The even and odd cut polytopes

Deza, M. and M. Laurent, The even and odd cut polytopes, Discrete Mathematics 119 (1993) 49966. The cut polytope P, is the convex hull of the incidence vectors of all cuts of the complete graph K, on n nodes. An even cut is a cut of even cardinality. For n odd, all cuts are even. For n even, we consider the even cut polytope EvP,, defined as the convex hull of the incidence vectors of all even cuts of K.. The vertex sets of both polytopes P, and EvP, come from the cycle sets of some binary matroids; a more general example is the even T-cut polytope. We study the facial structure of EvP. and of the closely related (via switching) odd cut polytope. We give the description of their symmetry groups, show that all facets of P, can be zero-lifted to EvP, for m > n + 5 and also exhibit two classes of facet inducing valid inequalities of EvP.. One is an even analogue of the hypermetric inequalities for P. and the other consists of inequalities violated by exactly one odd cut.