Collapsing and lifting for the cut cone

The cut polytope P,(G) of a graph G is the convex hull of the incidence vectors of all cuts of G; the cut cone C(G) of G is the cone generated by the incidence vectors of all cuts of G. We introduce the operation of collapsing an inequality valid over the cut cone C(K,) of the complete graph with n vertices: it consists of identifying vertices and adding the weights of the corresponding incident edges. Using collapsing and its inverse operation (lifting), we give several methods to find facets of C(K,). We also show how to construct facets of C(K.) from the difference of inequalities valid over C(K,). When G is an induced subgraph of a graph H, we give sufficient conditions to derive inequalities defining facets of P,(H) from inequalities defining facets of P,(G). Finally, the description (up to permutation) of the cut cone C(K,) is given.