"""Functions for generating line graphs.
"""
# Copyright (C) 2013 by
# Aric Hagberg <hagberg@lanl.gov>
# Dan Schult <dschult@colgate.edu>
# Pieter Swart <swart@lanl.gov>
# All rights reserved.
# BSD license.
__author__ = "\n".join(["Aric Hagberg (hagberg@lanl.gov)",
"Pieter Swart (swart@lanl.gov)",
"Dan Schult (dschult@colgate.edu)",
"chebee7i (chebee7i@gmail.com)"])
__all__ = ['line_graph']
[docs]def line_graph(G, create_using=None):
"""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.
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):
"""Return 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 : None
A digraph instance used to populate the line graph.
"""
if create_using is None:
L = G.fresh_copy()
else:
L = create_using
# 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):
"""Return 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 : None
A graph instance used to populate the line graph.
Notes
-----
The standard algorithm for line graphs of undirected graphs does not
produce self-loops.
"""
if create_using is None:
L = G.fresh_copy()
else:
L = create_using
# 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