Find the Simple Cycles in a Directed Graph

This problem, for me, looks very interesting. It was about to find a simple cycle (i.e. cycle where are not repeat nodes) in a directed graph.

My solution is going like this, i.e, this graph is a case problem:

I know that there is a cycle in a graph, when you can find "back edges" in a depth-first-search (dashed in my picture in DFSTree), and for a moment I can sure for a few cycles, but not for all, simple cycles. Because, the directed egdes so important to from a cycle, i.e (0123) != (0321)

I'm thinking in make a dfs for each node with back-edges, but I'm not sure, and it's not clear. So, I ask you, if you guide me. Thanks!.

Here is my count of simple loops for my case problem.

You may be interested in the algorithm presented in this paper:

Finding all the elementary circuits of a directed graph. Donald B. Johnson. SIAM J. COMPUT. Vol. 4, No. 1, March 1975

Abstract. An algorithm is presented which finds all the elementary circuits-of a directed graph in time bounded by O((n + e)(c + 1)) and space bounded by O(n + e), where there are n vertices, e edges and c elementary circuits in the graph. The algorithm resembles algorithms by Tiernan and Tarjan, but is faster because it considers each edge at most twice between any one circuit and the next in the output sequence.

The paper contains a complete algorithm.

StackExchange Computer Science Q#7216, answer score: 15