Vertex cycle cover

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.[1][2]

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.[3] However, adding the condition that each cycle should have length at least 3 makes the problem NP-hard.[4]

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.[5]

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.[6]

See also

References
  1. David Eppstein. "Partition a graph into node-disjoint cycles".
  2. Tutte, W. T. (1954), "A short proof of the factor theorem for finite graphs" (PDF), Canadian Journal of Mathematics, 6: 347–352, doi:10.4153/CJM-1954-033-3, MR 0063008.
  3. https://www.cs.cmu.edu/~avrim/451f13/recitation/rec1016.txt (problem 1)
  4. Garey and Johnson, Computers and intractability, GT13
  5. 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.
  6. Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties (1999) ISBN 3-540-65431-3 p.378, 379, citing Sahni, Sartaj; Gonzalez, Teofilo (1976), "P-complete approximation problems" (PDF), Journal of the ACM, 23 (3): 555–565, doi:10.1145/321958.321975, MR 0408313..
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