An interval digraph in relation to its associated bipartite graph

Abstract The intersection digraph of a family of ordered pairs of sets {( S v , T v ): v ∈ V } is the digraph D ( V , E ) such that uv ∈ E if and only if S u ∩ T v ≠0. Internal digraphs are those intersection digraphs for which the subsets are intervals on the real line. In a previous paper, they were characterized in terms of Ferrers digraphs and a close relationship was obtained between an interval digraph and a digraph of Ferrers dimension 2. In order to characterize a digraph D of Ferrers dimension 2, Cogis associated an undirected graph H ( D ) with D in a suitable way, the vertices of H ( D ) corresponding to the zeros of the adjacency matrix of D . He proved that D has Ferrers dimension at most 2 if and only if H ( D ) is bipartite. Depending on the above characterization, this paper first obtains some properties of a digraph of Ferrers dimension 2; then it is shown how the notion of interior edges is related to an interval digraph.