graph - How to find Minimum Directed Cycle (minimum total weight)?

Question

Here is an excise:

Let G be a weighted directed graph with n vertices and m edges, where all edges have positive weight. A directed cycle is a directe

Answer 1

Is my algorithm correct?

No. Let me give a counter example. Imagine you start DFS from u, there are two paths p1 and p2 from u to v and 1 path p3 from v back to u, p1 is shorter than p2.

Assume you start by taking the p2 path to v, and walk back to u by path p3. One cycle found but apparently it's not minimum. Then you continue exploring u by taking the p1 path, but since v is fully explored, the DFS ends without finding the minimum cycle.