"""Functions for generating line graphs."""
from itertools import combinations
from collections import defaultdict
from functools import partial
import networkx as nx
from networkx.utils import arbitrary_element
from networkx.utils.decorators import not_implemented_for
__all__ = ["line_graph", "inverse_line_graph"]
[docs]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
--------
>>> 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 _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 = partial(G.edges, keys=True) if G.is_multigraph() else G.edges
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.
get_edges = partial(G.edges, keys=True) if G.is_multigraph() else G.edges
# Determine if we include self-loops or not.
shift = 0 if selfloops else 1
# Introduce numbering of nodes
node_index = {n: i for i, n in enumerate(G)}
# Lift canonical representation of nodes to edges in line graph
edge_key_function = lambda edge: (node_index[edge[0]], node_index[edge[1]])
edges = set()
for u in G:
# Label nodes as a sorted tuple of nodes in original graph.
# Decide on representation of {u, v} as (u, v) or (v, u) depending on node_index.
# -> This ensures a canonical representation and avoids comparing values of different types.
nodes = [tuple(sorted(x[:2], key=node_index.get)) + x[2:] 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(
[
tuple(sorted((a, b), key=edge_key_function))
for b in nodes[i + shift :]
]
)
L.add_edges_from(edges)
return L
[docs]@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 G = L(H).
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 function raises 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:
return nx.empty_graph(1)
elif G.number_of_nodes() == 1:
v = arbitrary_element(G)
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 any(a_bit in b for a_bit in a):
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(f"Vertex {u} not in graph")
if v not in G[u]:
raise nx.NetworkXError(f"Edge ({u}, {v}) not in graph")
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(f"Vertex {u} not in graph")
for e in list(combinations(T, 2)):
if e[0] not in G[e[1]]:
raise nx.NetworkXError(f"Edge ({e[0]}, {e[1]}) not in graph")
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(G.edges())
else:
e = starting_edge
if e[1] not in G[e[0]]:
msg = f"starting_edge ({e[0]}, {e[1]}) is not in the Graph"
raise nx.NetworkXError(msg)
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