Singly connected directed graphs

Question

According to the definition available in CLRS 3rd edition, a singly connected directed graph is the one where for every pair of vertices (u,v) there is at most 1 unique path from u->v. Now most of the answers that I have read, state that we run DFS from every vertex in the graph and if in any case we find a Cross edge or a Forward edge, then the graph is not singly connected. I can understand the concept for forward edges, but running this algo on this graph

1 --> 2 <-- 3 will give us the result that it is NOT singly connected whereas this graph is singly connected. We have a cross edge from 3 -> 2 or 1 -> 2 depending upon which vertext started this entire procedure (1 or 3) . If we start the DFS from vertex 2, then we have 2 cross edges 1 -> 2 and 3 -> 2.Can anyone clarify please ?

Answer 1

The answer that suggests running DFS from every node means you should stop the DFS once you cannot continue (no outgoing edges left), and not start from a different node.

In this case, in your example, you will start (w.l.o) from 1, discover 2, and you are done. No back edgees
Next, is a completely new DFS, start from 3, discover 2, and done, again without back edges.

The idea is basically verifying the attribute by definition. You do a DFS from each node u until you either find that for each v there is at most one path from u to v (DFS is exhausted) or you found at some point a 2nd path from u to v, and then you are done.