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On kernel-perfect critical digraphs. (English) Zbl 0593.05034
Let G be a digraph. An independent set N of vertices of G is called a kernel if for every vertex z of G, \(z\not\in N\), there exists a vertex \(v\in N\) such that (z,v) is an edge of G. A digraph is said to be kernel- perfect (KP) if each of its proper induced subdigraphs has a kernel. A KP digraph which has no kernel is called KP-critical. In the paper some properties of KP-critical digraphs are studied and a construction of such digraphs is given. In a addition, the sufficient conditions for a digraph to be KP are investigated.
Reviewer: P.Horák

05C20 Directed graphs (digraphs), tournaments
05C35 Extremal problems in graph theory
Full Text: DOI
[1] Berge, C., Graphs and hypergraphs, (1973), North-Holland Amsterdam · Zbl 0483.05029
[2] Duchet, P., Representation; noyaux en theorie des graphes et hypergraphes, (), Paris
[3] Duchet, P., Graphes noyau-parfaits, Ann. discrete math., 9, 93-101, (1980) · Zbl 0462.05033
[4] Duchet, P.; Meyniel, H., A note on kernel-critical graphs, Discrete math., 33, 103-105, (1981) · Zbl 0456.05032
[5] Erdös, P., Problems and results in number theory and graph theory, (), 3-21
[6] Galeana-Sánchez, H., A counterexample to a conjecture of meyniel on kernel-perfect graphs, Discrete math., 41, 105-107, (1982) · Zbl 0484.05035
[7] Galeana-Sánchez, H.; Neumann-Lara, V., On kernels and semikernels of digraphs, Discrete math., 48, 67-76, (1984) · Zbl 0529.05024
[8] H. Galeana-Sánchez and V. Neumann-Lara, Extending kernel-perfect digraphs to kernel-perfect critical digraphs, preprint. · Zbl 0748.05060
[9] Harary, F.; Norman, R.Z.; Cartwright, D., Structural models, (1965), Wiley New York · Zbl 0139.41503
[10] Jacob, H., Etude theórique du noyau d’un graphe, ()
[11] H. Meyniel, Extension du nombre chromatique et du nombre de stabilité, preprint.
[12] Neumann-Lara, V., Seminúcleos de una digráfica, An. inst. mat. univ. nac. autónoma México II, (1971)
[13] Neumann-Lara, V., The dichromatic number of a digraph, J. combin. theory ser. B, 33, 3, 265-270, (1982) · Zbl 0506.05031
[14] Richardson, M., On weakly ordered systems, Bull. amer. math. soc., 52, 113, (1946) · Zbl 0060.06506
[15] Richardson, M., Solutions of irreflexive relations, Ann. math., 58, 2, 573, (1953) · Zbl 0053.02902
[16] Richardson, M., Extension theorems for solutions of irreflexive relations, (), 649 · Zbl 0053.02903
[17] J. Von Neumann and O. Morgenstern, Theory of Games and Economic Behavior (Princeton Univ. Press, Princeton). · Zbl 0063.05930
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