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## Discussiones Mathematicae Graph Theory

2011 | 31 | 2 | 293-312
Tytuł artykułu

### k-kernels in generalizations of transitive digraphs

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Let D be a digraph, V(D) and A(D) will denote the sets of vertices and arcs of D, respectively.
A (k,l)-kernel N of D is a k-independent set of vertices (if u,v ∈ N, u ≠ v, then d(u,v), d(v,u) ≥ k) and l-absorbent (if u ∈ V(D)-N then there exists v ∈ N such that d(u,v) ≤ l). A k-kernel is a (k,k-1)-kernel. Quasi-transitive, right-pretransitive and left-pretransitive digraphs are generalizations of transitive digraphs. In this paper the following results are proved: Let D be a right-(left-) pretransitive strong digraph such that every directed triangle of D is symmetrical, then D has a k-kernel for every integer k ≥ 3; the result is also valid for non-strong digraphs in the right-pretransitive case. We also give a proof of the fact that every quasi-transitive digraph has a (k,l)-kernel for every integers k > l ≥ 3 or k = 3 and l = 2.
Słowa kluczowe
EN
Kategorie tematyczne
Wydawca
Czasopismo
Rocznik
Tom
Numer
Strony
293-312
Opis fizyczny
Daty
wydano
2011
otrzymano
2009-11-12
poprawiono
2010-08-23
zaakceptowano
2010-08-24
Twórcy
• Instituto de Matemáticas, Universidad Nacional Autónoma de México, Ciudad Universitaria, México, D.F., C.P. 04510, México
autor
• Instituto de Matemáticas, Universidad Nacional Autónoma de México, Ciudad Universitaria, México, D.F., C.P. 04510, México
Bibliografia
• [1] J. Bang-Jensen and G. Gutin, Digraphs. Theory, Algorithms and Applications (Springer-Verlag Berlin Heidelberg New York, 2002).
• [2] J. Bang-Jensen and J. Huang, Quasi-transitive digraphs, J. Graph Theory 20 (1995) 141-161, doi: 10.1002/jgt.3190200205.
• [3] J. Bang-Jensen and J. Huang, Kings in quasi-transitive digraphs, Discrete Math. 185 (1998) 19-27, doi: 10.1016/S0012-365X(97)00179-9.
• [4] C. Berge, Graphs (North-Holland, Amsterdam New York, 1985).
• [5] C. Berge, Some classes of perfect graphs, in: Graph Theory and Theoretical Physics (Academic Press, London, 1967) 155-165, MR 38 No. 1017.
• [6] C. Berge and P. Duchet, Recent problems and results about kernels in directed graphs, Discrete Math. 86 (1990) 27-31, doi: 10.1016/0012-365X(90)90346-J.
• [7] E. Boros and V. Gurvich, Perfect graphs are kernel solvable, Discrete Math. 159 (1996) 35-55, doi: 10.1016/0012-365X(95)00096-F.
• [8] M. Chudnovsky, N. Robertson, P. Seymour and R. Thomas, The Strong Perfect Graph Theorem, Annals of Math. 164 (2006) 51-229.
• [9] R. Diestel, Graph Theory 3rd Edition (Springer-Verlag Berlin Heidelberg New York, 2005).
• [10] P. Duchet, Graphes Noyau-Parfaits, Annals of Discrete Math. 9 (1980) 93-101, doi: 10.1016/S0167-5060(08)70041-4.
• [11] H. Galeana-Sánchez, On the existence of kernels and h-kernels in directed graphs, Discrete Math. 110 (1992) 251-255, doi: 10.1016/0012-365X(92)90713-P.
• [12] H. Galeana-Sánchez and R. Rojas-Monroy, Kernels in quasi-transitive digraphs, Discrete Math. 306 (2006) 1969-1974, doi: 10.1016/j.disc.2006.02.015.
• [13] A. Ghouila-Houri, Caractérization des graphes non orientés dont on peut orienter les arretes de maniere a obtenir le graphe dune relation dordre, Comptes Rendus de l'Académie des Sciences Paris 254 (1962) 1370-1371.
• [14] I. Golfeder, (k,l)-kernels in quasi-transitive digraphs, Instituto de Matemáticas de la Universidad Nacional Autónoma de México, Publicación preliminar 866 (2009).
• [15] M. Kucharska and M. Kwaśnik, On (k,l)-kernels of special superdigraphs of Pₘ and Cₘ, Discuss. Math. Graph Theory 21 (2001) 95-109, doi: 10.7151/dmgt.1135.
• [16] M. Kwaśnik, On (k,l)-kernels on Graphs and Their Products, Doctoral dissertation (Technical University of Wrocław, Wrocław, 1980).
• [17] M. Kwaśnik, The Generalizaton of Richardson's Theorem, Discuss. Math. 4 (1981) 11-14.
• [18] V. Neumann-Lara, Semin'ucleos de una digráfica, Anales del Instituto de Matemáticas II (1971).
• [19] M. Richardson, On Weakly Ordered Systems, Bull. Amer. Math. Soc. 52 (1946) 113-116, doi: 10.1090/S0002-9904-1946-08518-3.
• [20] W. Szumny, A. Włoch and I. Włoch, On (k,l)-kernels in D-join of digraphs, Discuss. Math. Graph Theory 27 (2007) 457-470, doi: 10.7151/dmgt.1373.
• [21] W. Szumny, A. Włoch and I. Włoch, On the existence and on the number of (k,l)-kernels in the lexicographic product of graphs, Discrete Math. 308 (2008) 4616-4624, doi: 10.1016/j.disc.2007.08.078.
• [22] J. Von Neumann and O. Morgenstern, Theory of Games and Economic Behavior (Princeton University Press, Princeton, 1953).
• [23] A. Włoch and I. Włoch, On (k,l)-kernels in generalized products, Discrete Math. 164 (1997) 295-301.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1546
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