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 $K_{3,3}$-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.Comment: 15 pages, 2 figures, 1 tabl

Properties of normal cut polytopes

Motivated by a conjecture of Sturmfels and Sullivant we study normal cut polytopes. After a brief survey of known results for normal cut polytopes it is in particular shown that for simplicial and simple cut polytopes their cut algebras are normal and hence Cohen-Macaulay. Moreover, seminormality is considered. A proof is presented for the fact that the cut algebra of $K_5$ is not seminormal, not normal, and not Cohen-Macaulay. For normal Gorenstein cut algebras and other cases of interest we determine their canonical modules. The Castelnuovo-Mumford regularity of a cut algebra is computed for various types of graphs and bounds for it are provided if normality is assumed. As an application we classify all graphs for which the cut algebra has regularity less than or equal to 4.Comment: 23 pages; revised version, which completes the discussion of seminormal cut polytopes related to recent work of Lason and Michale