"""
Graph products.
"""
from itertools import product
import networkx as nx
from networkx.utils import not_implemented_for
__all__ = [
"tensor_product",
"cartesian_product",
"lexicographic_product",
"strong_product",
"power",
"rooted_product",
"corona_product",
"modular_product",
]
_G_H = {"G": 0, "H": 1}
def _dict_product(d1, d2):
return {k: (d1.get(k), d2.get(k)) for k in set(d1) | set(d2)}
# Generators for producing graph products
def _node_product(G, H):
for u, v in product(G, H):
yield ((u, v), _dict_product(G.nodes[u], H.nodes[v]))
def _directed_edges_cross_edges(G, H):
if not G.is_multigraph() and not H.is_multigraph():
for u, v, c in G.edges(data=True):
for x, y, d in H.edges(data=True):
yield (u, x), (v, y), _dict_product(c, d)
if not G.is_multigraph() and H.is_multigraph():
for u, v, c in G.edges(data=True):
for x, y, k, d in H.edges(data=True, keys=True):
yield (u, x), (v, y), k, _dict_product(c, d)
if G.is_multigraph() and not H.is_multigraph():
for u, v, k, c in G.edges(data=True, keys=True):
for x, y, d in H.edges(data=True):
yield (u, x), (v, y), k, _dict_product(c, d)
if G.is_multigraph() and H.is_multigraph():
for u, v, j, c in G.edges(data=True, keys=True):
for x, y, k, d in H.edges(data=True, keys=True):
yield (u, x), (v, y), (j, k), _dict_product(c, d)
def _undirected_edges_cross_edges(G, H):
if not G.is_multigraph() and not H.is_multigraph():
for u, v, c in G.edges(data=True):
for x, y, d in H.edges(data=True):
yield (v, x), (u, y), _dict_product(c, d)
if not G.is_multigraph() and H.is_multigraph():
for u, v, c in G.edges(data=True):
for x, y, k, d in H.edges(data=True, keys=True):
yield (v, x), (u, y), k, _dict_product(c, d)
if G.is_multigraph() and not H.is_multigraph():
for u, v, k, c in G.edges(data=True, keys=True):
for x, y, d in H.edges(data=True):
yield (v, x), (u, y), k, _dict_product(c, d)
if G.is_multigraph() and H.is_multigraph():
for u, v, j, c in G.edges(data=True, keys=True):
for x, y, k, d in H.edges(data=True, keys=True):
yield (v, x), (u, y), (j, k), _dict_product(c, d)
def _edges_cross_nodes(G, H):
if G.is_multigraph():
for u, v, k, d in G.edges(data=True, keys=True):
for x in H:
yield (u, x), (v, x), k, d
else:
for u, v, d in G.edges(data=True):
for x in H:
if H.is_multigraph():
yield (u, x), (v, x), None, d
else:
yield (u, x), (v, x), d
def _nodes_cross_edges(G, H):
if H.is_multigraph():
for x in G:
for u, v, k, d in H.edges(data=True, keys=True):
yield (x, u), (x, v), k, d
else:
for x in G:
for u, v, d in H.edges(data=True):
if G.is_multigraph():
yield (x, u), (x, v), None, d
else:
yield (x, u), (x, v), d
def _edges_cross_nodes_and_nodes(G, H):
if G.is_multigraph():
for u, v, k, d in G.edges(data=True, keys=True):
for x in H:
for y in H:
yield (u, x), (v, y), k, d
else:
for u, v, d in G.edges(data=True):
for x in H:
for y in H:
if H.is_multigraph():
yield (u, x), (v, y), None, d
else:
yield (u, x), (v, y), d
def _init_product_graph(G, H):
if G.is_directed() != H.is_directed():
msg = "G and H must be both directed or both undirected"
raise nx.NetworkXError(msg)
if G.is_multigraph() or H.is_multigraph():
GH = nx.MultiGraph()
else:
GH = nx.Graph()
if G.is_directed():
GH = GH.to_directed()
return GH
[docs]
@nx._dispatchable(graphs=_G_H, preserve_node_attrs=True, returns_graph=True)
def tensor_product(G, H):
r"""Returns the tensor product of G and H.
The tensor product $P$ of the graphs $G$ and $H$ has a node set that
is the Cartesian product of the node sets, $V(P)=V(G) \times V(H)$.
$P$ has an edge $((u,v), (x,y))$ if and only if $(u,x)$ is an edge in $G$
and $(v,y)$ is an edge in $H$.
