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Stable sets and polynomials. (English) Zbl 0792.05082

The author surveys various applications of methods involving nonlinear commutative algebra to the stable set problem for graphs. In particular, he discusses a procedure for generating the facets of the stable set polytope. If a class of graphs \(G\) is such that all the facets of the stable set polytopes can be generated this way in a bounded number of steps, then the stability numbers of these graphs \(G\) are computable in polynomial time. Perfect, \(t\)-perfect, and \(h\)-perfect graphs have this property.

MSC:

05C35 Extremal problems in graph theory
05C15 Coloring of graphs and hypergraphs
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[1] Alon, N.; Tarsi, M., Colorings and orientations of graphs, Combinatorica, 12, 125-134 (1992) · Zbl 0756.05049
[2] Andrásfai, B., On critical graphs, (Rosenstiehl, P., Int. Symp. on Theory of Graphs, Rome (1966), Gordon and Breach: Gordon and Breach New York), 9-19 · Zbl 0182.58001
[3] Balas, E.; Pulleyblank, W. R., The perfect matchable subgraph polytope of an arbitrary graph, Combinatorica, 9, 321-337 (1989) · Zbl 0723.05087
[4] Barahona, F.; Mahjoub, A. R., Compositions of graphs and polyhedra II: stable sets, (Res. Report 87464-OR (1987), Univ. of Bonn) · Zbl 0802.05068
[5] Berge, C., Graphs and Hypergraphs (1973), North-Holland: North-Holland Amsterdam · Zbl 0483.05029
[6] Chvátal, V., On certain polytopes associated with graphs, J. Combin. Theory Ser. B, 13, 138-154 (1975) · Zbl 0277.05139
[7] Erdős, P.; Gallai, T., On the minimal number of vertices representing the edges of a graph, MTA Mat. Kut. Int. Közl., 6, 181-203 (1961) · Zbl 0101.41001
[8] Erdős, P.; Hajnal, A.; Moon, J., A problem in graph theory, Amer. Math. Monthly, 71, 1107-1110 (1964) · Zbl 0126.39401
[9] Grötschel, M.; Lovász, L.; Schrijver, A., Relaxations of vertex packing, J. Combin. Theory Ser. B, 40, 330-343 (1986) · Zbl 0596.05052
[10] Grötschel, M.; Lovász, L.; Schrijver, A., Geometric Algorithms and Combinatorial Optimization (1988), Springer: Springer Berlin · Zbl 0634.05001
[11] Hajnal, A., A theorem on \(k\)-saturated graphs, Canad. J. Math., 17, 720-724 (1965) · Zbl 0129.39901
[12] König, D., Theorie der endlichen und unendlichen Graphen (1936), Akademische Verlagsgesellschaft: Akademische Verlagsgesellschaft Leipzig
[13] Li, S. R.; Li, W. W., Independence numbers of graphs and generators of ideals, Combinatorica, 1, 55-61 (1981) · Zbl 0524.05037
[14] Liu, W., Extended formulations and polyhedral projection, (Thesis (1988), Univ. of Waterloo)
[15] Lovász, L., Flats in matroids and geometric graphs, in: Combinatorial Surveys, (Proc. 6th British Comb. Conf. (1977), Academic Press: Academic Press New York), 45-86 · Zbl 0361.05027
[16] Lovász, L., Some finite basis theorems in graph theory, in: Combinatorics, Coll. Math. Soc. J. Bolyai, Vol. 18, 717-729 (1978) · Zbl 0384.05022
[17] Lovász, L., On the Shannon capacity of graphs, IEEE Trans. Inform. Theory, 25, 1-7 (1979) · Zbl 0395.94021
[18] Lovász, L.; Plummer, M. D., Matching Theory (1986), Akadémiai Kiadó: Akadémiai Kiadó Budapest, (North-Holland, Amsterdam) · Zbl 0618.05001
[19] L. Lovász and A. Schrijver, Matrix cones, projection representations, and stable set polyhedra, in: Polyhedral Combinatorics, DIMACS Series in Discrete Math. and Theor. Comp. Sci. I: 1-17.; L. Lovász and A. Schrijver, Matrix cones, projection representations, and stable set polyhedra, in: Polyhedral Combinatorics, DIMACS Series in Discrete Math. and Theor. Comp. Sci. I: 1-17. · Zbl 0745.05024
[20] Lovász, L.; Schrijver, A., Cones of matrices and setfunctions, and 0-1 optimization (A. Schrijver), SIAM J. Optim., 1, 166-190 (1990) · Zbl 0754.90039
[21] Motzkin, T. S.; Strauss, E. G., Maxima for graphs and a new proof of a theorem of Turán, Canad. J. Math., 17, 533-540 (1965) · Zbl 0129.39902
[22] Sewell, E. C., Stability critical graphs and the stable set polytope, (Res. Report 90-11 (1990), Cornell Comp. Opt. Project) · Zbl 0838.05068
[23] Sherali, H. D.; Adams, W. P., A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems, Siam J. Discrete Math., 3, 411-430 (1990) · Zbl 0712.90050
[24] Stanley, R. P., Combinatorics and Commutative Algebra (1983), Birkhäuser: Birkhäuser Boston · Zbl 0537.13009
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