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## Discussiones Mathematicae Graph Theory

2009 | 29 | 1 | 39-49
Tytuł artykułu

### k-Kernels and some operations in digraphs

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Let D be a digraph. V(D) denotes the set of vertices of D; a set N ⊆ V(D) is said to be a k-kernel of D if it satisfies the following two conditions: for every pair of different vertices u,v ∈ N it holds that every directed path between them has length at least k and for every vertex x ∈ V(D)-N there is a vertex y ∈ N such that there is an xy-directed path of length at most k-1. In this paper, we consider some operations on digraphs and prove the existence of k-kernels in digraphs formed by these operations from another digraphs.
Słowa kluczowe
EN
Kategorie tematyczne
Wydawca
Czasopismo
Rocznik
Tom
Numer
Strony
39-49
Opis fizyczny
Daty
wydano
2009
otrzymano
2007-09-12
poprawiono
2007-12-08
zaakceptowano
2008-12-29
Twórcy
• Instituto de Matemáticas, Universidad Nacional Autónoma de México, Ciudad Universitaria, México, D.F. 04510, México
autor
• Facultad de Ciencias, Universidad Nacional Autónoma de México, Ciudad Universitaria, Circuito Exterior, México, D.F. 04510, México
Bibliografia
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• [2] P. Duchet, Graphes noyau-parfaits, 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] P. Duchet and H. Meyniel, A note on kernel-critical graphs, Discrete Math. 33 (1981) 103-105, doi: 10.1016/0012-365X(81)90264-8.
• [5] H. Galeana-Sánchez, On the existence of (k,l)-kernels in digraphs, Discrete Math. 85 (1990) 99-102, doi: 10.1016/0012-365X(90)90167-G.
• [6] 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.
• [7] H. Galeana-Sánchez and V. Neumann-Lara, On kernel-perfect critical digraphs, Discrete Math. 59 (1986) 257-265, doi: 10.1016/0012-365X(86)90172-X.
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• [9] M. Harminc, On the (m,n)-basis of a digraph, Math. Slovaca 30 (1980) 401-404.
• [10] M. Harminc, Solutions and kernels of a directed graph, Math. Slovaca 32 (1982) 262-267.
• [11] M. Harminc and T. Olejníková, Binary operations on directed graphs and their solutions (Slovak with English and Russian summaries), Zb. Ved. Pr. VST, Košice (1984) 29-42.
• [12] M. Kucharska, On (k,l)-kernels of orientations of special graphs, Ars Combinatoria 60 (2001) 137-147.
• [13] M. Kucharska and M. Kwaśnik, On (k,l)-kernels of superdigraphs of Pₘ and Cₘ, Discuss. Math. Graph Theory 21 (2001) 95-109, doi: 10.7151/dmgt.1135.
• [14] M. Kwaśnik, The generalization of Richardson theorem, Discuss. Math. IV (1981) 11-13.
• [15] M. Kwaśnik, On (k,l)-kernels of exclusive disjunction, Cartesian sum and normal product of two directed graphs, Discuss. Math. V (1982) 29-34.
• [16] M. Blidia, P. Duchet, H. Jacob, F. Maffray and H. Meyniel, Some operations preserving the existence of kernels, Discrete Math. 205 (1999) 211-216, doi: 10.1016/S0012-365X(99)00026-6.
• [17] M. Richardson, Extensions theorems for solutions of irreflexive relations, Proc. Mat. Acad. Sci. 39 (1953) 649-655, doi: 10.1073/pnas.39.7.649.
• [18] M. Richardson, Solutions of irreflexive relations, Ann. Math. 58 (1953) 573-590, doi: 10.2307/1969755.
• [19] J. Topp, Kernels of digraphs formed by some unary operations from other digraphs, J. Rostock Math. Kolloq. 21 (1982) 73-81.
• [20] J. von Neumann and O. Morgenstern, Theory of games and economic behavior (Princeton University Press, Princeton, 1944).
• [21] A. Włoch and I. Włoch, On (k,l)-kernels in generalized products, Discrete Math. 164 (1997) 295-301, doi: 10.1016/S0012-365X(96)00064-7.
Typ dokumentu
Bibliografia
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