Line Directed Hypergraphs

Abstract

Abstract. In this article we generalize the concept of line digraphs to line dihypergraphs. We give some general properties in particular concerning connectivity parameters of dihypergraphs and their line dihypergraphs, like the fact that the arc connectivity of a line dihypergraph is greater than or equal to that of the original dihypergraph. Then we show that the De Bruijn and Kautz dihypergraphs (which are among the best known bus networks) are iterated line digraphs. Finally we give short proofs that they are highly connected.

Eulerian and Hamiltonian Dicycles in Directed Hypergraphs

“…The digraph-theoretic notions of connectivity can be extended to dihypergraphs [19]. We say that H is strongly (resp.…”

“…U (H) is weakly connected if its associated multigraph G U (H) (obtained by forgetting the orientation) is a connected multigraph (in Graph Theory this undirected graph is often called the underlying graph; we use here a different terminology as we already use the word underlying for the digraph associated to a dihypergraph). The digraph-theoretic notions of vertex-connectivity and arc-connectivity are also generalized by the dihypergraph-theoretic notions of vertex-connectivity and hyperarc-connectivity (see [19]). Unlike 1-arc connected digraphs, 1-hyperarc connected dihypergraphs are not always 1-vertex connected.…”

“…By using induction, one can prove that every digraph of the family is Hamiltonian. It was shown in [19] that de Bruijn and Kautz dihypergraphs are also stable by line dihypergraph operation. So, it is natural to wonder whether this property can be generalized to dihypergraphs.…”

“…Connectivity properties of generalized de Bruijn dihypergraphs have been studied in [19,20,21], but, to our best knowledge, their Hamiltonian and Eulerian properties have not been studied.…”