Facets for the cut cone I

We study facets of the cut cone C,, i.e., the cone of dimension 1⁄2n(n 1) generated by the cuts of the complete graph on n vertices. Actually, the study of the facets of the cut cone is equivalent in some sense to the study of the facets of the cut polytope. We present several operations on facets and, in particular, a "'lifting" procedure for constructing facets of C~ +1 from given facets of the lower dimensional cone C A. After reviewing hypermetric valid inequalities, we describe the new class of cycle inequalities and prove the facet property for several subclasses. The new class of parachute facets is developed and other known facets and valid inequalities are presented.