Depth-first search parenthesis property ($d(u)<d(v)<f(v)<f(u)$)

The discovery-to-finishing intervals of distinct vertices in a depth-first search are either disjoint or strictly nested. A recursive call completes before its caller returns; separate completed branches have disjoint intervals. Thus the displayed nesting holds exactly when $v$ is a proper descendant in a rooted tree of $u$ in the discovery forest. A directed path alone does not imply this nesting, because it can leave the current branch through an already discovered ancestor.