I'm working on a algorithm that creates a rooted directed graph where cycles exist, but no directed cycles.
So the set of vertices in this graph contain a root and leaves (just like a tree), it allows for cycles (unlike a tree), but there are no directed cycles. So a walk from the root will be guaranteed to end up in a leave vertex. But, different from a tree, there can be multiple paths from the root to a leave.
So I found that this set of graphs is a subset of rooted directed graphs, and that rooted directed trees are a subset of this set. It is not acyclic (an acyclic rooted directed graph is the same as a rooted directed tree i guess?).
Now I'm trying to document my algorithm, but I'm a bit at a loss of a good term for the graph. Is there an existing term that describes this particular subset of rooted directed graphs? Or am I just wrong is assuming that a rooted directed graph can contain directed cycles and did I already find the correct term?
There are directed acyclic graphs (DAGs, https://en.wikipedia.org/wiki/Directed_acyclic_graph), so your structure looks like a rooted DAG. Note that it seems that in “directed acyclic graph”, “acyclic” refers to directed cycles by default.