"""
Algorithms for chordal graphs.
A graph is chordal if every cycle of length at least 4 has a chord
(an edge joining two nodes not adjacent in the cycle).
https://en.wikipedia.org/wiki/Chordal_graph
"""
import sys
import networkx as nx
from networkx.algorithms.components import connected_components
from networkx.utils import arbitrary_element, not_implemented_for
__all__ = [
"is_chordal",
"find_induced_nodes",
"chordal_graph_cliques",
"chordal_graph_treewidth",
"NetworkXTreewidthBoundExceeded",
"complete_to_chordal_graph",
]
class NetworkXTreewidthBoundExceeded(nx.NetworkXException):
"""Exception raised when a treewidth bound has been provided and it has
been exceeded"""
[docs]
@not_implemented_for("directed")
@not_implemented_for("multigraph")
@nx._dispatchable
def is_chordal(G):
"""Checks whether G is a chordal graph.
A graph is chordal if every cycle of length at least 4 has a chord
(an edge joining two nodes not adjacent in the cycle).
Parameters
----------
G : graph
A NetworkX graph.
Returns
-------
chordal : bool
True if G is a chordal graph and False otherwise.
Raises
------
NetworkXNotImplemented
The algorithm does not support DiGraph, MultiGraph and MultiDiGraph.
Examples
--------
>>> e = [
... (1, 2),
... (1, 3),
... (2, 3),
... (2, 4),
... (3, 4),
... (3, 5),
... (3, 6),
... (4, 5),
... (4, 6),
... (5, 6),
... ]
>>> G = nx.Graph(e)
>>> nx.is_chordal(G)
True
Notes
-----
The routine tries to go through every node following maximum cardinality
search. It returns False when it finds that the separator for any node
is not a clique. Based on the algorithms in [1]_.
Self loops are ignored.
References
----------
.. [1] R. E. Tarjan and M. Yannakakis, Simple linear-time algorithms
to test chordality of graphs, test acyclicity of hypergraphs, and
selectively reduce acyclic hypergraphs, SIAM J. Comput., 13 (1984),
pp. 566–579.
"""
if len(G.nodes) <= 3:
return True
return len(_find_chordality_breaker(G)) == 0
[docs]
@nx._dispatchable
def find_induced_nodes(G, s, t, treewidth_bound=sys.maxsize):
"""Returns the set of induced nodes in the path from s to t.
Parameters
----------
G : graph
A chordal NetworkX graph
s : node
Source node to look for induced nodes
t : node
Destination node to look for induced nodes
treewidth_bound: float
Maximum treewidth acceptable for the graph H. The search
for induced nodes will end as soon as the treewidth_bound is exceeded.
Returns
-------
induced_nodes : Set of nodes
The set of induced nodes in the path from s to t in G
Raises
------
NetworkXError
The algorithm does not support DiGraph, MultiGraph and MultiDiGraph.
If the input graph is an instance of one of these classes, a
:exc:`NetworkXError` is raised.
The algorithm can only be applied to chordal graphs. If the input
graph is found to be non-chordal, a :exc:`NetworkXError` is raised.
Examples
--------
>>> G = nx.Graph()
>>> G = nx.generators.classic.path_graph(10)
>>> induced_nodes = nx.find_induced_nodes(G, 1, 9, 2)
>>> sorted(induced_nodes)
[1, 2, 3, 4, 5, 6, 7, 8, 9]
Notes
-----
G must be a chordal graph and (s,t) an edge that is not in G.
If a treewidth_bound is provided, the search for induced nodes will end
as soon as the treewidth_bound is exceeded.
The algorithm is inspired by Algorithm 4 in [1]_.
A formal definition of induced node can also be found on that reference.
Self Loops are ignored
References
----------
.. [1] Learning Bounded Treewidth Bayesian Networks.
Gal Elidan, Stephen Gould; JMLR, 9(Dec):2699--2731, 2008.
http://jmlr.csail.mit.edu/papers/volume9/elidan08a/elidan08a.pdf
"""
if not is_chordal(G):
raise nx.NetworkXError("Input graph is not chordal.")
