"""
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]
@nx._dispatchable
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.
Graphs with isolated vertices (i.e. vertices with zero degree) are not
considered to have Eulerian circuits. Therefore, if the graph is not
connected (or not strongly connected, for directed graphs), this function
returns False.
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
If you prefer to allow graphs with isolated vertices to have Eulerian circuits,
you can first remove such vertices and then call `is_eulerian` as below example shows.
>>> G = nx.Graph([(0, 1), (1, 2), (0, 2)])
>>> G.add_node(3)
>>> nx.is_eulerian(G)
False
>>> G.remove_nodes_from(list(nx.isolates(G)))
>>> nx.is_eulerian(G)
True
"""
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]
@nx._dispatchable
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.
See Also
--------
has_eulerian_path
is_eulerian
"""
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]
@nx._dispatchable
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]
@nx._dispatchable
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 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 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.
Graphs with isolated vertices (i.e. vertices with zero degree) are not considered
to have an Eulerian path. Therefore, if the graph is not connected (or not strongly
connected, for directed graphs), this function returns False.
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.
Examples
--------
If you prefer to allow graphs with isolated vertices to have Eulerian path,
you can first remove such vertices and then call `has_eulerian_path` as below example shows.
>>> G = nx.Graph([(0, 1), (1, 2), (0, 2)])
>>> G.add_node(3)
>>> nx.has_eulerian_path(G)
False
>>> G.remove_nodes_from(list(nx.isolates(G)))
>>> nx.has_eulerian_path(G)
True
See Also
--------
is_eulerian
eulerian_path
"""
if nx.is_eulerian(G):
return True
if G.is_directed():
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]
@nx._dispatchable
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")
@nx._dispatchable(returns_graph=True)
def eulerize(G):
"""Transforms a graph into an Eulerian graph.
If `G` is Eulerian the result is `G` as a MultiGraph, otherwise the result is a smallest
(in terms of the number of edges) multigraph whose underlying simple graph is `G`.
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 the number of vertices in a graph + 1 as an upper bound on
# the maximum length of a path in G
upper_bound_on_max_path_length = len(G) + 1
# use "len(G) + 1 - len(P)",
# where P is a shortest path between vertices n and m,
# 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=upper_bound_on_max_path_length - 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
