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2011 | 31 | 2 | 293-312
Tytuł artykułu

k-kernels in generalizations of transitive digraphs

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.
Wydawca
Rocznik
Tom
31
Numer
2
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
  • 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).
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  • [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.
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Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1546
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