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Discussiones Mathematicae Graph Theory 29(2) (2009) 313-335
DOI: https://doi.org/10.7151/dmgt.1449

DIRECTED HYPERGRAPHS: A TOOL FOR RESEARCHING DIGRAPHS AND HYPERGRAPHS

Hortensia Galeana-Sánchez  and  Martín Manrique

Instituto de Matemáticas
Universidad Nacional Autónoma de México
Ciudad Universitaria, México, D.F., 04510, Mexico

e-mail:	hgaleana@matem.unam.mx
e-mail:	martin@matem.unam.mx
	martin.manrique@gmail.com

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.

Keywords: hypergraph, strongly independent set, transversal set, kernel.

2000 Mathematics Subject Classification: 05C20, 05C65, 05C69.

References

[1]	J. Bang-Jensen and G. Gutin, Digraphs. Theory, Algorithms and Applications (Springer-Verlag London, London, UK, 2001).
[2]	C. Berge, The Theory of Graphs (Dover Publications, New York, USA, 2001).
[3]	C. Berge, Hypergraphs. Combinatorics of Finite Sets (Elsevier Science Publishers, Amsterdam, Holland, 1989).
[4]	J.A. Bondy and U.S.R. Murty, Graph Theory with Applications (MacMillan Press, London, UK, 1976).
[5]	M. Borowiecki, Connected Bijection Method in Hypergraph Theory and Some Results Concerning the Structure of Graphs and Hypergraphs (Wydawnictwo Uczelniane, Zielona Góra, Poland, 1979).
[6]	G. Chartrand and L. Lesniak, Graphs and Digraphs (Wadsworth Inc, Belmont, USA, 1986).
[7]	P. Duchet, Graphes noyau-parfaites, Ann. Discrete Math. 9 (1980) 93-101, doi: 10.1016/S0167-5060(08)70041-4.
[8]	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.
[9]	H. Galeana-Sánchez and V. Neumann-Lara, On Kernels and Semikernels of Digraphs, Discrete Math. 48 (1984) 67-76, doi: 10.1016/0012-365X(84)90131-6.
[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.
[11]	T. Haynes, S. Hedetniemi and P. Slater, Domination in Graphs (Marcel Dekker Inc. New York, USA, 1998).
[12]	V. Neumann-Lara, Seminúcleos de una digráfica, An. Inst. Mat. UNAM, México, II (1984) 67-76.
[13]	M. Richardson, Solutions of Irreflexive Relations, Ann. Math. USA 58 (1953) p. 573, doi: 10.2307/1969755.
[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.

Received 26 June 2007
Revised 8 June 2009
Accepted 8 June 2009