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In this paper, we prove the following sufficient condition for the existence of k-kernels in digraphs: Let D be a digraph whose asymmetrical part is strongly conneted and such that every directed triangle has at least two symmetrical arcs. If every directed cycle γ of D with l(γ) ≢ 0 (mod k), k ≥ 2 satisfies at least one of the following properties: (a) γ has two symmetrical arcs, (b) γ has four short chords. Then D has a k-kernel.
This result generalizes some previous results on the existence of kernels and k-kernels in digraphs. In particular, it generalizes the following Theorem of M. Kwaśnik [5]: Let D be a strongly connected digraph, if every directed cycle of D has length ≡ 0 (mod k), k ≥ 2. Then D has a k-kernel.
This result generalizes some previous results on the existence of kernels and k-kernels in digraphs. In particular, it generalizes the following Theorem of M. Kwaśnik [5]: Let D be a strongly connected digraph, if every directed cycle of D has length ≡ 0 (mod k), k ≥ 2. Then D has a k-kernel.
Kategorie tematyczne
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Czasopismo
Rocznik
Tom
Numer
Strony
197-204
Opis fizyczny
Daty
wydano
1998
otrzymano
1997-11-17
poprawiono
1998-03-10
Twórcy
autor
- Instituto de Matemáticas, UNAM, Ciudad Universitaria, Circuito Exterior, 04510 México, D.F., México
autor
- Departamento de Matemáticas, Facultad de Ciencias, UNAM, Ciudad Universitaria, Circuito Exterior, 04510 México, D.F., México
Bibliografia
- [1] C. Berge, Graphs and hypergraphs (North-Holland, Amsterdan, 1973).
- [2] P. Duchet, Graphes Noyau-Porfaits, Ann. Discrete Math. 9 (1980) 93-101, doi: 10.1016/S0167-5060(08)70041-4.
- [3] P. Duchet, A sufficient condition for a digraph to be kernel-perfect, J. Graph Theory 11 (1987) 81-85, doi: 10.1002/jgt.3190110112.
- [4] H. Galeana-Sánchez, On the existence of kernels and k-kernels in directed graphs, Discrete Math. 110 (1992) 251-255, doi: 10.1016/0012-365X(92)90713-P.
- [5] M. Kwaśnik, The generalization of Richardson theorem, Discussiones Math. IV (1981) 11-14.
- [6] M. Kwaśnik, On (k,l)-kernels of exclusive disjunction, cartesian sum and normal product of two directed graphs, Discussiones Math. V (1982) 29-34.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1075