453 lines
13 KiB
Python
453 lines
13 KiB
Python
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"""
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Eulerian circuits and graphs.
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"""
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from itertools import combinations
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import networkx as nx
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from ..utils import arbitrary_element, not_implemented_for
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__all__ = [
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"is_eulerian",
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"eulerian_circuit",
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"eulerize",
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"is_semieulerian",
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"has_eulerian_path",
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"eulerian_path",
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]
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def is_eulerian(G):
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"""Returns True if and only if `G` is Eulerian.
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A graph is *Eulerian* if it has an Eulerian circuit. An *Eulerian
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circuit* is a closed walk that includes each edge of a graph exactly
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once.
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Graphs with isolated vertices (i.e. vertices with zero degree) are not
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considered to have Eulerian circuits. Therefore, if the graph is not
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connected (or not strongly connected, for directed graphs), this function
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returns False.
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Parameters
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----------
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G : NetworkX graph
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A graph, either directed or undirected.
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Examples
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--------
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>>> nx.is_eulerian(nx.DiGraph({0: [3], 1: [2], 2: [3], 3: [0, 1]}))
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True
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>>> nx.is_eulerian(nx.complete_graph(5))
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True
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>>> nx.is_eulerian(nx.petersen_graph())
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False
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If you prefer to allow graphs with isolated vertices to have Eulerian circuits,
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you can first remove such vertices and then call `is_eulerian` as below example shows.
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>>> G = nx.Graph([(0, 1), (1, 2), (0, 2)])
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>>> G.add_node(3)
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>>> nx.is_eulerian(G)
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False
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>>> G.remove_nodes_from(list(nx.isolates(G)))
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>>> nx.is_eulerian(G)
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True
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"""
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if G.is_directed():
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# Every node must have equal in degree and out degree and the
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# graph must be strongly connected
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return all(
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G.in_degree(n) == G.out_degree(n) for n in G
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) and nx.is_strongly_connected(G)
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# An undirected Eulerian graph has no vertices of odd degree and
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# must be connected.
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return all(d % 2 == 0 for v, d in G.degree()) and nx.is_connected(G)
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def is_semieulerian(G):
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"""Return True iff `G` is semi-Eulerian.
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G is semi-Eulerian if it has an Eulerian path but no Eulerian circuit.
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See Also
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--------
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has_eulerian_path
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is_eulerian
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"""
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return has_eulerian_path(G) and not is_eulerian(G)
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def _find_path_start(G):
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"""Return a suitable starting vertex for an Eulerian path.
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If no path exists, return None.
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"""
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if not has_eulerian_path(G):
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return None
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if is_eulerian(G):
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return arbitrary_element(G)
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if G.is_directed():
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v1, v2 = (v for v in G if G.in_degree(v) != G.out_degree(v))
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# Determines which is the 'start' node (as opposed to the 'end')
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if G.out_degree(v1) > G.in_degree(v1):
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return v1
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else:
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return v2
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else:
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# In an undirected graph randomly choose one of the possibilities
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start = [v for v in G if G.degree(v) % 2 != 0][0]
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return start
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def _simplegraph_eulerian_circuit(G, source):
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if G.is_directed():
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degree = G.out_degree
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edges = G.out_edges
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else:
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degree = G.degree
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edges = G.edges
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vertex_stack = [source]
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last_vertex = None
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while vertex_stack:
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current_vertex = vertex_stack[-1]
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if degree(current_vertex) == 0:
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if last_vertex is not None:
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yield (last_vertex, current_vertex)
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last_vertex = current_vertex
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vertex_stack.pop()
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else:
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_, next_vertex = arbitrary_element(edges(current_vertex))
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vertex_stack.append(next_vertex)
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G.remove_edge(current_vertex, next_vertex)
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def _multigraph_eulerian_circuit(G, source):
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if G.is_directed():
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degree = G.out_degree
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edges = G.out_edges
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else:
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degree = G.degree
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edges = G.edges
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vertex_stack = [(source, None)]
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last_vertex = None
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last_key = None
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while vertex_stack:
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current_vertex, current_key = vertex_stack[-1]
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if degree(current_vertex) == 0:
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if last_vertex is not None:
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yield (last_vertex, current_vertex, last_key)
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last_vertex, last_key = current_vertex, current_key
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vertex_stack.pop()
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else:
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triple = arbitrary_element(edges(current_vertex, keys=True))
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_, next_vertex, next_key = triple
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vertex_stack.append((next_vertex, next_key))
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G.remove_edge(current_vertex, next_vertex, next_key)
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def eulerian_circuit(G, source=None, keys=False):
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"""Returns an iterator over the edges of an Eulerian circuit in `G`.
