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 . As an application of these techniques we sharpen a result of Surányi  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.
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics
- Computational Mathematics