Finding all cycles in a directed graph

Question

How can I find (iterate over) ALL the cycles in a directed graph from/to a given node?

For example, I want something like this:

A->B->A
A->B         


        

Answer 1

There are two steps (algorithms) involved in finding all cycles in a DAG.

The first step is to use Tarjan's algorithm to find the set of strongly connected components.

1. Start from any arbitrary vertex.
2. DFS from that vertex. For each node x, keep two numbers, dfs_index[x] and dfs_lowval[x]. dfs_index[x] stores when that node is visited, while dfs_lowval[x] = min(dfs_low[k]) where k is all the children of x that is not the directly parent of x in the dfs-spanning tree.
3. All nodes with the same dfs_lowval[x] are in the same strongly connected component.

The second step is to find cycles (paths) within the connected components. My suggestion is to use a modified version of Hierholzer's algorithm.

The idea is:

1. Choose any starting vertex v, and follow a trail of edges from that vertex until you return to v. It is not possible to get stuck at any vertex other than v, because the even degree of all vertices ensures that, when the trail enters another vertex w there must be an unused edge leaving w. The tour formed in this way is a closed tour, but may not cover all the vertices and edges of the initial graph.
2. As long as there exists a vertex v that belongs to the current tour but that has adjacent edges not part of the tour, start another trail from v, following unused edges until you return to v, and join the tour formed in this way to the previous tour.

Here is the link to a Java implementation with a test case:

http://stones333.blogspot.com/2013/12/find-cycles-in-directed-graph-dag.html