On the diameter of cut polytopes

Abstract

Given an undirected graph G with node set V and edge set E, the cut polytope CUT(G) is defined as the convex hull of the incidence vectors of the cuts in G. For a given integer k∈{1,2,…,⌊|V|2⌋}, the uniform cut polytope CUT=k(G) is defined as the convex hull of the incidence vectors of the cuts which correspond to a bipartition of the node set into sets with cardinalities k and |V|−k.In this paper, we study the diameter of these two families of polytopes. For a connected graph G, we prove that the diameter of CUT(G) is at most |V|−1, improving on the bound of |E| given by F. Barahona and A.R. Mahjoub. We also give its exact value for trees and complete bipartite graphs. Then, with respect to uniform cut polytopes, we introduce sufficient conditions for adjacency of two vertices in CUT=k(G), we study some particular cases and provide some connections with other partition polytopes from the literature.