Hamilton Decomposition Decision Problem
If we request that each $|V_i| \ge 3$, then this is the 2-factor problem,
see the book Combinatorial Optimization by Schrijver. If you allow $|V_i| = 2$,
then we can solve this by replacing each undirected edge by two directed ones and compute what is called a cycle-cover. This can be done in polynomial time by reduction to bipartite matching.