529 lines
18 KiB
Python
529 lines
18 KiB
Python
# Copyright (C) 2013-2019 by
|
|
# Aric Hagberg <hagberg@lanl.gov>
|
|
# Dan Schult <dschult@colgate.edu>
|
|
# Pieter Swart <swart@lanl.gov>
|
|
# All rights reserved.
|
|
# BSD license.
|
|
#
|
|
# Authors: James Clough <james.clough91@gmail.com>
|
|
# Aric Hagberg <hagberg@lanl.gov>
|
|
# Pieter Swart <swart@lanl.gov>
|
|
# Dan Schult <dschult@colgate.edu>
|
|
# chebee7i <chebee7i@gmail.com>
|
|
"""Functions for generating line graphs."""
|
|
from itertools import combinations
|
|
from collections import defaultdict
|
|
|
|
import networkx as nx
|
|
from networkx.utils import arbitrary_element, generate_unique_node
|
|
from networkx.utils.decorators import *
|
|
|
|
__all__ = ['line_graph', 'inverse_line_graph']
|
|
|
|
|
|
def line_graph(G, create_using=None):
|
|
r"""Returns the line graph of the graph or digraph `G`.
|
|
|
|
The line graph of a graph `G` has a node for each edge in `G` and an
|
|
edge joining those nodes if the two edges in `G` share a common node. For
|
|
directed graphs, nodes are adjacent exactly when the edges they represent
|
|
form a directed path of length two.
|
|
|
|
The nodes of the line graph are 2-tuples of nodes in the original graph (or
|
|
3-tuples for multigraphs, with the key of the edge as the third element).
|
|
|
|
For information about self-loops and more discussion, see the **Notes**
|
|
section below.
|
|
|
|
Parameters
|
|
----------
|
|
G : graph
|
|
A NetworkX Graph, DiGraph, MultiGraph, or MultiDigraph.
|
|
create_using : NetworkX graph constructor, optional (default=nx.Graph)
|
|
Graph type to create. If graph instance, then cleared before populated.
|
|
|
|
Returns
|
|
-------
|
|
L : graph
|
|
The line graph of G.
|
|
|
|
Examples
|
|
--------
|
|
>>> import networkx as nx
|
|
>>> G = nx.star_graph(3)
|
|
>>> L = nx.line_graph(G)
|
|
>>> print(sorted(map(sorted, L.edges()))) # makes a 3-clique, K3
|
|
[[(0, 1), (0, 2)], [(0, 1), (0, 3)], [(0, 2), (0, 3)]]
|
|
|
|
Notes
|
|
-----
|
|
Graph, node, and edge data are not propagated to the new graph. For
|
|
undirected graphs, the nodes in G must be sortable, otherwise the
|
|
constructed line graph may not be correct.
|
|
|
|
*Self-loops in undirected graphs*
|
|
|
|
For an undirected graph `G` without multiple edges, each edge can be
|
|
written as a set `\{u, v\}`. Its line graph `L` has the edges of `G` as
|
|
its nodes. If `x` and `y` are two nodes in `L`, then `\{x, y\}` is an edge
|
|
in `L` if and only if the intersection of `x` and `y` is nonempty. Thus,
|
|
the set of all edges is determined by the set of all pairwise intersections
|
|
of edges in `G`.
|
|
|
|
Trivially, every edge in G would have a nonzero intersection with itself,
|
|
and so every node in `L` should have a self-loop. This is not so
|
|
interesting, and the original context of line graphs was with simple
|
|
graphs, which had no self-loops or multiple edges. The line graph was also
|
|
meant to be a simple graph and thus, self-loops in `L` are not part of the
|
|
standard definition of a line graph. In a pairwise intersection matrix,
|
|
this is analogous to excluding the diagonal entries from the line graph
|
|
definition.
|
|
|
|
Self-loops and multiple edges in `G` add nodes to `L` in a natural way, and
|
|
do not require any fundamental changes to the definition. It might be
|
|
argued that the self-loops we excluded before should now be included.
|
|
However, the self-loops are still "trivial" in some sense and thus, are
|
|
usually excluded.
