On the Existence of (k,l)-Kernels in Infinite Digraphs: A Survey

H. Galeana-Sánchez; C. Hernández-Cruz

Discussiones Mathematicae Graph Theory (2014)

  • Volume: 34, Issue: 3, page 431-466
  • ISSN: 2083-5892

Abstract

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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

How to cite

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H. Galeana-Sánchez, and C. Hernández-Cruz. "On the Existence of (k,l)-Kernels in Infinite Digraphs: A Survey." Discussiones Mathematicae Graph Theory 34.3 (2014): 431-466. <http://eudml.org/doc/268081>.

@article{H2014,
abstract = {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},
author = {H. Galeana-Sánchez, C. Hernández-Cruz},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {kernel; k-kernel; (k; l)-kernel; infinite digraph; -kernel; -kernel},
language = {eng},
number = {3},
pages = {431-466},
title = {On the Existence of (k,l)-Kernels in Infinite Digraphs: A Survey},
url = {http://eudml.org/doc/268081},
volume = {34},
year = {2014},
}

TY - JOUR
AU - H. Galeana-Sánchez
AU - C. Hernández-Cruz
TI - On the Existence of (k,l)-Kernels in Infinite Digraphs: A Survey
JO - Discussiones Mathematicae Graph Theory
PY - 2014
VL - 34
IS - 3
SP - 431
EP - 466
AB - 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
LA - eng
KW - kernel; k-kernel; (k; l)-kernel; infinite digraph; -kernel; -kernel
UR - http://eudml.org/doc/268081
ER -

References

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