Adjacency of vertices of the complete pre-order polytope

A complete pre-order (c.p.o.) on a finite set V is a relation on V that is transitive and total. We define the c.p.o. polytope P-n as the convex hull of the incidence vectors of all c.p.o.'s on a set with n elements. We study the adjacency relation of the vertices of the c.p.o. polytope P-n, n greater than or equal to 3. The main results are: (1) if A, B are c.p.o.s with \A Delta B\=2, then x(A) and x(B) are adjacent in P-n if and only if the digraph induced by A Delta B is connected; (2) if A subset of or equal to B are c.p.o.'s then x(A) and x(B) are adjacent in P-n if and only if the digraph induced by B\A is connected; (3) the diameter of the c.p.o. polytope is equal to 2.