# Find all cycles in an undirected graph

For directed graphs, we have a built-in command all_simple_cycles to search for specific length cycles or all cycles. However, it is indeed strange that there is no readily available command for undirected graphs. Although, we can convert an undirected graph into a directed graph by assigning two directions to each edge. For example:

g = graphs.CompleteGraph(4).to_directed();
all_cycles=g.all_simple_cycles()
print(all_cycles)


[[0, 1, 0], [0, 2, 0], [0, 3, 0], [1, 2, 1], [1, 3, 1], [2, 3, 2], [0, 1, 2, 0], [0, 1, 3, 0], [0, 2, 1, 0], [0, 2, 3, 0], [0, 3, 1, 0], [0, 3, 2, 0], [1, 2, 3, 1], [1, 3, 2, 1]]

allcycles = []
for array in all_dcycles:
if len(array)>=4:
reversed_array = array[::-1]
if reversed_array not in allcycles:
allcycles.append(array)
print(allcycles)


[[0, 2, 1, 0], [0, 3, 1, 0], [0, 3, 2, 0], [1, 3, 2, 1], [0, 3, 2, 1, 0], [0, 2, 3, 1, 0], [0, 3, 1, 2, 0]]

Reversing lists and eliminating all 2-cycles seems tedious. Does SageMath provide a direct command for undirected graphs to find all cycles or cycles of a specific length? It's somewhat similar to Mathematica's FindCycle (FindCycle[CompleteGraph[4], Infinity, All]).

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There is no need to reverse and eliminate cycles - instead, to fix a cycle direction it's enough to compare the second vertex with the second but last vertex:

g = graphs.CompleteGraph(4)
undirected_cycles = [ c for c in g.to_directed().all_simple_cycles() if len(c)>=4 and c[1]<c[-2] ]


Alternatively, all simple cycles can be constructed from .cycle_basis():

def gen_simple_cycles(G):
C = [frozenset(tuple(sorted(e[:2])) for e in c) for c in G.cycle_basis(output='edge')]
for S in Subsets(C):
T = set()
for c in S:
T = T.symmetric_difference(c)
H = Graph(T, format='list_of_edges')
if H.is_eulerian() and max(H.degree(),default=0)==2:
yield H.eulerian_circuit(return_vertices=True)[1]

list( gen_simple_cycles( graphs.CompleteGraph(4) ) )

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Last updated: Jun 21 '23