Evaluate Division

Start by clarifying the exact contract for Evaluate Division: input representation, mutation rules, duplicate behavior, and what should happen on invalid or empty input. The core idea is weighted graph traversal with path-product invariant. Work a tiny example aloud and name the invariant after each step, because that is the proof that the approach stays correct as the input grows.

A dependable implementation plan is to add reciprocal weighted edges; reject unknown variables; traverse each query with fresh visited set; multiply weights; return failure when disconnected. Unknown x/x should not return one. Keep the code aligned with that invariant instead of adding special cases that silently change the contract. Test the smallest input, a normal case, duplicates or ties where relevant, an adversarial boundary, and a case that exercises the failure path. The expected complexity is O(V + E) per query and O(V + E) space. Explain both time and auxiliary memory, including recursion or helper structures; do not hide sorting, copying, or repeated scans. If the title omits critical constraints, say what you need to know before committing to a more specialized solution.