"""
Eulerian circuits and graphs.
"""
from itertools import combinations
import networkx as nx
from ..utils import arbitrary_element, not_implemented_for
__all__ = [
"is_eulerian",
"eulerian_circuit",
"eulerize",
"is_semieulerian",
"has_eulerian_path",
"eulerian_path",
]
[docs]def is_eulerian(G):
"""Returns True if and only if `G` is Eulerian.
A graph is *Eulerian* if it has an Eulerian circuit. An *Eulerian
circuit* is a closed walk that includes each edge of a graph exactly
once.
Parameters
----------
G : NetworkX graph
A graph, either directed or undirected.
Examples
--------
>>> nx.is_eulerian(nx.DiGraph({0: [3], 1: [2], 2: [3], 3: [0, 1]}))
True
>>> nx.is_eulerian(nx.complete_graph(5))
True
>>> nx.is_eulerian(nx.petersen_graph())
False
Notes
-----
If the graph is not connected (or not strongly connected, for
directed graphs), this function returns False.
"""
if G.is_directed():
# Every node must have equal in degree and out degree and the
# graph must be strongly connected
return all(
G.in_degree(n) == G.out_degree(n) for n in G
) and nx.is_strongly_connected(G)
# An undirected Eulerian graph has no vertices of odd degree and
# must be connected.
return all(d % 2 == 0 for v, d in G.degree()) and nx.is_connected(G)
[docs]def is_semieulerian(G):
"""Return True iff `G` is semi-Eulerian.
G is semi-Eulerian if it has an Eulerian path but no Eulerian circuit.
"""
return has_eulerian_path(G) and not is_eulerian(G)
def _find_path_start(G):
"""Return a suitable starting vertex for an Eulerian path.
If no path exists, return None.
"""
if not has_eulerian_path(G):
return None
if is_eulerian(G):
return arbitrary_element(G)
if G.is_directed():
v1, v2 = (v for v in G if G.in_degree(v) != G.out_degree(v))
# Determines which is the 'start' node (as opposed to the 'end')
if G.out_degree(v1) > G.in_degree(v1):
return v1
else:
return v2
else:
# In an undirected graph randomly choose one of the possibilities
start = [v for v in G if G.degree(v) % 2 != 0][0]
return start
def _simplegraph_eulerian_circuit(G, source):
if G.is_directed():
degree = G.out_degree
edges = G.out_edges
else:
degree = G.degree
edges = G.edges
vertex_stack = [source]
last_vertex = None
while vertex_stack:
current_vertex = vertex_stack[-1]
if degree(current_vertex) == 0:
if last_vertex is not None:
yield (last_vertex, current_vertex)
last_vertex = current_vertex
vertex_stack.pop()
else:
_, next_vertex = arbitrary_element(edges(current_vertex))
vertex_stack.append(next_vertex)
G.remove_edge(current_vertex, next_vertex)
def _multigraph_eulerian_circuit(G, source):
if G.is_directed():
degree = G.out_degree
edges = G.out_edges
else:
degree = G.degree
edges = G.edges
vertex_stack = [(source, None)]
last_vertex = None
last_key = None
while vertex_stack:
current_vertex, current_key = vertex_stack[-1]
if degree(current_vertex) == 0:
if last_vertex is not None:
yield (last_vertex, current_vertex, last_key)
last_vertex, last_key = current_vertex, current_key
vertex_stack.pop()
else:
triple = arbitrary_element(edges(current_vertex, keys=True))
_, next_vertex, next_key = triple
vertex_stack.append((next_vertex, next_key))
G.remove_edge(current_vertex, next_vertex, next_key)
[docs]def eulerian_circuit(G, source=None, keys=False):
"""Returns an iterator over the edges of an Eulerian circuit in `G`.
An *Eulerian circuit* is a closed walk that includes each edge of a
graph exactly once.
