Geodesic
On the existence of (k,l)-kernels in infinite digraphs: A survey
Discussiones Mathematicae. Graph Theory, Tome 34 (2014) no. 3, pp. 431-466

Voir la notice de l'article provenant de la source Library of Science

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 (if u, v ∈ N, u 6= 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) set of vertices. A k-kernel is a (k, k −1)-kernel. This work is a survey of results proving sufficient conditions for the existence of (k, l)-kernels in infinite digraphs. Despite all the previous work in this direction was done for (2, 1)-kernels, we present many original results concerning (k, l)-kernels for distinct values of k and l. The original results are sufficient conditions for the existence of (k, l)- kernels in diverse families of infinite digraphs. Among the families that we study are: transitive digraphs, quasi-transitive digraphs, right/left pretransitive digraphs, cyclically k-partite digraphs, κ-strong digraphs, k-transitive digraphs, k-quasi-transitive digraphs.
Keywords: kernel, k-kernel, infinite digraph, (k, l)-kernel 
@article{DMGT_2014_34_3_a0,
     author = {Galeana-S\'anchez, H. and Hern\'andez-Cruz, C.},
     title = {On the existence of (\protect\emph{k},\protect\emph{l})-kernels in infinite digraphs: {A} survey},
     journal = {Discussiones Mathematicae. Graph Theory},
     pages = {431--466},
     publisher = {mathdoc},
     volume = {34},
     number = {3},
     year = {2014},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMGT_2014_34_3_a0/}
}
TY  - JOUR
AU  - Galeana-Sánchez, H.
AU  - Hernández-Cruz, C.
TI  - On the existence of (k,l)-kernels in infinite digraphs: A survey
JO  - Discussiones Mathematicae. Graph Theory
PY  - 2014
SP  - 431
EP  - 466
VL  - 34
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DMGT_2014_34_3_a0/
LA  - en
ID  - DMGT_2014_34_3_a0
ER  - 
%0 Journal Article
%A Galeana-Sánchez, H.
%A Hernández-Cruz, C.
%T On the existence of (k,l)-kernels in infinite digraphs: A survey
%J Discussiones Mathematicae. Graph Theory
%D 2014
%P 431-466
%V 34
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DMGT_2014_34_3_a0/
%G en
%F DMGT_2014_34_3_a0
Galeana-Sánchez, H.; Hernández-Cruz, C. On the existence of (k,l)-kernels in infinite digraphs: A survey. Discussiones Mathematicae. Graph Theory, Tome 34 (2014) no. 3, pp. 431-466. http://geodesic.mathdoc.fr/item/DMGT_2014_34_3_a0/