One-third-integrality in the max-cut problem

Given a graph G = (V; E), the metric polytope S(G) is deened by the inequalities x(F) ? x(C n F) jFj ? 1 for F C; jFj odd ; C cycle of G, and 0 x e 1 for e 2 E. Optimization over S(G) provides an approximation for the max-cut problem. The graph G is called 1 d-integral if all the vertices of S(G) have their coordinates in f i d j 0 i dg. We prove that the class of 1 d-integral graphs is closed under minors, and we present several minimal forbidden minors for 1 3-integrality. In particular, we characterize the 1 3-integral graphs on 7 nodes. We study several operations preserving 1 d-integrality, in particular, the k-sum operation for 0 k 3. We prove that series parallel graphs are characterized by the following stronger property. All vertices of the polytope S(G)\fx j ` x ug are 1 3-integral for every choice of 1 3-integral bounds`, u on the edges of G.