Parameters
----------
G : NetworkX graph
A graph, either directed or undirected.
source : node, optional
Starting node for circuit.
keys : bool
If False, edges generated by this function will be of the form
``(u, v)``. Otherwise, edges will be of the form ``(u, v, k)``.
This option is ignored unless `G` is a multigraph.
Returns
-------
edges : iterator
An iterator over edges in the Eulerian circuit.
Raises
------
NetworkXError
If the graph is not Eulerian.
See Also
--------
is_eulerian
Notes
-----
This is a linear time implementation of an algorithm adapted from [1]_.
For general information about Euler tours, see [2]_.
References
----------
.. [1] J. Edmonds, E. L. Johnson.
Matching, Euler tours and the Chinese postman.
Mathematical programming, Volume 5, Issue 1 (1973), 111-114.
.. [2] https://en.wikipedia.org/wiki/Eulerian_path
Examples
--------
To get an Eulerian circuit in an undirected graph::
>>> G = nx.complete_graph(3)
>>> list(nx.eulerian_circuit(G))
[(0, 2), (2, 1), (1, 0)]
>>> list(nx.eulerian_circuit(G, source=1))
[(1, 2), (2, 0), (0, 1)]
To get the sequence of vertices in an Eulerian circuit::
>>> [u for u, v in nx.eulerian_circuit(G)]
[0, 2, 1]
"""
if not is_eulerian(G):
raise nx.NetworkXError("G is not Eulerian.")
if G.is_directed():
G = G.reverse()
else:
G = G.copy()
if source is None:
source = arbitrary_element(G)
if G.is_multigraph():
for u, v, k in _multigraph_eulerian_circuit(G, source):
if keys:
yield u, v, k
else:
yield u, v
else:
yield from _simplegraph_eulerian_circuit(G, source)
[docs]def has_eulerian_path(G, source=None):
"""Return True iff `G` has an Eulerian path.
An Eulerian path is a path in a graph which uses each edge of a graph
exactly once. If `source` is specified, then this function checks
whether an Eulerian path that starts at node `source` exists.
A directed graph has an Eulerian path iff:
- at most one vertex has out_degree - in_degree = 1,
- at most one vertex has in_degree - out_degree = 1,
- every other vertex has equal in_degree and out_degree,
- and all of its vertices with nonzero degree belong to a
single connected component of the underlying undirected graph.
If `source` is not None, an Eulerian path starting at `source` exists if no
other node has out_degree - in_degree = 1. This is equivalent to either
there exists an Eulerian circuit or `source` has out_degree - in_degree = 1
and the conditions above hold.
An undirected graph has an Eulerian path iff:
- exactly zero or two vertices have odd degree,
- and all of its vertices with nonzero degree belong to a
- single connected component.
If `source` is not None, an Eulerian path starting at `source` exists if
either there exists an Eulerian circuit or `source` has an odd degree and the
conditions above hold.
Parameters
----------
G : NetworkX Graph
The graph to find an euler path in.
source : node, optional
Starting node for path.
Returns
-------
Bool : True if G has an Eulerian path.
See Also
--------
is_eulerian
eulerian_path
"""
if nx.is_eulerian(G):
return True
if G.is_directed():
# Remove isolated nodes (if any) without altering the input graph
nodes_remove = [v for v in G if G.in_degree[v] == 0 and G.out_degree[v] == 0]
if nodes_remove:
G = G.copy()
G.remove_nodes_from(nodes_remove)
ins = G.in_degree
outs = G.out_degree
# Since we know it is not eulerian, outs - ins must be 1 for source
if source is not None and outs[source] - ins[source] != 1:
return False
unbalanced_ins = 0
unbalanced_outs = 0
for v in G:
if ins[v] - outs[v] == 1:
unbalanced_ins += 1
elif outs[v] - ins[v] == 1:
unbalanced_outs += 1
elif ins[v] != outs[v]:
return False
return (
unbalanced_ins <= 1 and unbalanced_outs <= 1 and nx.is_weakly_connected(G)
)
else:
# We know it is not eulerian, so degree of source must be odd.
if source is not None and G.degree[source] % 2 != 1:
return False
# Sum is 2 since we know it is not eulerian (which implies sum is 0)
return sum(d % 2 == 1 for v, d in G.degree()) == 2 and nx.is_connected(G)
[docs]def eulerian_path(G, source=None, keys=False):
"""Return an iterator over the edges of an Eulerian path in `G`.
