Critical Facets of the Stable Set Polytope

Dedicated to the memory of Paul Erdős

A facet of the stable set polytope of a graph G can be viewed as a generalization of the notion of an -critical graph. We extend several results from the theory of -critical graphs to facets. The defect of a nontrivial, full-dimensional facet of the stable set polytope of a graph G is defined by . We prove the upper bound for the degree of any node u in a critical facet-graph, and show that can occur only when . 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 -critical graph has defect and contains nodes of degree , then the graph is an odd subdivision of .

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Received October 23, 1998

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Lipták, L., Lovász, L. Critical Facets of the Stable Set Polytope. Combinatorica 21, 61–88 (2001).

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  • AMS Subject Classification (2000) Classes:  05C69; 90C57