CXXGraph库提供了一组检测周期的算法和函数。

要获得完整的算法解释，请访问wiki。

首先，你并不是真的想要找出所有的循环因为如果有1个，那么就会有无穷多个循环。比如A-B-A, A-B-A- b - a等等。或者可以将2个循环组合成一个8-like循环等等……有意义的方法是寻找所有所谓的简单循环——那些除了开始/结束点之外不交叉的循环。如果你愿意，你可以生成简单循环的组合。

One of the baseline algorithms for finding all simple cycles in a directed graph is this: Do a depth-first traversal of all simple paths (those that do not cross themselves) in the graph. Every time when the current node has a successor on the stack a simple cycle is discovered. It consists of the elements on the stack starting with the identified successor and ending with the top of the stack. Depth first traversal of all simple paths is similar to depth first search but you do not mark/record visited nodes other than those currently on the stack as stop points.

The brute force algorithm above is terribly inefficient and in addition to that generates multiple copies of the cycles. It is however the starting point of multiple practical algorithms which apply various enhancements in order to improve performance and avoid cycle duplication. I was surprised to find out some time ago that these algorithms are not readily available in textbooks and on the web. So I did some research and implemented 4 such algorithms and 1 algorithm for cycles in undirected graphs in an open source Java library here : http://code.google.com/p/niographs/ .

顺便说一句，因为我提到了无向图:它们的算法是不同的。构建一棵生成树，然后每一条不属于树的边与树中的一些边一起形成一个简单的循环。这样发现的循环形成了所谓的循环基。所有的简单循环都可以通过组合两个或多个不同的基循环来找到。更多细节请参见:http://dspace.mit.edu/bitstream/handle/1721.1/68106/FTL_R_1982_07.pdf。

澄清:

Strongly Connected Components will find all subgraphs that have at least one cycle in them, not all possible cycles in the graph. e.g. if you take all strongly connected components and collapse/group/merge each one of them into one node (i.e. a node per component), you'll get a tree with no cycles (a DAG actually). Each component (which is basically a subgraph with at least one cycle in it) can contain many more possible cycles internally, so SCC will NOT find all possible cycles, it will find all possible groups that have at least one cycle, and if you group them, then the graph will not have cycles. to find all simple cycles in a graph, as others mentioned, Johnson's algorithm is a candidate.

从节点X开始，检查所有子节点(如果无方向，父节点和子节点是等价的)。将这些子节点标记为X的子节点。对于任何这样的子节点A，标记它的子节点是A的子节点，X'，其中X'标记为2步远。)如果您稍后点击X并将其标记为X的子节点”，这意味着X处于3节点周期中。回溯到它的父节点很容易(因为算法不支持这一点，所以你可以找到任何一个有X'的父节点)。

注意:如果图是无向的或者有任何双向边，这个算法会变得更复杂，假设你不想在一个周期内两次遍历同一条边。

我在搜索中找到了这个页面，由于循环与强连接组件不相同，我继续搜索，最后，我找到了一个高效的算法，它列出了有向图的所有(基本)循环。这篇论文来自唐纳德·b·约翰逊(Donald B. Johnson)，可以在以下链接中找到:

http://www.cs.tufts.edu/comp/150GA/homeworks/hw1/Johnson%2075.PDF

java实现可以在下面找到:

http://normalisiert.de/code/java/elementaryCycles.zip

约翰逊算法的Mathematica演示可以在这里找到，实现可以从右边下载(“下载作者代码”)。

注:实际上，这个问题有很多算法。本文列举了其中一些:

http://dx.doi.org/10.1137/0205007

根据文章，Johnson的算法是最快的。

在无向图的情况下，最近发表的一篇论文(无向图中循环和st路径的最优列表)提供了一个渐近最优解。你可以在这里阅读http://arxiv.org/abs/1205.2766或http://dl.acm.org/citation.cfm?id=2627951 我知道它不能回答你的问题，但由于你的问题标题没有提到方向，它可能仍然对谷歌搜索有用