The edge from v to a child u is a bridge exactly when u's subtree cannot reach v or an ancestor of v. The point is to make the formal expression readable enough to audit line by line.
The data
Connectivity asks how many vertices or edges must be removed to disconnect a graph \(G\). Depth-first search timestamps \(\operatorname{tin}(v)\) and low-link values \(\operatorname{low}(v)\) expose local cut structure.
The typography mirrors the proof: first declare \(\mathsf D\), then state \(\mathsf C\). The symbol \(\Longrightarrow\) below is a logical dependency, not extra mathematical structure.
Derivation
Here is a concrete symbolic test. Reading it from left to right reveals which transformation is reversible and which is only an implication.
Invariant content
The formula is reusable precisely because it says which pieces are structural and which belong only to the worked example.
Scope
Bridge and articulation criteria depend on DFS-tree relationships. Applying them to an arbitrary spanning tree gives false positives.
The important habit is to remember what was fixed before the calculation began and what was proved only afterward. The final display preserves that order.