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.

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.

Contents

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.[1]

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.[2]

See also

References

  1. ^ 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.
  2. ^ Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties (1999) ISBN 3540654313 p.378, 379, citing Sahni, S. K., and Gonzalez, T. F. (1976), P-complete approximation problems, J. ACM 23, 555-565.