networkx.algorithms.cycles.simple_cycles¶

simple_cycles(G)[source]¶

Find simple cycles (elementary circuits) of a directed graph.

A simple cycle, or elementary circuit, is a closed path where no node appears twice. Two elementary circuits are distinct if they are not cyclic permutations of each other.

This is a nonrecursive, iterator/generator version of Johnson’s algorithm 1. There may be better algorithms for some cases 2 3.

Parameters: G (NetworkX DiGraph) – A directed graph
Returns: cycle_generator – A generator that produces elementary cycles of the graph. Each cycle is represented by a list of nodes along the cycle.
Return type: generator

Examples

>>> edges = [(0, 0), (0, 1), (0, 2), (1, 2), (2, 0), (2, 1), (2, 2)]
>>> G = nx.DiGraph(edges)
>>> len(list(nx.simple_cycles(G)))
5

To filter the cycles so that they don’t include certain nodes or edges, copy your graph and eliminate those nodes or edges before calling

>>> copyG = G.copy()
>>> copyG.remove_nodes_from([1])
>>> copyG.remove_edges_from([(0, 1)])
>>> len(list(nx.simple_cycles(copyG)))
3

Notes

The implementation follows pp. 79-80 in 1.

The time complexity is \(O((n+e)(c+1))\) for \(n\) nodes, \(e\) edges and \(c\) elementary circuits.

References

1(1,2): Finding all the elementary circuits of a directed graph. D. B. Johnson, SIAM Journal on Computing 4, no. 1, 77-84, 1975. https://doi.org/10.1137/0204007
2: Enumerating the cycles of a digraph: a new preprocessing strategy. G. Loizou and P. Thanish, Information Sciences, v. 27, 163-182, 1982.
3: A search strategy for the elementary cycles of a directed graph. J.L. Szwarcfiter and P.E. Lauer, BIT NUMERICAL MATHEMATICS, v. 16, no. 2, 192-204, 1976.