Whitney's connectivity inequalities for directed hypergraphs

Keywords: Strong and weak vertex elimination, strong connectedness, unilateral connectedness, directed hypergraph, total degree, connectivity indices.


Whitney’s inequality established an important connection between vertex and edge connectivity and the degree of a graph, which was later generalized to digraphs and to undirected hypergraphs. Here we show, using the most common definitions of connectedness for directed hypergraphs, that an analogous result holds directed hypergraphs. It relates the vertex connectivity under strong vertex elimination, edge connectivity under weak edge elimination, and a suitable degree-like parameter and is a proper generalization of the situation in both digraphs and undirected hypergraphs. We furthermore relate the connectivity parameters of directed hypergraphs with those of its directed bipartite König representation.