|
|
|
|
*Self-loops in directed graphs*
|
|
|
|
For a directed graph `G` without multiple edges, each edge can be written
|
|
as a tuple `(u, v)`. Its line graph `L` has the edges of `G` as its
|
|
nodes. If `x` and `y` are two nodes in `L`, then `(x, y)` is an edge in `L`
|
|
if and only if the tail of `x` matches the head of `y`, for example, if `x
|
|
= (a, b)` and `y = (b, c)` for some vertices `a`, `b`, and `c` in `G`.
|
|
|
|
Due to the directed nature of the edges, it is no longer the case that
|
|
every edge in `G` should have a self-loop in `L`. Now, the only time
|
|
self-loops arise is if a node in `G` itself has a self-loop. So such
|
|
self-loops are no longer "trivial" but instead, represent essential
|
|
features of the topology of `G`. For this reason, the historical
|
|
development of line digraphs is such that self-loops are included. When the
|
|
graph `G` has multiple edges, once again only superficial changes are
|
|
required to the definition.
|
|
|
|
References
|
|
----------
|
|
* Harary, Frank, and Norman, Robert Z., "Some properties of line digraphs",
|
|
Rend. Circ. Mat. Palermo, II. Ser. 9 (1960), 161--168.
|
|
* Hemminger, R. L.; Beineke, L. W. (1978), "Line graphs and line digraphs",
|
|
in Beineke, L. W.; Wilson, R. J., Selected Topics in Graph Theory,
|
|
Academic Press Inc., pp. 271--305.
|
|
|
|
"""
|
|
if G.is_directed():
|
|
L = _lg_directed(G, create_using=create_using)
|
|
else:
|
|
L = _lg_undirected(G, selfloops=False, create_using=create_using)
|
|
return L
|
|
|
|
|
|
def _node_func(G):
|
|
"""Returns a function which returns a sorted node for line graphs.
|
|
|
|
When constructing a line graph for undirected graphs, we must normalize
|
|
the ordering of nodes as they appear in the edge.
|
|
|
|
"""
|
|
if G.is_multigraph():
|
|
def sorted_node(u, v, key):
|
|
return (u, v, key) if u <= v else (v, u, key)
|
|
else:
|
|
def sorted_node(u, v):
|
|
return (u, v) if u <= v else (v, u)
|
|
return sorted_node
|
|
|
|
|
|
def _edge_func(G):
|
|
"""Returns the edges from G, handling keys for multigraphs as necessary.
|
|
|
|
"""
|
|
if G.is_multigraph():
|
|
def get_edges(nbunch=None):
|
|
return G.edges(nbunch, keys=True)
|
|
else:
|
|
def get_edges(nbunch=None):
|
|
return G.edges(nbunch)
|
|
return get_edges
|
|
|
|
|
|
def _sorted_edge(u, v):
|
|
"""Returns a sorted edge.
|
|
|
|
During the construction of a line graph for undirected graphs, the data
|
|
structure can be a multigraph even though the line graph will never have
|
|
multiple edges between its nodes. For this reason, we must make sure not
|
|
to add any edge more than once. This requires that we build up a list of
|
|
edges to add and then remove all duplicates. And so, we must normalize
|
|
the representation of the edges.
|
|
|
|
"""
|
|
return (u, v) if u <= v else (v, u)
|
|
|
|
|
|
def _lg_directed(G, create_using=None):
|
|
"""Returns the line graph L of the (multi)digraph G.
|
|
|
|
Edges in G appear as nodes in L, represented as tuples of the form (u,v)
|
|
or (u,v,key) if G is a multidigraph. A node in L corresponding to the edge
|
|
(u,v) is connected to every node corresponding to an edge (v,w).
|
|
|
|
Parameters
|
|
----------
|
|
G : digraph
|
|
A directed graph or directed multigraph.
|
|
create_using : NetworkX graph constructor, optional
|
|
Graph type to create. If graph instance, then cleared before populated.
|
|
Default is to use the same graph class as `G`.
|
|
|
|
"""
|
|
L = nx.empty_graph(0, create_using, default=G.__class__)
|
|
|
|
# Create a graph specific edge function.
|
|
get_edges = _edge_func(G)
|
|
|
|
for from_node in get_edges():
|
|
# from_node is: (u,v) or (u,v,key)
|
|
L.add_node(from_node)
|
|
for to_node in get_edges(from_node[1]):
|
|
L.add_edge(from_node, to_node)
|
|
|
|
return L
|
|
|
|
|
|
def _lg_undirected(G, selfloops=False, create_using=None):
|
|
"""Returns the line graph L of the (multi)graph G.
