# On the Complexity of the 3-Kernel Problem in Some Classes of Digraphs

• Volume: 34, Issue: 1, page 167-185
• ISSN: 2083-5892

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## Abstract

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Let D be a digraph with the vertex set V (D) and the arc set A(D). A subset N of V (D) is k-independent if for every pair of vertices u, v ∈ N, we have d(u, v), d(v, u) ≥ k; it is l-absorbent if for every u ∈ V (D) − N there exists v ∈ N such that d(u, v) ≤ l. A k-kernel of D is a k-independent and (k − 1)-absorbent subset of V (D). A 2-kernel is called a kernel. It is known that the problem of determining whether a digraph has a kernel (“the kernel problem”) is NP-complete, even in quite restricted families of digraphs. In this paper we analyze the computational complexity of the corresponding 3-kernel problem, restricted to three natural families of digraphs. As a consequence of one of our main results we prove that the kernel problem remains NP-complete when restricted to 3-colorable digraphs.

## How to cite

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Pavol Hell, and César Hernández-Cruz. "On the Complexity of the 3-Kernel Problem in Some Classes of Digraphs." Discussiones Mathematicae Graph Theory 34.1 (2014): 167-185. <http://eudml.org/doc/267793>.

@article{PavolHell2014,
abstract = {Let D be a digraph with the vertex set V (D) and the arc set A(D). A subset N of V (D) is k-independent if for every pair of vertices u, v ∈ N, we have d(u, v), d(v, u) ≥ k; it is l-absorbent if for every u ∈ V (D) − N there exists v ∈ N such that d(u, v) ≤ l. A k-kernel of D is a k-independent and (k − 1)-absorbent subset of V (D). A 2-kernel is called a kernel. It is known that the problem of determining whether a digraph has a kernel (“the kernel problem”) is NP-complete, even in quite restricted families of digraphs. In this paper we analyze the computational complexity of the corresponding 3-kernel problem, restricted to three natural families of digraphs. As a consequence of one of our main results we prove that the kernel problem remains NP-complete when restricted to 3-colorable digraphs.},
author = {Pavol Hell, César Hernández-Cruz},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {kernel; 3-kernel; NP-completeness; multipartite tournament; cyclically 3-partite digraphs; k-quasi-transitive digraph},
language = {eng},
number = {1},
pages = {167-185},
title = {On the Complexity of the 3-Kernel Problem in Some Classes of Digraphs},
url = {http://eudml.org/doc/267793},
volume = {34},
year = {2014},
}

TY - JOUR
AU - Pavol Hell
AU - César Hernández-Cruz
TI - On the Complexity of the 3-Kernel Problem in Some Classes of Digraphs
JO - Discussiones Mathematicae Graph Theory
PY - 2014
VL - 34
IS - 1
SP - 167
EP - 185
AB - Let D be a digraph with the vertex set V (D) and the arc set A(D). A subset N of V (D) is k-independent if for every pair of vertices u, v ∈ N, we have d(u, v), d(v, u) ≥ k; it is l-absorbent if for every u ∈ V (D) − N there exists v ∈ N such that d(u, v) ≤ l. A k-kernel of D is a k-independent and (k − 1)-absorbent subset of V (D). A 2-kernel is called a kernel. It is known that the problem of determining whether a digraph has a kernel (“the kernel problem”) is NP-complete, even in quite restricted families of digraphs. In this paper we analyze the computational complexity of the corresponding 3-kernel problem, restricted to three natural families of digraphs. As a consequence of one of our main results we prove that the kernel problem remains NP-complete when restricted to 3-colorable digraphs.
LA - eng
KW - kernel; 3-kernel; NP-completeness; multipartite tournament; cyclically 3-partite digraphs; k-quasi-transitive digraph
UR - http://eudml.org/doc/267793
ER -

## References

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