H = nx.Graph(G)
H.add_edge(s, t)
induced_nodes = set()
triplet = _find_chordality_breaker(H, s, treewidth_bound)
while triplet:
(u, v, w) = triplet
induced_nodes.update(triplet)
for n in triplet:
if n != s:
H.add_edge(s, n)
triplet = _find_chordality_breaker(H, s, treewidth_bound)
if induced_nodes:
# Add t and the second node in the induced path from s to t.
induced_nodes.add(t)
for u in G[s]:
if len(induced_nodes & set(G[u])) == 2:
induced_nodes.add(u)
break
return induced_nodes
[docs]
@nx._dispatchable
def chordal_graph_cliques(G):
"""Returns all maximal cliques of a chordal graph.
The algorithm breaks the graph in connected components and performs a
maximum cardinality search in each component to get the cliques.
Parameters
----------
G : graph
A NetworkX graph
Yields
------
frozenset of nodes
Maximal cliques, each of which is a frozenset of
nodes in `G`. The order of cliques is arbitrary.
Raises
------
NetworkXError
The algorithm does not support DiGraph, MultiGraph and MultiDiGraph.
The algorithm can only be applied to chordal graphs. If the input
graph is found to be non-chordal, a :exc:`NetworkXError` is raised.
Examples
--------
>>> e = [
... (1, 2),
... (1, 3),
... (2, 3),
... (2, 4),
... (3, 4),
... (3, 5),
... (3, 6),
... (4, 5),
... (4, 6),
... (5, 6),
... (7, 8),
... ]
>>> G = nx.Graph(e)
>>> G.add_node(9)
>>> cliques = [c for c in chordal_graph_cliques(G)]
>>> cliques[0]
frozenset({1, 2, 3})
"""
for C in (G.subgraph(c).copy() for c in connected_components(G)):
if C.number_of_nodes() == 1:
if nx.number_of_selfloops(C) > 0:
raise nx.NetworkXError("Input graph is not chordal.")
yield frozenset(C.nodes())
else:
unnumbered = set(C.nodes())
v = arbitrary_element(C)
unnumbered.remove(v)
numbered = {v}
clique_wanna_be = {v}
while unnumbered:
v = _max_cardinality_node(C, unnumbered, numbered)
unnumbered.remove(v)
numbered.add(v)
new_clique_wanna_be = set(C.neighbors(v)) & numbered
sg = C.subgraph(clique_wanna_be)
if _is_complete_graph(sg):
new_clique_wanna_be.add(v)
if not new_clique_wanna_be >= clique_wanna_be:
yield frozenset(clique_wanna_be)
clique_wanna_be = new_clique_wanna_be
else:
raise nx.NetworkXError("Input graph is not chordal.")
yield frozenset(clique_wanna_be)
[docs]
@nx._dispatchable
def chordal_graph_treewidth(G):
"""Returns the treewidth of the chordal graph G.
Parameters
----------
G : graph
A NetworkX graph
Returns
-------
treewidth : int
The size of the largest clique in the graph minus one.
Raises
------
NetworkXError
The algorithm does not support DiGraph, MultiGraph and MultiDiGraph.
The algorithm can only be applied to chordal graphs. If the input
graph is found to be non-chordal, a :exc:`NetworkXError` is raised.
Examples
--------
>>> e = [
... (1, 2),
... (1, 3),
... (2, 3),
... (2, 4),
... (3, 4),
... (3, 5),
... (3, 6),
... (4, 5),
... (4, 6),
... (5, 6),
... (7, 8),
... ]
>>> G = nx.Graph(e)
>>> G.add_node(9)
>>> nx.chordal_graph_treewidth(G)
3
References
----------
.. [1] https://en.wikipedia.org/wiki/Tree_decomposition#Treewidth
"""
if not is_chordal(G):
raise nx.NetworkXError("Input graph is not chordal.")
max_clique = -1
for clique in nx.chordal_graph_cliques(G):
max_clique = max(max_clique, len(clique))
return max_clique - 1
def _is_complete_graph(G):
"""Returns True if G is a complete graph."""
if nx.number_of_selfloops(G) > 0:
raise nx.NetworkXError("Self loop found in _is_complete_graph()")
n = G.number_of_nodes()
if n < 2:
return True
e = G.number_of_edges()
max_edges = (n * (n - 1)) / 2
return e == max_edges
def _find_missing_edge(G):
"""Given a non-complete graph G, returns a missing edge."""
nodes = set(G)
for u in G:
missing = nodes - set(list(G[u].keys()) + [u])
if missing:
return (u, missing.pop())
def _max_cardinality_node(G, choices, wanna_connect):
"""Returns a the node in choices that has more connections in G
to nodes in wanna_connect.
