I came up with this problem myself just now, but I don't know how to solve it besides brutal implement. Here's the thing.
Assume there're n vertexes, they form a number of trees,(initially n). Then process three kinds of operations q+c+m times.
Operation 1: query x: print the current root of the tree which contains vertex x.
Operation 2: cut x: cut subtree x from it's original tree, keep its structure and make it become a new tree.
Operation 3: merge x y: let tree x become a subtree under vertex y.
Let's say n<2e5 and q+c+m<2e5, what's the best solution can you get?
just link cut tree $$$O(N*log^2N)$$$