Parameters
----------
G : NetworkX Graph
The graph in which to look for an eulerian path.
source : node or None (default: None)
The node at which to start the search. None means search over all
starting nodes.
keys : Bool (default: False)
Indicates whether to yield edge 3-tuples (u, v, edge_key).
The default yields edge 2-tuples
Yields
------
Edge tuples along the eulerian path.
Warning: If `source` provided is not the start node of an Euler path
will raise error even if an Euler Path exists.
"""
if not has_eulerian_path(G, source):
raise nx.NetworkXError("Graph has no Eulerian paths.")
if G.is_directed():
G = G.reverse()
if source is None or nx.is_eulerian(G) is False:
source = _find_path_start(G)
if G.is_multigraph():
for u, v, k in _multigraph_eulerian_circuit(G, source):
if keys:
yield u, v, k
else:
yield u, v
else:
yield from _simplegraph_eulerian_circuit(G, source)
else:
G = G.copy()
if source is None:
source = _find_path_start(G)
if G.is_multigraph():
if keys:
yield from reversed(
[(v, u, k) for u, v, k in _multigraph_eulerian_circuit(G, source)]
)
else:
yield from reversed(
[(v, u) for u, v, k in _multigraph_eulerian_circuit(G, source)]
)
else:
yield from reversed(
[(v, u) for u, v in _simplegraph_eulerian_circuit(G, source)]
)
[docs]@not_implemented_for("directed")
def eulerize(G):
"""Transforms a graph into an Eulerian graph
Parameters
----------
G : NetworkX graph
An undirected graph
Returns
-------
G : NetworkX multigraph
Raises
------
NetworkXError
If the graph is not connected.
See Also
--------
is_eulerian
eulerian_circuit
References
----------
.. [1] J. Edmonds, E. L. Johnson.
Matching, Euler tours and the Chinese postman.
Mathematical programming, Volume 5, Issue 1 (1973), 111-114.
.. [2] https://en.wikipedia.org/wiki/Eulerian_path
.. [3] http://web.math.princeton.edu/math_alive/5/Notes1.pdf
Examples
--------
>>> G = nx.complete_graph(10)
>>> H = nx.eulerize(G)
>>> nx.is_eulerian(H)
True
"""
if G.order() == 0:
raise nx.NetworkXPointlessConcept("Cannot Eulerize null graph")
if not nx.is_connected(G):
raise nx.NetworkXError("G is not connected")
odd_degree_nodes = [n for n, d in G.degree() if d % 2 == 1]
G = nx.MultiGraph(G)
if len(odd_degree_nodes) == 0:
return G
# get all shortest paths between vertices of odd degree
odd_deg_pairs_paths = [
(m, {n: nx.shortest_path(G, source=m, target=n)})
for m, n in combinations(odd_degree_nodes, 2)
]
# use inverse path lengths as edge-weights in a new graph
# store the paths in the graph for easy indexing later
Gp = nx.Graph()
for n, Ps in odd_deg_pairs_paths:
for m, P in Ps.items():
if n != m:
Gp.add_edge(m, n, weight=1 / len(P), path=P)
# find the minimum weight matching of edges in the weighted graph
best_matching = nx.Graph(list(nx.max_weight_matching(Gp)))
# duplicate each edge along each path in the set of paths in Gp
for m, n in best_matching.edges():
path = Gp[m][n]["path"]
G.add_edges_from(nx.utils.pairwise(path))
return G