Graphic vertices of the metric polytope

The metric polytope ,//~. is defined by the triangle inequalities: xij-Xik-Xjk <~ 0 and xii + Xik + Xjk ~< 2 for all triples i,j, k of(1 ..... n}. The integral vertices of ~¢¢~. are the incidence vectors of the cuts of the complete graph Kn. Therefore, ~¢~. is a relaxation of the cut polytope of K~. We study here the fractional vertices of ~¢¢~. Many of them are constructed from graphs; this is the case for the one-third-integral vertices. One-third-integral vertices are, in a sense, the simplest fractional vertices of ~'~, as ./t'~ has no half-integral vertices. Several constructions for one-third-integral vertices are presented. In particular, the graphic vertices arising from the suspension of a tree are characterized. We describe the symmetries of ~¢/~. and obtain that the vertices are partitioned into switching classes. With the exception of the cuts which are pairwise adjacent, it is shown that no two vertices of the same switching class are adjacent on ~¢~n. The question of adjacency of the fractional vertices to the integral ones is also addressed. All the vertices of ,//~. for n ~< 6 are described.