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. 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]
If the cycles of the cover have no edges in common, the cover is called edge-disjoint or simply disjoint cycle cover.
Similar definitions may be introduced for digraphs, in terms of directed cycles.
Properties and applications
Permanent
The permanent of a (0,1)-matrix is equal to the number of cycle covers of a directed graph with this adjacency matrix. This fact is used in a simplified proof of the fact that computation of the permanent is #P-complete.[2]
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.[3]
See also
- Edge cycle cover, a collection of cycles covering all edges of G
References
- ↑ 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.
- ↑ 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.
- ↑ Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties (1999) ISBN 3-540-65431-3 p.378, 379, citing Sahni, S. K., and Gonzalez, T. F. (1976), P-complete approximation problems, J. ACM 23, 555-565.