Critical Facets of the Stable Set Polytope L´aszl´oLipt´ak L´aszl´oLov´asz Department of Mathematics Yale University New Haven, CT 06520 Dedicated to the memory of PaulErd˝os 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 ∑ v∈V a(v)x v ≤ 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 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 Sur´ anyi [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 . 1 Introduction Let G =(V,E) be a simple graph on n nodes. Let α(G) denote the maximum size of an independent set of nodes in G. The graph G is called α-critical if deleting any edge increases α(G), and (to exclude some trivial complications) G has no isolated node. Since every connected component of an α-critical graph is also α-critical, we often restrict our attention to connected α-critical graphs. The theory of α-critical graphs was initiated by Erd˝os and Gallai [4], and contains a variety of interesting structural results (see [8] for a survey). The 1