def bridges(G, root=None):
"""Generate all bridges in a graph.
A *bridge* in a graph is an edge whose removal causes the number of
connected components of the graph to increase. Equivalently, a bridge is an
edge that does not belong to any cycle.
Parameters
----------
G : undirected graph
root : node (optional)
A node in the graph `G`. If specified, only the bridges in the
connected component containing this node will be returned.
Yields
------
e : edge
An edge in the graph whose removal disconnects the graph (or
causes the number of connected components to increase).
Raises
------
NodeNotFound
If `root` is not in the graph `G`.
Examples
--------
The barbell graph with parameter zero has a single bridge:
>>> G = nx.barbell_graph(10, 0)
>>> list(nx.bridges(G))
[(9, 10)]
Notes
-----
This is an implementation of the algorithm described in _[1]. An edge is a
bridge if and only if it is not contained in any chain. Chains are found
using the :func:`networkx.chain_decomposition` function.
Ignoring polylogarithmic factors, the worst-case time complexity is the
same as the :func:`networkx.chain_decomposition` function,
$O(m + n)$, where $n$ is the number of nodes in the graph and $m$ is
the number of edges.
References
----------
.. [1] https://en.wikipedia.org/wiki/Bridge_%28graph_theory%29#Bridge-Finding_with_Chain_Decompositions
"""
chains = nx.chain_decomposition(G, root=root)
chain_edges = set(chain.from_iterable(chains))
for u, v in G.edges():
if (u, v) not in chain_edges and (v, u) not in chain_edges:
yield u, v
def test_barbell_graph(self):
# The (3, 0) barbell graph has two triangles joined by a single edge.
G = nx.barbell_graph(3, 0)
chains = list(nx.chain_decomposition(G, root=0))
expected = [
[(0, 1), (1, 2), (2, 0)],
[(3, 4), (4, 5), (5, 3)],
]
self.assertEqual(len(chains), len(expected))
for chain in chains:
self.assertContainsChain(chain, expected)
def test_disconnected_graph_root_node(self):
"""Test for a single component of a disconnected graph."""
G = nx.barbell_graph(3, 0)
H = nx.barbell_graph(3, 0)
mapping = dict(zip(range(6), 'abcdef'))
nx.relabel_nodes(H, mapping, copy=False)
G = nx.union(G, H)
chains = list(nx.chain_decomposition(G, root='a'))
expected = [
[('a', 'b'), ('b', 'c'), ('c', 'a')],
[('d', 'e'), ('e', 'f'), ('f', 'd')],
]
self.assertEqual(len(chains), len(expected))
for chain in chains:
self.assertContainsChain(chain, expected)
def test_disconnected_graph(self):
"""Test for a graph with multiple connected components."""
G = nx.barbell_graph(3, 0)
H = nx.barbell_graph(3, 0)
mapping = dict(zip(range(6), 'abcdef'))
nx.relabel_nodes(H, mapping, copy=False)
G = nx.union(G, H)
chains = list(nx.chain_decomposition(G))
expected = [
[(0, 1), (1, 2), (2, 0)],
[(3, 4), (4, 5), (5, 3)],
[('a', 'b'), ('b', 'c'), ('c', 'a')],
[('d', 'e'), ('e', 'f'), ('f', 'd')],
]
self.assertEqual(len(chains), len(expected))
for chain in chains:
self.assertContainsChain(chain, expected)
def bridges(G, root=None):
"""Generate all bridges in a graph.
A *bridge* in a graph is an edge whose removal causes the number of
connected components of the graph to increase.
Parameters
----------
G : undirected graph
root : node (optional)
A node in the graph `G`. If specified, only the bridges in the
connected component containing this node will be returned.
Yields
------
e : edge
An edge in the graph whose removal disconnects the graph (or
causes the number of connected components to increase).
Raises
------
NodeNotFound
If `root` is not in the graph `G`.
Examples
--------
The barbell graph with parameter zero has a single bridge::
>>> G = nx.barbell_graph(10, 0)
>>> list(nx.bridges(G))
[(9, 10)]
Notes
-----
This implementation uses the :func:`networkx.chain_decomposition`
function, so it shares its worst-case time complexity, :math:`O(m +
n)`, ignoring polylogarithmic factors, where *n* is the number of
nodes in the graph and *m* is the number of edges.