|
|
|
|
Edges in G appear as nodes in L, represented as sorted tuples of the form
|
|
(u,v), or (u,v,key) if G is a multigraph. A node in L corresponding to
|
|
the edge {u,v} is connected to every node corresponding to an edge that
|
|
involves u or v.
|
|
|
|
Parameters
|
|
----------
|
|
G : graph
|
|
An undirected graph or multigraph.
|
|
selfloops : bool
|
|
If `True`, then self-loops are included in the line graph. If `False`,
|
|
they are excluded.
|
|
create_using : NetworkX graph constructor, optional (default=nx.Graph)
|
|
Graph type to create. If graph instance, then cleared before populated.
|
|
|
|
Notes
|
|
-----
|
|
The standard algorithm for line graphs of undirected graphs does not
|
|
produce self-loops.
|
|
|
|
"""
|
|
L = nx.empty_graph(0, create_using, default=G.__class__)
|
|
|
|
# Graph specific functions for edges and sorted nodes.
|
|
get_edges = _edge_func(G)
|
|
sorted_node = _node_func(G)
|
|
|
|
# Determine if we include self-loops or not.
|
|
shift = 0 if selfloops else 1
|
|
|
|
edges = set([])
|
|
for u in G:
|
|
# Label nodes as a sorted tuple of nodes in original graph.
|
|
nodes = [sorted_node(*x) for x in get_edges(u)]
|
|
|
|
if len(nodes) == 1:
|
|
# Then the edge will be an isolated node in L.
|
|
L.add_node(nodes[0])
|
|
|
|
# Add a clique of `nodes` to graph. To prevent double adding edges,
|
|
# especially important for multigraphs, we store the edges in
|
|
# canonical form in a set.
|
|
for i, a in enumerate(nodes):
|
|
edges.update([_sorted_edge(a, b) for b in nodes[i + shift:]])
|
|
|
|
L.add_edges_from(edges)
|
|
return L
|
|
|
|
|
|
@not_implemented_for('directed')
|
|
@not_implemented_for('multigraph')
|
|
def inverse_line_graph(G):
|
|
""" Returns the inverse line graph of graph G.
|
|
|
|
If H is a graph, and G is the line graph of H, such that H = L(G).
|
|
Then H is the inverse line graph of G.
|
|
|
|
Not all graphs are line graphs and these do not have an inverse line graph.
|
|
In these cases this generator returns a NetworkXError.
|
|
|
|
Parameters
|
|
----------
|
|
G : graph
|
|
A NetworkX Graph
|
|
|
|
Returns
|
|
-------
|
|
H : graph
|
|
The inverse line graph of G.
|
|
|
|
Raises
|
|
------
|
|
NetworkXNotImplemented
|
|
If G is directed or a multigraph
|
|
|
|
NetworkXError
|
|
If G is not a line graph
|
|
|
|
Notes
|
|
-----
|
|
This is an implementation of the Roussopoulos algorithm.
|
|
|
|
If G consists of multiple components, then the algorithm doesn't work.
|
|
You should invert every component seperately:
|
|
|
|
>>> K5 = nx.complete_graph(5)
|
|
>>> P4 = nx.Graph([('a', 'b'), ('b', 'c'), ('c', 'd')])
|
|
>>> G = nx.union(K5, P4)
|
|
>>> root_graphs = []
|
|
>>> for comp in nx.connected_components(G):
|
|
... root_graphs.append(nx.inverse_line_graph(G.subgraph(comp)))
|
|
>>> len(root_graphs)
|
|
2
|
|
|
|
References
|
|
----------
|
|
* Roussopolous, N, "A max {m, n} algorithm for determining the graph H from
|
|
its line graph G", Information Processing Letters 2, (1973), 108--112.
|
|
|
|
"""
|
|
if G.number_of_nodes() == 0:
|
|
a = generate_unique_node()
|
|
H = nx.Graph()
|
|
H.add_node(a)
|
|
return H
|
|
elif G.number_of_nodes() == 1:
|
|
v = list(G)[0]
|
|
a = (v, 0)
|
|
b = (v, 1)
|
|
H = nx.Graph([(a,b)])
|
|
return H
|
|
elif G.number_of_nodes() > 1 and G.number_of_edges() == 0:
|
|
msg = (
|
|
"inverse_line_graph() doesn't work on an edgeless graph. "
|
|
"Please use this function on each component seperately."
