Term for a "rooted" directional graph? Consider an acyclic directed graph in which a traversal from any node in the graph must eventually end at some terminal node R.  Borrowing from tree-based vocabulary, I would tend call this the "root" of the graph, since all traversals eventually terminate there.  However, the term "rooted graph" seems to already be in use for something else.

Is there a term I can use to describe this property?  The best way I can describe the graph at present is a "multi-parented directed tree" -- all nodes point toward the root, but, unlike a directed tree, nodes may have more than one parent. I don't like this terminology, though, as it seems like an abuse of the term "tree".

I considered "hierarchical graph", but a node can have more than one outgoing vertex, so I think that may disqualify it from being hierarchical.
I don't know if these graphs have a common name. But I think, one would call vertices with no outgoing edge rather a sink than a root.

This paper studies acyclic orientations of grid graphs. They call an orientation, which would correspond to the graphs you consider, acyclic orientations with a unique sink.