A characterization of interval catch digraphs (Q1118608)

In the catch digraph of a family of sets \(S_ x\) with base points \(p_ x\in S_ x\), two distinct vertices x and y are joined by an arc iff \(p_ y\in S_ x\). The author shows that a (finite or infinite) digraph is the catch digraph of a family of pointed intervals iff it contains no set of three vertices which fulfil: any two of them are weakly connected by a chain where no initial endpoint of an arc precedes the third vertex. Furthermore, he shows that all these digraphs have weakly triangulated (and thus perfect) underlying graphs.