|
|
)
|
|
raise nx.NetworkXError(msg)
|
|
|
|
starting_cell = _select_starting_cell(G)
|
|
P = _find_partition(G, starting_cell)
|
|
# count how many times each vertex appears in the partition set
|
|
P_count = {u: 0 for u in G.nodes()}
|
|
for p in P:
|
|
for u in p:
|
|
P_count[u] += 1
|
|
|
|
if max(P_count.values()) > 2:
|
|
msg = "G is not a line graph (vertex found in more " \
|
|
"than two partition cells)"
|
|
raise nx.NetworkXError(msg)
|
|
W = tuple([(u,) for u in P_count if P_count[u] == 1])
|
|
H = nx.Graph()
|
|
H.add_nodes_from(P)
|
|
H.add_nodes_from(W)
|
|
for a, b in combinations(H.nodes(), 2):
|
|
if len(set(a).intersection(set(b))) > 0:
|
|
H.add_edge(a, b)
|
|
return H
|
|
|
|
|
|
def _triangles(G, e):
|
|
""" Return list of all triangles containing edge e"""
|
|
u, v = e
|
|
if u not in G:
|
|
raise nx.NetworkXError("Vertex %s not in graph" % u)
|
|
if v not in G[u]:
|
|
raise nx.NetworkXError("Edge (%s, %s) not in graph" % (u, v))
|
|
triangle_list = []
|
|
for x in G[u]:
|
|
if x in G[v]:
|
|
triangle_list.append((u, v, x))
|
|
return triangle_list
|
|
|
|
|
|
def _odd_triangle(G, T):
|
|
""" Test whether T is an odd triangle in G
|
|
|
|
Parameters
|
|
----------
|
|
G : NetworkX Graph
|
|
T : 3-tuple of vertices forming triangle in G
|
|
|
|
Returns
|
|
-------
|
|
True is T is an odd triangle
|
|
False otherwise
|
|
|
|
Raises
|
|
------
|
|
NetworkXError
|
|
T is not a triangle in G
|
|
|
|
Notes
|
|
-----
|
|
An odd triangle is one in which there exists another vertex in G which is
|
|
adjacent to either exactly one or exactly all three of the vertices in the
|
|
triangle.
|
|
|
|
"""
|
|
for u in T:
|
|
if u not in G.nodes():
|
|
raise nx.NetworkXError("Vertex %s not in graph" % u)
|
|
for e in list(combinations(T, 2)):
|
|
if e[0] not in G[e[1]]:
|
|
raise nx.NetworkXError("Edge (%s, %s) not in graph" % (e[0], e[1]))
|
|
|
|
T_neighbors = defaultdict(int)
|
|
for t in T:
|
|
for v in G[t]:
|
|
if v not in T:
|
|
T_neighbors[v] += 1
|
|
for v in T_neighbors:
|
|
if T_neighbors[v] in [1, 3]:
|
|
return True
|
|
return False
|
|
|
|
|
|
def _find_partition(G, starting_cell):
|
|
""" Find a partition of the vertices of G into cells of complete graphs
|
|
|
|
Parameters
|
|
----------
|
|
G : NetworkX Graph
|
|
starting_cell : tuple of vertices in G which form a cell
|
|
|
|
Returns
|
|
-------
|
|
List of tuples of vertices of G
|
|
|
|
Raises
|
|
------
|
|
NetworkXError
|
|
If a cell is not a complete subgraph then G is not a line graph
|
|
"""
|
|
G_partition = G.copy()
|
|
P = [starting_cell] # partition set
|
|
G_partition.remove_edges_from(list(combinations(starting_cell, 2)))
|
|
# keep list of partitioned nodes which might have an edge in G_partition
|
|
partitioned_vertices = list(starting_cell)
|
|
while G_partition.number_of_edges() > 0:
|
|
# there are still edges left and so more cells to be made
|
|
u = partitioned_vertices[-1]
|
|
deg_u = len(G_partition[u])
|
|
if deg_u == 0:
|
|
# if u has no edges left in G_partition then we have found
|
|
# all of its cells so we do not need to keep looking
|
|
partitioned_vertices.pop()
|
|
else:
|
|
# if u still has edges then we need to find its other cell
|
|
# this other cell must be a complete subgraph or else G is
|
|
