all)
  • Discrete Mathematics and Combinatorics

Cite this

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

In: Combinatorica, Vol. 21, No. 1, 2001, p. 61-88.

Research output: Contribution to journalArticle

Lipták, László ; Lovász, L. / Critical facets of the stable set polytope. In: Combinatorica. 2001 ; Vol. 21, No. 1. pp. 61-88.
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