go-rollback-dynamic-vertex-disjoint-path-certificates

Answer

I've produced the solution. Here's a summary of what I did and delivered.

Deliverable

~/solution.md — a self-contained markdown with title, root-cause analysis, exact fix (full Go code), and verification.

Key findings (root cause)

The task is vertex 2-connectivity (biconnectivity), and the natural rollback-DSU toolkit fails for three independent reasons:

1. A vertex DSU collapses blocks that only share an articulation point. Biconnected components partition edges, not vertices. Example {2,4}×2, {1,4}×3, {2,3} is falsely answered yes for (1,2) even though 4 is a cut vertex.
2. The vertex-splitting reduction to undirected edge connectivity is invalid. Splitting each vertex into in/out and each undirected edge into two undirected edges lets one original edge be traversed in opposite directions, creating artificial cycles (false certificate).
3. Spanning-forest path merging can leave the block. For vertex biconnectivity the tree path between two vertices of a block can be routed through an articulation point outside the block, so merging the tree path is wrong.

Exact fix

• Offline time-segment tree over the operation indices (edge instances get [add, delete) intervals), so adversarial deletions and arbitrary rollback are handled without rebuilding.
• Rollback block-cut forest with explicit block nodes and shared articulation vertices, keeping every tree root a vertex and handling the subtle case where LCA(u,v) is itself a block (par[B*] = par[L]).
• Certificate extraction via a 2-unit vertex-capacitated max-flow on the directed split graph (the only correct split).

Verification

• Built and ran the clean 3-file solution; sample and trap regressions match expected output.
• ./rollback selftest runs 5000 randomized suites (self-loops, multi-edges, adversarial deletes, articulation points, rollback pressure), comparing every answer against a brute-force max-flow oracle and validating every certificate. All pass deterministically across repeated runs.
• During development I isolated the nondeterministic failure (map-iteration order exposing the LCA-is-a-block bug) and fixed it; the final code is order-independent.