"""
max_number = -1
for x in choices:
number = len([y for y in G[x] if y in wanna_connect])
if number > max_number:
max_number = number
max_cardinality_node = x
return max_cardinality_node
def _find_chordality_breaker(G, s=None, treewidth_bound=sys.maxsize):
"""Given a graph G, starts a max cardinality search
(starting from s if s is given and from an arbitrary node otherwise)
trying to find a non-chordal cycle.
If it does find one, it returns (u,v,w) where u,v,w are the three
nodes that together with s are involved in the cycle.
It ignores any self loops.
"""
if len(G) == 0:
raise nx.NetworkXPointlessConcept("Graph has no nodes.")
unnumbered = set(G)
if s is None:
s = arbitrary_element(G)
unnumbered.remove(s)
numbered = {s}
current_treewidth = -1
while unnumbered: # and current_treewidth <= treewidth_bound:
v = _max_cardinality_node(G, unnumbered, numbered)
unnumbered.remove(v)
numbered.add(v)
clique_wanna_be = set(G[v]) & numbered
sg = G.subgraph(clique_wanna_be)
if _is_complete_graph(sg):
# The graph seems to be chordal by now. We update the treewidth
current_treewidth = max(current_treewidth, len(clique_wanna_be))
if current_treewidth > treewidth_bound:
raise nx.NetworkXTreewidthBoundExceeded(
f"treewidth_bound exceeded: {current_treewidth}"
)
else:
# sg is not a clique,
# look for an edge that is not included in sg
(u, w) = _find_missing_edge(sg)
return (u, v, w)
return ()
[docs]
@not_implemented_for("directed")
@nx._dispatchable(returns_graph=True)
def complete_to_chordal_graph(G):
"""Return a copy of G completed to a chordal graph
Adds edges to a copy of G to create a chordal graph. A graph G=(V,E) is
called chordal if for each cycle with length bigger than 3, there exist
two non-adjacent nodes connected by an edge (called a chord).
Parameters
----------
G : NetworkX graph
Undirected graph
Returns
-------
H : NetworkX graph
The chordal enhancement of G
alpha : Dictionary
The elimination ordering of nodes of G
Notes
-----
There are different approaches to calculate the chordal
enhancement of a graph. The algorithm used here is called
MCS-M and gives at least minimal (local) triangulation of graph. Note
that this triangulation is not necessarily a global minimum.
https://en.wikipedia.org/wiki/Chordal_graph
References
----------
.. [1] Berry, Anne & Blair, Jean & Heggernes, Pinar & Peyton, Barry. (2004)
Maximum Cardinality Search for Computing Minimal Triangulations of
Graphs. Algorithmica. 39. 287-298. 10.1007/s00453-004-1084-3.
Examples
--------
>>> from networkx.algorithms.chordal import complete_to_chordal_graph
>>> G = nx.wheel_graph(10)
>>> H, alpha = complete_to_chordal_graph(G)
"""
H = G.copy()
alpha = {node: 0 for node in H}
if nx.is_chordal(H):
return H, alpha
chords = set()
weight = {node: 0 for node in H.nodes()}
unnumbered_nodes = list(H.nodes())
for i in range(len(H.nodes()), 0, -1):
# get the node in unnumbered_nodes with the maximum weight
z = max(unnumbered_nodes, key=lambda node: weight[node])
unnumbered_nodes.remove(z)
alpha[z] = i
update_nodes = []
for y in unnumbered_nodes:
if G.has_edge(y, z):
update_nodes.append(y)
else:
# y_weight will be bigger than node weights between y and z
y_weight = weight[y]
lower_nodes = [
node for node in unnumbered_nodes if weight[node] < y_weight
]
if nx.has_path(H.subgraph(lower_nodes + [z, y]), y, z):
update_nodes.append(y)
chords.add((z, y))
# during calculation of paths the weights should not be updated
for node in update_nodes:
weight[node] += 1
H.add_edges_from(chords)
return H, alpha
