In mathematics, a vertex cycle cover (commonly called simply cycle cover) of a graph G is a set of cycles which are subgraphs of G and contain all vertices of G.
If the cycles of the cover have no vertices in common, the cover is called vertex-disjoint or sometimes simply disjoint cycle cover. This is sometimes known as exact vertex 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. Finding a vertex-disjoint cycle cover of a directed graph can also be performed in polynomial time by a similar reduction to perfect matching.https://www.cs.cmu.edu/~avrim/451f13/recitation/rec1016.txt (problem 1) However, adding the condition that each cycle should have length at least 3 makes the problem NP-hard.Garey and Johnson, Computers and intractability, GT13
Properties and applications
Permanent 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.Ben-Dor, Amir and Halevi, Shai. (1993). "Zero-one permanent is #P-complete, a simpler proof". Proceedings of the 2nd Israel Symposium on the Theory and Computing Systems, 108-117.
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.Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties (1999) p.378, 379, citing .
See also Edge cycle cover, a collection of cycles covering all edges of G
References