A single depth-first search collects all the information needed to answer the queries. While searching, we store the following numbers for each vertex:

• Discovery time – discrete time index at which we started processing the vertex
• Finishing time – time index when we finished processing the vertex
• Depth – the depth of the vertex in the DFS tree
• lowlink – the vertex with the least discovery time, that is reachable from the current vertex or from its descendants (via a single back edge)

These numbers allow us to construct two useful functions:

• ~\text{is_descendant}(A, B)~ – is vertex ~A~ in the subtree rooted at ~B~
• ~\text{find_related_child}(A, B)~ – when ~A~ is in vertex ~B~'s subtree, find the immediate child of ~B~ such that ~A~ is a descendant of that child (in other words, if ~B~ has multiple children, find which of those subtrees ~A~ is in)

With some case analysis, it is possible to answer the queries using the described functions.

For an in-depth discussion of depth-first search and its applications, see "Introduction to Algorithms" by Cormen et al. This problem is closely related to finding articulation points and bridges in a graph.