Authors’ abstract: The intersection digraph of a family of ordered pairs of sets \(\{(S_ v,T_ v):v \in V\}\) is the digraph \(D(V,E)\) such that \(uv \in E\) if and only if \(S_ u \cap T_ v \neq \varnothing\). Interval 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.