how to define this directed graph satisfying these conditions?

Question

I want to know the definition of a type of directed graph that satisfies these conditions:

1) this is a directed graph;

2) there is a directed spanning tree in this graph;

3) there is not any directed circle.

Here is an example:

Is there a mathematical term that can define this graph?

I found two notions: "directed tree" and "aborescence". But both the two require that there is exactly one directed path from the root to any other node. So they are not appropriate. Do you have other ideas?

Thanks.

Answer 1

There isn't necessarily a single term that satisfies all of your criteria under all discussions.

https://en.wikipedia.org/wiki/Rooted_graph

Rooted DAG (directed acyclic graph), often (almost always) has the condition that the graph is reachable from the root vertex (you would technically qualify that the first time in a paper if you wanted to be formal).

Alternatively it's a DAG with a topological order, although that admits multiple roots (which may be something you want). https://en.wikipedia.org/wiki/Topological_sorting

Technically if you have a tree, you have no cycles, so there is only one path, what you are describing here is a DAG with additional properties.