Let's define a directed graph as a 4-tuple $E = (E^0, E^1, r, s)$ where $E^0$ is the set of vertices, $E^1$ is the set of edges, $r: E^1 \to E^0$ is a map assigning a range to every edge, and $s: E^1 \to E^0$ is a map assigning a source to every edge. A sink is a vertex $v$ for which $s^{-1}(v) = \varnothing$.
I'm looking for a standard name for this kind of directed tree graph:
- $E^0$ is finite
- $E$ has exactly one sink $v_0 \in E^0$.
- $E$ is connected (this means the corresponding undirected graph is connected).
- $E$ is acyclic (this means there are no directed cycles).
- $|s^{-1}(v)| = 1$ for every $v \in E^0 \smallsetminus \{v_0\}$.
- $|r^{-1}(v_0)| \ge 1$.
For example
$$\begin{align} \bullet \longrightarrow \bullet \longrightarrow &\bullet \longrightarrow \bullet^{v_0} \\& \underset{\bullet}\uparrow \end{align}$$.
I think it is possible to omit the connectedness condition (it is implied by its being acyclic and having only one sink). Are there any other conditions which may safely be omitted and deduced from the others?
I just discovered that if you reverse all of your edges, you have an arborescence, which according to the article is also called a directed rooted tree, an out-arborescence, and an out-tree. The article notes that what you have has been called an in-tree and an anti-arborescence, among other things.