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

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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


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.


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