Vertex cycle cover
If the cycles of the cover have no vertices in common, the cover is called vertex-disjoint or sometimes simply disjoint cycle cover. In this case the set of the cycles constitutes a spanning subgraph of G. A disjoint cycle cover of an undirected graph (if it exists) can be found in polynomial time by transforming the problem into a problem of finding a perfect matching in a larger graph. 
If the cycles of the cover have no edges in common, the cover is called edge-disjoint or simply disjoint cycle cover.
Similar definitions exist for digraphs, in terms of directed cycles.
Properties and applications
The permanent of a (0,1)-matrix is equal to the number of vertex-disjoint cycle covers of a directed graph with this adjacency matrix. This fact is used in a simplified proof showing that computing the permanent is #P-complete.
Minimal disjoint cycle covers
The problems of finding a vertex disjoint and edge disjoint cycle covers with minimal number of cycles are NP-complete. The problems are not in complexity class APX. The variants for digraphs are not in APX either.
- Edge cycle cover, a collection of cycles covering all edges of G
- Tutte, W. T. (1954), "A short proof of the factor theorem for finite graphs", Canadian Journal of Mathematics, 6: 347–352, doi:10.4153/CJM-1954-033-3, MR 0063008.
- http://www.cs.cmu.edu/~avrim/451f13/recitation/rec1016.txt (problem 1)
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