Every Path Goes Through

It links each vertex to its immediate dominator, the closest vertex through which every $s$-to-$v$ path passes. A single DFS with low-link values builds it in $O(n+m)$, and the subtree of $v$ is exactly the set of vertices that become unreachable once $v$ is removed.

It links each vertex to its immediate dominator, the closest vertex through which every $s$-to-$v$ path passes. Lengauer-Tarjan builds it in $O(m,\alpha(n,m))$, and the subtree of $v$ is exactly the set of vertices that become unreachable once $v$ is removed.

It links each vertex to its parent in a BFS tree from $s$, the closest vertex from which $v$ was discovered. Lengauer-Tarjan builds it in $O(m,\alpha(n,m))$, and the subtree of $v$ is exactly the set of vertices that become unreachable once $v$ is removed.

It is the block-cut tree of the underlying undirected graph, linking each vertex to its nearest articulation point. Lengauer-Tarjan builds it in $O(m,\alpha(n,m))$, and the subtree of $v$ is exactly the set of vertices that become unreachable once $v$ is removed.