An undirected graph can only be converted to a directed graph if there is no bridge in the graph.
I can only prove one side:
Suppose there is no bridge in the graph. so for vertices u and v We can remove the edge between them. and still there are some paths (paths) in the graph that we can reach from u (so the graph is still connected). Since there is no bridge in the diagram and each vertex pair can be reached without the boundary in between, we can convert the undirected graph to a directed graph.
how about the other side? How can I prove that?
Can I say this: because the graph is undirected and can be converted to a digrap for any pair of vertices u and v There is a path (or maybe some paths) we can reach v from u and vice versa, except for the directional path between them (the edge between them). so we can remove the directional path between them. So there is no bridge edge in the graph.