I would like to know if there is a name for the following thing:
Let $G$ be a directed graph, and $Y$ a subset of vertices of $G$ such that every vertex of $G$ can be reached by a path starting at a vertex in $Y$.
Is there a standard name in graph theory for a set $Y$ with this property? What if $Y$ is of minimal size?
I do not know that there is a name for such a set in the literature.
If I'm not mistaken, I think this set $Y$ will consist solely of those vertices with in-degree $0$ (since $G$ is directed). Indeed, if they have in-degree $0$ they are not reachable from any other vertex. On the other hand, if some vertex does not have in-degree $0$, it is reachable from another vertex, so it's pointless taking that vertex in $Y$.
Cycles in the graph might cause problems though, so you will need to take a vertex from each cycle too to be safe (see the example below, where the only vertex with in-degree $0$ does not suffice). Cycles poke a hole in the argument I made above, that "you can keep going back till you find a vertex of in-degree $0$".
If $G$ is undirected, the problem is trivial; $Y$ will consist of only one vertex (or one for each component if $G$ is disconnected).