Strongly connected orientations of undirected graphs
If $uv$ is a bridge then any orientation of&nbsp;$G$ either includes the edge $u\to v$ and has no path from $v$ to&nbsp;$u$, or vice-versa.

If $G$ has no bridge, then it is $2$-connected. A graph is $2$-connected if, and only if, it can be decomposed as $C\cup P_1 \cup\dots \cup P_k$ where $C$&nbsp;is a cycle and each $P_i$&nbsp;is a path (possibly just a single edge) that intersects $C\cup P_1\cup\dots\cup P_{i-1}$ at both its endpoints and nowhere else. Now construct your strongly connected orientation from this decomposition.