# Directed hypergraphs: a tool for researching digraphs and hypergraphs

• Volume: 29, Issue: 2, page 313-335
• ISSN: 2083-5892

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## Abstract

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In this paper we introduce the concept of directed hypergraph. It is a generalisation of the concept of digraph and is closely related with hypergraphs. The basic idea is to take a hypergraph, partition its edges non-trivially (when possible), and give a total order to such partitions. The elements of these partitions are called levels. In order to preserve the structure of the underlying hypergraph, we ask that only vertices which belong to exactly the same edges may be in the same level of any edge they belong to. Some little adjustments are needed to avoid directed walks within a single edge of the underlying hypergraph, and to deal with isolated vertices. The concepts of independent set, absorbent set, and transversal set are inherited directly from digraphs. As a consequence of our results on this topic, we have found both a class of kernel-perfect digraphs with odd cycles and a class of hypergraphs which have a strongly independent transversal set.

## How to cite

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Hortensia Galeana-Sánchez, and Martín Manrique. "Directed hypergraphs: a tool for researching digraphs and hypergraphs." Discussiones Mathematicae Graph Theory 29.2 (2009): 313-335. <http://eudml.org/doc/270515>.

@article{HortensiaGaleana2009,
abstract = { In this paper we introduce the concept of directed hypergraph. It is a generalisation of the concept of digraph and is closely related with hypergraphs. The basic idea is to take a hypergraph, partition its edges non-trivially (when possible), and give a total order to such partitions. The elements of these partitions are called levels. In order to preserve the structure of the underlying hypergraph, we ask that only vertices which belong to exactly the same edges may be in the same level of any edge they belong to. Some little adjustments are needed to avoid directed walks within a single edge of the underlying hypergraph, and to deal with isolated vertices. The concepts of independent set, absorbent set, and transversal set are inherited directly from digraphs. As a consequence of our results on this topic, we have found both a class of kernel-perfect digraphs with odd cycles and a class of hypergraphs which have a strongly independent transversal set. },
author = {Hortensia Galeana-Sánchez, Martín Manrique},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {hypergraph; strongly independent set; transversal set; kernel},
language = {eng},
number = {2},
pages = {313-335},
title = {Directed hypergraphs: a tool for researching digraphs and hypergraphs},
url = {http://eudml.org/doc/270515},
volume = {29},
year = {2009},
}

TY - JOUR
AU - Hortensia Galeana-Sánchez
AU - Martín Manrique
TI - Directed hypergraphs: a tool for researching digraphs and hypergraphs
JO - Discussiones Mathematicae Graph Theory
PY - 2009
VL - 29
IS - 2
SP - 313
EP - 335
AB - In this paper we introduce the concept of directed hypergraph. It is a generalisation of the concept of digraph and is closely related with hypergraphs. The basic idea is to take a hypergraph, partition its edges non-trivially (when possible), and give a total order to such partitions. The elements of these partitions are called levels. In order to preserve the structure of the underlying hypergraph, we ask that only vertices which belong to exactly the same edges may be in the same level of any edge they belong to. Some little adjustments are needed to avoid directed walks within a single edge of the underlying hypergraph, and to deal with isolated vertices. The concepts of independent set, absorbent set, and transversal set are inherited directly from digraphs. As a consequence of our results on this topic, we have found both a class of kernel-perfect digraphs with odd cycles and a class of hypergraphs which have a strongly independent transversal set.
LA - eng
KW - hypergraph; strongly independent set; transversal set; kernel
UR - http://eudml.org/doc/270515
ER -

## References

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10. [10] H. Galeana-Sánchez and V. Neumann-Lara, On Kernel-imperfect Critical Digraphs, Discrete Math. 59 (1986) 257-265, doi: 10.1016/0012-365X(86)90172-X. Zbl0593.05034
11. [11] T. Haynes, S. Hedetniemi and P. Slater, Domination in Graphs (Marcel Dekker Inc. New York, USA, 1998). Zbl0890.05002
12. [12] V. Neumann-Lara, Seminúcleos de una digráfica, An. Inst. Mat. UNAM, México, II (1984) 67-76.
13. [13] M. Richardson, Solutions of Irreflexive Relations, Ann. Math. USA 58 (1953) p. 573, doi: 10.2307/1969755.
14. [14] M. Richardson, Extension Theorems for Solutions of Irreflexive Relations, Proc. Math. Acad. Sci. USA 39 (1953) p. 649, doi: 10.1073/pnas.39.7.649. Zbl0053.02903

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