Critical Facets of the Stable Set Polytope

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 ∑ v∈V a(v)xv ≤ b of the stable set polytope of a graph G is defined by δ = ∑ v∈V a(v)−2b. We prove the upper bound a(u) + δ for the degree of any node u in a critical facetgraph, 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.