"""
chains = nx.chain_decomposition(G, root=root)
chain_edges = set(chain.from_iterable(chains))
for u, v in G.edges():
if (u, v) not in chain_edges and (v, u) not in chain_edges:
yield u, v
def test_disconnected_graph(self):
"""Test for a graph with multiple connected components."""
G = nx.barbell_graph(3, 0)
H = nx.barbell_graph(3, 0)
mapping = dict(zip(range(6), 'abcdef'))
nx.relabel_nodes(H, mapping, copy=False)
G = nx.union(G, H)
chains = list(nx.chain_decomposition(G))
expected = [
[(0, 1), (1, 2), (2, 0)],
[(3, 4), (4, 5), (5, 3)],
[('a', 'b'), ('b', 'c'), ('c', 'a')],
[('d', 'e'), ('e', 'f'), ('f', 'd')],
]
self.assertEqual(len(chains), len(expected))
for chain in chains:
self.assertContainsChain(chain, expected)
def bridges(G, root=None):
"""Generate all bridges in a graph.
A *bridge* in a graph is an edge whose removal causes the number of
connected components of the graph to increase.
Parameters
----------
G : undirected graph
root : node (optional)
A node in the graph `G`. If specified, only the bridges in the
connected component containing this node will be returned.
Yields
------
e : edge
An edge in the graph whose removal disconnects the graph (or
causes the number of connected components to increase).
Raises
------
NodeNotFound
If `root` is not in the graph `G`.
Examples
--------
The barbell graph with parameter zero has a single bridge::
>>> G = nx.barbell_graph(10, 0)
>>> list(nx.bridges(G))
[(9, 10)]
Notes
-----
This implementation uses the :func:`networkx.chain_decomposition`
function, so it shares its worst-case time complexity, :math:`O(m +
n)`, ignoring polylogarithmic factors, where *n* is the number of
nodes in the graph and *m* is the number of edges.
"""
chains = nx.chain_decomposition(G, root=root)
chain_edges = set(chain.from_iterable(chains))
for u, v in G.edges():
if (u, v) not in chain_edges and (v, u) not in chain_edges:
yield u, v
def test_decomposition(self):
edges = [
# DFS tree edges.
(1, 2), (2, 3), (3, 4), (3, 5), (5, 6), (6, 7), (7, 8), (5, 9),
(9, 10),
# Nontree edges.
(1, 3), (1, 4), (2, 5), (5, 10), (6, 8)
]
G = nx.Graph(edges)
expected = [
[(1, 3), (3, 2), (2, 1)],
[(1, 4), (4, 3)],
[(2, 5), (5, 3)],
[(5, 10), (10, 9), (9, 5)],
[(6, 8), (8, 7), (7, 6)],
]
chains = list(nx.chain_decomposition(G, root=1))
self.assertEqual(len(chains), len(expected))
def chain(self):
"""
Return a list of tuples of nodes in the serial chain, rooted at datum
if datum is in the chain.
Warning
-------
Do not prerve the edges orientations
"""
# set root
if self.datum in self.nodes:
root = self.datum
else:
