I'm wondering if there is an established name for vertices of a finite directed graph that are reachable from a directed cycle. These also can be described as endpoints of arbitratily long directed paths or even as endpoints of directed paths that are as long as the number of vertices in the graph. They are also the endpoints of a left infinite directed paths in the graph.
These crop up a lot in what I am working on now and so I need to name them. I was going to call them essential vertices but I prefer to use established nomenclature if possible.
In my context the digraphs have no sinks and so these vertices are on bi-infinite paths.