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An *Eulerian circuit* is a closed walk that includes each edge of a
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graph exactly once.
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Parameters
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----------
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G : NetworkX graph
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A graph, either directed or undirected.
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source : node, optional
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Starting node for circuit.
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keys : bool
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If False, edges generated by this function will be of the form
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``(u, v)``. Otherwise, edges will be of the form ``(u, v, k)``.
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This option is ignored unless `G` is a multigraph.
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Returns
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-------
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edges : iterator
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An iterator over edges in the Eulerian circuit.
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Raises
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------
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NetworkXError
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If the graph is not Eulerian.
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See Also
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--------
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is_eulerian
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Notes
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-----
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This is a linear time implementation of an algorithm adapted from [1]_.
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For general information about Euler tours, see [2]_.
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References
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----------
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.. [1] J. Edmonds, E. L. Johnson.
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Matching, Euler tours and the Chinese postman.
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Mathematical programming, Volume 5, Issue 1 (1973), 111-114.
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.. [2] https://en.wikipedia.org/wiki/Eulerian_path
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Examples
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--------
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To get an Eulerian circuit in an undirected graph::
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>>> G = nx.complete_graph(3)
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>>> list(nx.eulerian_circuit(G))
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[(0, 2), (2, 1), (1, 0)]
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>>> list(nx.eulerian_circuit(G, source=1))
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[(1, 2), (2, 0), (0, 1)]
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To get the sequence of vertices in an Eulerian circuit::
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>>> [u for u, v in nx.eulerian_circuit(G)]
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[0, 2, 1]
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"""
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if not is_eulerian(G):
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raise nx.NetworkXError("G is not Eulerian.")
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if G.is_directed():
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G = G.reverse()
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else:
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G = G.copy()
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if source is None:
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source = arbitrary_element(G)
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if G.is_multigraph():
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for u, v, k in _multigraph_eulerian_circuit(G, source):
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if keys:
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yield u, v, k
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else:
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yield u, v
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else:
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yield from _simplegraph_eulerian_circuit(G, source)
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def has_eulerian_path(G, source=None):
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"""Return True iff `G` has an Eulerian path.
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An Eulerian path is a path in a graph which uses each edge of a graph
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exactly once. If `source` is specified, then this function checks
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whether an Eulerian path that starts at node `source` exists.
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A directed graph has an Eulerian path iff:
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- at most one vertex has out_degree - in_degree = 1,
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- at most one vertex has in_degree - out_degree = 1,
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- every other vertex has equal in_degree and out_degree,
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- and all of its vertices belong to a single connected
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component of the underlying undirected graph.
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If `source` is not None, an Eulerian path starting at `source` exists if no
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other node has out_degree - in_degree = 1. This is equivalent to either
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there exists an Eulerian circuit or `source` has out_degree - in_degree = 1
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and the conditions above hold.
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An undirected graph has an Eulerian path iff:
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- exactly zero or two vertices have odd degree,
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- and all of its vertices belong to a single connected component.
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If `source` is not None, an Eulerian path starting at `source` exists if
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either there exists an Eulerian circuit or `source` has an odd degree and the
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conditions above hold.
