I am a software engineer. I'm dealing with a data structure which represents a digraph of a very specific structure. I am wondering if this class of graph has been identified and studied as I need to do a fair bit of work with it and would love to not reinvent the wheel if not necessary.

The structure of the graph is as follows. It is a digraph wherein each vertex has at most one edge leading from it. Note that a vertex is allowed to have zero edges leading from it as well. There is no restriction on the edges that lead into a vertex, nor is there a restriction on the graph being cyclic.