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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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[1] F. Barahona, ”On the computational complexity of Ising spin glass models,”Journal of Physics A Mathematics and General 15 (1982) 3241–3250.
[2] F. Barahona, ”The max cut problem in graphs not contractible toK 5,”Operations Research Letters 2 (1983) 107–111. · Zbl 0525.90094
[3] F. Barahona, M. Grötschel and A.R. Mahjoub, ”Facets of the bipartite subgraph polytope,”Mathematics of Operations Research 10 (1985) 340–358. · Zbl 0578.05056
[4] M.R. Garey and D.S. Johnson,Computers and Intractability: A Guide to the Theory of NP-Completeness (Freeman, San Francisco, 1979). · Zbl 0411.68039
[5] M. Grötschel, L. Lovász and A. Schrijver, ”The ellipsoid method and its consequences in combinatorial optimization,”Combinatorica 1 (1981) 169–197. · Zbl 0492.90056
[6] M. Grötschel and G.L. Nemhauser, ”A polynomial algorithm for the max-cut problem on graphs without long odd cycles”,Math. Programming 29 (1984) 28–40. · Zbl 0532.90074
[7] F.O. Hadlock, ”Finding a maximum cut of planar graph in polynomial time”,SIAM Journal on Computing 4 (1975) 221–225. · Zbl 0321.05120
[8] F. Harary, ”On the notion of balance of a signed graph,”The Michigan Mathematic Journal 2 (1953) 143–146. · Zbl 0056.42103
[9] P.D. Seymour, ”Matroids and multicommodity flows,”European Journal of Combinatorics 2 (1981) 257–290. · Zbl 0479.05023
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