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On the cut polytope. (English) Zbl 0616.90058
The cut polytope $$P_ C(G)$$ of a graph $$G=(V,E)$$ is the convex hull of the incidence vectors of all edge sets of cuts of G. We show some classes of facet-defining inequalities of $$P_ C(G)$$. We describe three methods with which new facet-defining inequalities of $$P_ C(G)$$ can be constructed from known ones. In particular, we show that inequalities associated with chordless cycles define facets of this polytope; moreover, for these inequalities a polynomial algorithm to solve the separation problem is presented. We characterize the facet defining inequalities of $$P_ C(G)$$ if G is not contractible to $$K_ 5$$. We give a simple characterization of adjacency in $$P_ C(G)$$ and prove that for complete graphs this polytope has diameter one and that $$P_ C(G)$$ has the Hirsch property. A relationship between $$P_ C(G)$$ and the convex hull of incidence vectors of balancing edge sets of a signed graph is studied.

##### MSC:
 90C27 Combinatorial optimization 52Bxx Polytopes and polyhedra 90C35 Programming involving graphs or networks
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##### References:
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