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.

The top graph is vertex covered by 3 cycles, and the cycles share both vertices (vertex 3) and edges (edge between 4 and 6).

The middle graph is covered by 2 cycles, and while there is a vertex overlap (vertex 3), no edges are used twice, making the covering an edge-disjoint.

The bottom graph has a covering where no vertex or edge is shared between the cycles, making the covering both edge-disjoint and vertex-disjoint.

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

• Edge cycle cover, a collection of cycles covering all edges of G