What's the most time efficient way to find the number of nodes reachable from each root and no other root?
A rough sketch, you can fill in the details:

Using Strongly Connected Components (SCCs), you can reduce the graph to a DAG. If you run topological sort on this DAG, all incoming edges to a node will be processed before any outgoing edges. Then just store at each node which root has reached it, or if none/more than one roots have reached it.