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Graphs with isolated vertices (i.e. vertices with zero degree) are not considered
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to have an Eulerian path. Therefore, if the graph is not connected (or not strongly
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connected, for directed graphs), this function returns False.
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Parameters
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----------
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G : NetworkX Graph
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The graph to find an euler path in.
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source : node, optional
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Starting node for path.
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Returns
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-------
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Bool : True if G has an Eulerian path.
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Examples
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--------
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If you prefer to allow graphs with isolated vertices to have Eulerian path,
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you can first remove such vertices and then call `has_eulerian_path` as below example shows.
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>>> G = nx.Graph([(0, 1), (1, 2), (0, 2)])
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>>> G.add_node(3)
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>>> nx.has_eulerian_path(G)
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False
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>>> G.remove_nodes_from(list(nx.isolates(G)))
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>>> nx.has_eulerian_path(G)
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True
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See Also
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--------
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is_eulerian
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eulerian_path
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"""
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if nx.is_eulerian(G):
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return True
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if G.is_directed():
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ins = G.in_degree
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outs = G.out_degree
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# Since we know it is not eulerian, outs - ins must be 1 for source
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if source is not None and outs[source] - ins[source] != 1:
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return False
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unbalanced_ins = 0
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unbalanced_outs = 0
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for v in G:
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if ins[v] - outs[v] == 1:
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unbalanced_ins += 1
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elif outs[v] - ins[v] == 1:
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unbalanced_outs += 1
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elif ins[v] != outs[v]:
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return False
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return (
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unbalanced_ins <= 1 and unbalanced_outs <= 1 and nx.is_weakly_connected(G)
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)
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else:
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# We know it is not eulerian, so degree of source must be odd.
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if source is not None and G.degree[source] % 2 != 1:
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return False
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# Sum is 2 since we know it is not eulerian (which implies sum is 0)
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return sum(d % 2 == 1 for v, d in G.degree()) == 2 and nx.is_connected(G)
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def eulerian_path(G, source=None, keys=False):
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"""Return an iterator over the edges of an Eulerian path in `G`.
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Parameters
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----------
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G : NetworkX Graph
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The graph in which to look for an eulerian path.
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source : node or None (default: None)
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The node at which to start the search. None means search over all
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starting nodes.
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keys : Bool (default: False)
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Indicates whether to yield edge 3-tuples (u, v, edge_key).
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The default yields edge 2-tuples
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Yields
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------
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Edge tuples along the eulerian path.
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Warning: If `source` provided is not the start node of an Euler path
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will raise error even if an Euler Path exists.
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"""
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if not has_eulerian_path(G, source):
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raise nx.NetworkXError("Graph has no Eulerian paths.")
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if G.is_directed():
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G = G.reverse()
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if source is None or nx.is_eulerian(G) is False:
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source = _find_path_start(G)
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if G.is_multigraph():
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for u, v, k in _multigraph_eulerian_circuit(G, source):
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if keys:
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yield u, v, k
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else:
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yield u, v
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else:
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yield from _simplegraph_eulerian_circuit(G, source)
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else:
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G = G.copy()
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if source is None:
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source = _find_path_start(G)
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if G.is_multigraph():
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if keys:
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yield from reversed(
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[(v, u, k) for u, v, k in _multigraph_eulerian_circuit(G, source)]
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)
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else:
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yield from reversed(
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[(v, u) for u, v, k in _multigraph_eulerian_circuit(G, source)]
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)
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else:
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yield from reversed(
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[(v, u) for u, v in _simplegraph_eulerian_circuit(G, source)]
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)
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@not_implemented_for("directed")
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def eulerize(G):
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"""Transforms a graph into an Eulerian graph
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Parameters
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----------
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G : NetworkX graph
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An undirected graph
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Returns
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||
|
|
-------
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||
|
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G : NetworkX multigraph
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||
|
|
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||
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Raises
|
||
|
|
------
|
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|
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NetworkXError
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||
|
|
If the graph is not connected.
|
||
|
|
|
||
|
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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
|