Cut polytopes of minor-free graphs

The cut polytope of a graph G is the convex hull of the indicator vectors of all cuts in G and is closely related to the MAxCUT problem. We give the facet-description of cut polytopes of K-3,K-3-minor-free graphs and introduce an algorithm solving MAxCUT on those graphs, which only requires the running time of planar MAxCUT. Moreover, starting a systematic geometric study of cut polytopes, we classify graphs admitting a simple or simplicial cut polytope.