Is there an efficient algorithm for this vertex cycle cover problem?
Let a cycle cover of a directed graph be a collection of vertex-disjoint cycles such that each vertex is in exactly one cycle. So if I understand you correctly, given a directed graph $G$, you want a maximum cycle cover, where each cycle is of length at least $k$ (perhaps for $k=2$, or $k=3$ at least). A cycle cover where each cycle is of length at least $k$ is called a $k$-cycle cover.

Deciding if a digraph $G$ has a 2-cycle cover is solvable in polynomial time. The problem of deciding whether $G$ has a 3-cycle cover is NP-complete. The corresponding optimization problem (i.e., finding a maximum weight 3-cycle cover) is APX-complete, and has an $((3/5)-\epsilon)$-approximation algorithm (for any $\epsilon &gt; 0$). The good news here is that one can also compute a maximum weight 2-cycle cover in polynomial time (provided edge weights are nonnegative).

For details and proofs of the above claims, see [1].



[1] Bläser, Markus, and Bodo Manthey. "Two approximation algorithms for 3-cycle covers." Approximation Algorithms for Combinatorial Optimization. Springer Berlin Heidelberg, 2002. 40-50.