root = None
# Convert graph type to undirected graph with no parallel edge.
simple_graph = nwxGraph(self)
# Return chain decomposition
return list(chain_decomposition(simple_graph, root=root))[0]
f.write(wd + "\n")
# path btw two nodes
pf = nx.shortest_path(G=G_skip, source="nemzet", target="okos")
pn = nx.shortest_path(G=G_skip, source="nő", target="okos")
print(pf, pn)
pf = nx.shortest_path(G=G_embed, source="nemzet", target="okos")
pn = nx.shortest_path(G=G_embed, source="nő", target="okos")
print(pf, pn)
sp_ffi = nx.shortest_simple_paths(G=G_embed, source="férfi", target="okos")
sp_no = nx.shortest_simple_paths(G=G_embed, source="nő", target="okos")
# chain decomposition
ff_chain = nx.chain_decomposition(G=G_embed, root="férfi")
ff_chain = [e for e in ff_chain if 40 > len(e) > 10]
fn = max([len(e) for e in ff_chain])
fi = [len(e) for e in ff_chain].index(fn)
no_chain = nx.chain_decomposition(G=G_embed, root="nő")
no_chain = [e for e in no_chain if 40 > len(e) > 10]
nn = max([len(e) for e in no_chain])
no = [len(e) for e in no_chain].index(nn)
ff_path = ff_chain[fi]
no_path = no_chain[no]
G = nx.Graph()
for e in ff_path:
G.add_edge(e[0], e[1])
def crust_algorithm(self, show=False):
# Guaranteed to result in the proper contour, but may not contain all points in specific, rare pixel
# configurations
def _simplices2edges(_deltri, _crust_set):
unique_edges = {'points': [], 'indices': []}
for simplex in _deltri.simplices:
del_edges_i = combinations(simplex, 2)
for _del_edge_i in del_edges_i:
_del_edge_i = np.sort(_del_edge_i)
_del_edge = _crust_set[_del_edge_i, :]
unique_edges['points'].append(_del_edge)
unique_edges['indices'].append(_del_edge_i)
unique_edges['points'] = np.unique(unique_edges['points'], axis=0)
unique_edges['indices'] = np.unique(unique_edges['indices'],
axis=0)
return unique_edges
vor = Voronoi(self.edge_points)
list_edge_points = self.edge_points.tolist()
crust_set = np.unique(np.concatenate((self.edge_points, vor.vertices)),
axis=0)
deltri = Delaunay(crust_set)
all_edges = _simplices2edges(deltri, crust_set)
crust_edges = {'points': [], 'indices': []}
for del_edge, edge_id in zip(all_edges['points'],
all_edges['indices']):
if del_edge[0].tolist() in list_edge_points and del_edge[1].tolist(
) in list_edge_points:
crust_edges['points'].append(del_edge)
crust_edges['indices'].append(tuple(edge_id))
G = nx.Graph()
G.add_nodes_from([tuple(point) for point in crust_set])
G.add_edges_from(crust_edges['indices'])
point_id = int(np.min(np.array(crust_edges['indices'])[:, 0]))
try:
cd = list(nx.chain_decomposition(G, point_id))[0]
except IndexError: # raised when there is no chain. The leaves of the graph must be connected.
leaves = np.array([v for v, d in list(G.degree) if d == 1])
leaf_dists = cdist(crust_set[leaves], crust_set[leaves])
closest_leaves = []
for leaf in leaf_dists:
closest_leaves.append(
min(enumerate(leaf),
key=lambda x: x[1] if x[1] > 0 else float('inf'))[0])
leaf_edges = [[
leaf1, leaf2
] for leaf1, leaf2 in zip(leaves, leaves[closest_leaves])]
leaf_edges = sorted(leaf_edges, key=lambda x: x[0])
unique_edges = np.unique(leaf_edges).reshape((-1, 2))
unique_edges = [tuple(edge) for edge in unique_edges]
G.add_edges_from(list(unique_edges))
cd = list(nx.chain_decomposition(G, point_id))[0]
graph_path = [x[0] for x in list(cd)]
G.remove_edges_from(crust_edges['indices'])
nx.add_path(G, graph_path)
_sorted_points = np.array(crust_set[graph_path])
if show:
plt.subplot(121)
plt.plot(self.edge_points[:, 0], self.edge_points[:, 1], 'r.')
for crust_edge in crust_edges['points']:
plt.plot(crust_edge[:, 0], crust_edge[:, 1], 'b')
plt.title('Found edges')
plt.subplot(122)
plt.title('Crust algorithm result')
plt.plot(_sorted_points[:, 0], _sorted_points[:, 1], 'b')
plt.plot(self.edge_points[:, 0], self.edge_points[:, 1], 'r.')
plt.scatter(_sorted_points[0][0],
_sorted_points[0][1],
c='g',
marker='d',
s=80)
plt.scatter(_sorted_points[-1][0],
_sorted_points[-1][1],
c='k',
marker='d')
plt.show()
return _sorted_points
import networkx as nx
g = nx.Graph()
g.add_edge(1, 4)
g.add_edge(1, 5)
g.add_edge(1, 2)
g.add_edge(4, 5)
g.add_edge(2, 5)
g.add_edge(3, 5)
g.add_edge(3, 4)
g.add_edge(3, 2)
g.add_edge(3, 6)
g.add_edge(6, 7)
g.add_edge(7, 8)
g.add_edge(6, 8)
chains = nx.chain_decomposition(g)
print('\n'.join(map(str, list(chains))))