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improve method cycle_basis for graphs with multiple edges #36493

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merged 3 commits into from
Oct 31, 2023
Merged

improve method cycle_basis for graphs with multiple edges #36493

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dd2044f
improve cycle basis for multigraphs
dcoudert Oct 21, 2023
2a4c180
Merge branch 'develop' into graphs/cycle_basis_multigraphs
dcoudert Oct 21, 2023
00c5af4
remove some useless # needs networkx
dcoudert Oct 24, 2023
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42 changes: 30 additions & 12 deletions src/sage/graphs/generic_graph.py
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Original file line number Diff line number Diff line change
Expand Up @@ -5150,13 +5150,13 @@ def cycle_basis(self, output='vertex'):
sage: G = Graph([(0, 2, 'a'), (0, 2, 'b'), (0, 1, 'c'), (1, 2, 'd')],
....: multiedges=True)
sage: G.cycle_basis() # needs networkx
[[0, 2], [2, 1, 0]]
[[2, 0], [2, 0, 1]]
sage: G.cycle_basis(output='edge') # needs networkx
[[(0, 2, 'b'), (2, 0, 'a')], [(2, 1, 'd'), (1, 0, 'c'), (0, 2, 'a')]]
[[(0, 2, 'b'), (2, 0, 'a')], [(1, 2, 'd'), (2, 0, 'a'), (0, 1, 'c')]]
sage: H = Graph([(1, 2), (2, 3), (2, 3), (3, 4), (1, 4),
....: (1, 4), (4, 5), (5, 6), (4, 6), (6, 7)], multiedges=True)
sage: H.cycle_basis() # needs networkx
[[1, 4], [2, 3], [4, 3, 2, 1], [6, 5, 4]]
[[4, 1], [3, 2], [4, 1, 2, 3], [6, 4, 5]]

Disconnected graph::

Expand All @@ -5168,14 +5168,14 @@ def cycle_basis(self, output='vertex'):
('Really ?', 'Wuuhuu', None),
('Wuuhuu', 'Hey', None)],
[(0, 2, 'a'), (2, 0, 'b')],
[(0, 2, 'b'), (1, 0, 'c'), (2, 1, 'd')]]
[(0, 1, 'c'), (1, 2, 'd'), (2, 0, 'b')]]

Graph that allows multiple edges but does not contain any::

sage: G = graphs.CycleGraph(3)
sage: G.allow_multiple_edges(True)
sage: G.cycle_basis() # needs networkx
[[2, 1, 0]]
[[2, 0, 1]]

Not yet implemented for directed graphs::

Expand All @@ -5192,11 +5192,11 @@ def cycle_basis(self, output='vertex'):
sage: G = Graph([(1, 2, 'a'), (2, 3, 'b'), (2, 3, 'c'),
....: (3, 4, 'd'), (3, 4, 'e'), (4, 1, 'f')], multiedges=True)
sage: G.cycle_basis() # needs networkx
[[2, 3], [4, 3, 2, 1], [4, 3, 2, 1]]
[[3, 2], [4, 1, 2, 3], [4, 1, 2, 3]]
sage: G.cycle_basis(output='edge') # needs networkx
[[(2, 3, 'c'), (3, 2, 'b')],
[(4, 3, 'd'), (3, 2, 'b'), (2, 1, 'a'), (1, 4, 'f')],
[(4, 3, 'e'), (3, 2, 'b'), (2, 1, 'a'), (1, 4, 'f')]]
[(3, 4, 'd'), (4, 1, 'f'), (1, 2, 'a'), (2, 3, 'b')],
[(3, 4, 'e'), (4, 1, 'f'), (1, 2, 'a'), (2, 3, 'b')]]
"""
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Then for these cases, you may remove # needs networkx?

if output not in ['vertex', 'edge']:
raise ValueError('output must be either vertex or edge')
Expand All @@ -5211,14 +5211,32 @@ def cycle_basis(self, output='vertex'):
[])

from sage.graphs.graph import Graph
from itertools import pairwise
T = Graph(self.min_spanning_tree(), multiedges=True, format='list_of_edges')
H = self.copy()
H.delete_edges(T.edge_iterator())
root = next(T.vertex_iterator())
rank = dict(T.breadth_first_search(root, report_distance=True))
parent = {v: u for u, v in T.breadth_first_search(root, edges=True)}
L = []
for e in H.edge_iterator():
T.add_edge(e)
L.append(T.is_tree(certificate=True, output=output)[1])
T.delete_edge(e)
# Search for the nearest common ancestor of e[0] and e[1] in T
P = [e[0]]
Q = [e[1]]
while P[-1] != Q[-1]:
# If rank[P[-1]] > rank[Q[-1]], we extend the path P.
# If rank[P[-1]] < rank[Q[-1]], we extend the path Q.
# In case of equality, we extend both paths.
diff = rank[P[-1]] - rank[Q[-1]]
if diff >= 0:
P.append(parent[P[-1]])
if diff <= 0:
Q.append(parent[Q[-1]])

cycle = Q + P[-2::-1]
if output == 'edge':
cycle = [e] + [(x, y, T.edge_label(x, y)[0]) for x, y in pairwise(cycle)]
L.append(cycle)
return L

# second case: there are no multiple edges
Expand Down Expand Up @@ -5289,7 +5307,7 @@ def minimum_cycle_basis(self, algorithm=None, weight_function=None, by_weight=Fa
TESTS::

sage: g = Graph([(0, 1, 1), (1, 2, 'a')])
sage: g.min_spanning_tree(by_weight=True)
sage: g.minimum_cycle_basis(by_weight=True)
Traceback (most recent call last):
...
ValueError: the weight function cannot find the weight of (1, 2, 'a')
Expand Down