# not a line graph
|
|
new_cell = [u] + list(G_partition[u])
|
|
for u in new_cell:
|
|
for v in new_cell:
|
|
if (u != v) and (v not in G_partition[u]):
|
|
msg = "G is not a line graph" \
|
|
"(partition cell not a complete subgraph)"
|
|
raise nx.NetworkXError(msg)
|
|
P.append(tuple(new_cell))
|
|
G_partition.remove_edges_from(list(combinations(new_cell, 2)))
|
|
partitioned_vertices += new_cell
|
|
return P
|
|
|
|
|
|
def _select_starting_cell(G, starting_edge=None):
|
|
""" Select a cell to initiate _find_partition
|
|
|
|
Parameters
|
|
----------
|
|
G : NetworkX Graph
|
|
starting_edge: an edge to build the starting cell from
|
|
|
|
Returns
|
|
-------
|
|
Tuple of vertices in G
|
|
|
|
Raises
|
|
------
|
|
NetworkXError
|
|
If it is determined that G is not a line graph
|
|
|
|
Notes
|
|
-----
|
|
If starting edge not specified then pick an arbitrary edge - doesn't
|
|
matter which. However, this function may call itself requiring a
|
|
specific starting edge. Note that the r, s notation for counting
|
|
triangles is the same as in the Roussopoulos paper cited above.
|
|
"""
|
|
if starting_edge is None:
|
|
e = arbitrary_element(list(G.edges()))
|
|
else:
|
|
e = starting_edge
|
|
if e[0] not in G[e[1]]:
|
|
msg = 'starting_edge (%s, %s) is not in the Graph'
|
|
raise nx.NetworkXError(msg % e)
|
|
e_triangles = _triangles(G, e)
|
|
r = len(e_triangles)
|
|
if r == 0:
|
|
# there are no triangles containing e, so the starting cell is just e
|
|
starting_cell = e
|
|
elif r == 1:
|
|
# there is exactly one triangle, T, containing e. If other 2 edges
|
|
# of T belong only to this triangle then T is starting cell
|
|
T = e_triangles[0]
|
|
a, b, c = T
|
|
# ab was original edge so check the other 2 edges
|
|
ac_edges = [x for x in _triangles(G, (a, c))]
|
|
bc_edges = [x for x in _triangles(G, (b, c))]
|
|
if len(ac_edges) == 1:
|
|
if len(bc_edges) == 1:
|
|
starting_cell = T
|
|
else:
|
|
return _select_starting_cell(G, starting_edge=(b, c))
|
|
else:
|
|
return _select_starting_cell(G, starting_edge=(a, c))
|
|
else:
|
|
# r >= 2 so we need to count the number of odd triangles, s
|
|
s = 0
|
|
odd_triangles = []
|
|
for T in e_triangles:
|
|
if _odd_triangle(G, T):
|
|
s += 1
|
|
odd_triangles.append(T)
|
|
if r == 2 and s == 0:
|
|
# in this case either triangle works, so just use T
|
|
starting_cell = T
|
|
elif r - 1 <= s <= r:
|
|
# check if odd triangles containing e form complete subgraph
|
|
# there must be exactly s+2 of them
|
|
# and they must all be connected
|
|
triangle_nodes = set([])
|
|
for T in odd_triangles:
|
|
for x in T:
|
|
triangle_nodes.add(x)
|
|
if len(triangle_nodes) == s + 2:
|
|
for u in triangle_nodes:
|
|
for v in triangle_nodes:
|
|
if u != v and (v not in G[u]):
|
|
msg = "G is not a line graph (odd triangles " \
|
|
"do not form complete subgraph)"
|
|
raise nx.NetworkXError(msg)
|
|
# otherwise then we can use this as the starting cell
|
|
starting_cell = tuple(triangle_nodes)
|
|
else:
|
|
msg = "G is not a line graph (odd triangles " \
|
|
"do not form complete subgraph)"
|
|
raise nx.NetworkXError(msg)
|
|
else:
|
|
msg = "G is not a line graph (incorrect number of " \
|
|
"odd triangles around starting edge)"
|
|
raise nx.NetworkXError(msg)
|
|
return starting_cell
|