k-Kernels and some operations in digraphs

Hortensia Galeana-Sanchez ; Laura Pastrana

Discussiones Mathematicae Graph Theory, Tome 29 (2009), p. 39-49 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Détail

Publié le : 2009-01-01

Zbl 1213.05113

EUDML-ID : urn:eudml:doc:270303

Bibliographie

[000] [1] C. Berge, Graphs (North-Holland Mathematical Library, 6 North Holland, Amsterdam, 1985).

[001] [2] P. Duchet, Graphes noyau-parfaits, Ann. Discrete Math. 9 (1980) 93-101, doi: 10.1016/S0167-5060(08)70041-4. | Zbl 0462.05033

[002] [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. | Zbl 0607.05036

[003] [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. | Zbl 0456.05032

[004] [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. | Zbl 0729.05020

[005] [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. | Zbl 0770.05050

[006] [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. | Zbl 0593.05034

[007] [8] H. Galeana-Sánchez and V. Neumann-Lara, Extending kernel perfect digraphs to kernel perfect critical digraphs, Discrete Math. 94 (1991) 181-187, doi: 10.1016/0012-365X(91)90023-U. | Zbl 0748.05060

[008] [9] M. Harminc, On the (m,n)-basis of a digraph, Math. Slovaca 30 (1980) 401-404. | Zbl 0452.05028

[009] [10] M. Harminc, Solutions and kernels of a directed graph, Math. Slovaca 32 (1982) 262-267. | Zbl 0491.05029

[010] [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. | Zbl 0586.05018

[011] [12] M. Kucharska, On (k,l)-kernels of orientations of special graphs, Ars Combinatoria 60 (2001) 137-147. | Zbl 1068.05504

[012] [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.

[013] [14] M. Kwaśnik, The generalization of Richardson theorem, Discuss. Math. IV (1981) 11-13. | Zbl 0509.05048

[014] [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. | Zbl 0508.05038

[015] [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. | Zbl 0936.05047

[016] [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. | Zbl 0053.02903

[017] [18] M. Richardson, Solutions of irreflexive relations, Ann. Math. 58 (1953) 573-590, doi: 10.2307/1969755. | Zbl 0053.02902

[018] [19] J. Topp, Kernels of digraphs formed by some unary operations from other digraphs, J. Rostock Math. Kolloq. 21 (1982) 73-81. | Zbl 0559.05032

[019] [20] J. von Neumann and O. Morgenstern, Theory of games and economic behavior (Princeton University Press, Princeton, 1944). | Zbl 0063.05930

[020] [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.