Characterization of bridges

Statement

Let $G$ be a graph and let $e$ be an edge of $G$. Then $e$ is a bridge of $G$ if and only if $e$ is not an edge of any cycle of $G$.

Remarks

Proof idea: let $e = uv$. If $e$ lies on a cycle, the rest of the cycle is a path from $u$ to $v$ in $G - e$. Conversely, a path from $u$ to $v$ in $G - e$ together with $e$ gives a cycle. A consequence: a graph is a forest if and only if every edge is a bridge.

Explanations

L1 human

An edge on a cycle is never a bridge: if you delete it, you can still go around the other way. An edge on no cycle has no detour, so deleting it splits the graph.