On the cut polytope.

*(English)*Zbl 0616.90058The 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 |

##### Keywords:

max cut problem; facets of polyhedra; polyhedral combinatorics; cut polytope; facet-defining inequalities; adjacency
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\textit{F. Barahona} and \textit{A. R. Mahjoub}, Math. Program. 36, 157--173 (1986; Zbl 0616.90058)

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##### References:

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