Tensor product is sometimes also referred to as the categorical product,
direct product, cardinal product or conjunction.
Parameters
----------
G, H: graphs
Networkx graphs.
Returns
-------
P: NetworkX graph
The tensor product of G and H. P will be a multi-graph if either G
or H is a multi-graph, will be a directed if G and H are directed,
and undirected if G and H are undirected.
Raises
------
NetworkXError
If G and H are not both directed or both undirected.
Notes
-----
Node attributes in P are two-tuple of the G and H node attributes.
Missing attributes are assigned None.
Examples
--------
>>> G = nx.Graph()
>>> H = nx.Graph()
>>> G.add_node(0, a1=True)
>>> H.add_node("a", a2="Spam")
>>> P = nx.tensor_product(G, H)
>>> list(P)
[(0, 'a')]
Edge attributes and edge keys (for multigraphs) are also copied to the
new product graph
"""
GH = _init_product_graph(G, H)
GH.add_nodes_from(_node_product(G, H))
GH.add_edges_from(_directed_edges_cross_edges(G, H))
if not GH.is_directed():
GH.add_edges_from(_undirected_edges_cross_edges(G, H))
return GH
[docs]
@nx._dispatchable(graphs=_G_H, preserve_node_attrs=True, returns_graph=True)
def cartesian_product(G, H):
r"""Returns the Cartesian product of G and H.
The Cartesian product $P$ of the graphs $G$ and $H$ has a node set that
is the Cartesian product of the node sets, $V(P)=V(G) \times V(H)$.
$P$ has an edge $((u,v),(x,y))$ if and only if either $u$ is equal to $x$
and both $v$ and $y$ are adjacent in $H$ or if $v$ is equal to $y$ and
both $u$ and $x$ are adjacent in $G$.
Parameters
----------
G, H: graphs
Networkx graphs.
Returns
-------
P: NetworkX graph
The Cartesian product of G and H. P will be a multi-graph if either G
or H is a multi-graph. Will be a directed if G and H are directed,
and undirected if G and H are undirected.
Raises
------
NetworkXError
If G and H are not both directed or both undirected.
Notes
-----
Node attributes in P are two-tuple of the G and H node attributes.
Missing attributes are assigned None.
Examples
--------
>>> G = nx.Graph()
>>> H = nx.Graph()
>>> G.add_node(0, a1=True)
>>> H.add_node("a", a2="Spam")
>>> P = nx.cartesian_product(G, H)
>>> list(P)
[(0, 'a')]
Edge attributes and edge keys (for multigraphs) are also copied to the
new product graph
"""
GH = _init_product_graph(G, H)
GH.add_nodes_from(_node_product(G, H))
GH.add_edges_from(_edges_cross_nodes(G, H))
GH.add_edges_from(_nodes_cross_edges(G, H))
return GH
[docs]
@nx._dispatchable(graphs=_G_H, preserve_node_attrs=True, returns_graph=True)
def lexicographic_product(G, H):
r"""Returns the lexicographic product of G and H.
The lexicographical product $P$ of the graphs $G$ and $H$ has a node set
that is the Cartesian product of the node sets, $V(P)=V(G) \times V(H)$.
$P$ has an edge $((u,v), (x,y))$ if and only if $(u,v)$ is an edge in $G$
or $u==v$ and $(x,y)$ is an edge in $H$.
Parameters
----------
G, H: graphs
Networkx graphs.
Returns
-------
P: NetworkX graph
The Cartesian product of G and H. P will be a multi-graph if either G
or H is a multi-graph. Will be a directed if G and H are directed,
and undirected if G and H are undirected.
Raises
------
NetworkXError
If G and H are not both directed or both undirected.
Notes
-----
Node attributes in P are two-tuple of the G and H node attributes.
Missing attributes are assigned None.
Examples
--------
>>> G = nx.Graph()
>>> H = nx.Graph()
>>> G.add_node(0, a1=True)
>>> H.add_node("a", a2="Spam")
>>> P = nx.lexicographic_product(G, H)
>>> list(P)
[(0, 'a')]
Edge attributes and edge keys (for multigraphs) are also copied to the
new product graph
"""
GH = _init_product_graph(G, H)
GH.add_nodes_from(_node_product(G, H))
# Edges in G regardless of H designation
GH.add_edges_from(_edges_cross_nodes_and_nodes(G, H))
# For each x in G, only if there is an edge in H
GH.add_edges_from(_nodes_cross_edges(G, H))
return GH
[docs]
@nx._dispatchable(graphs=_G_H, preserve_node_attrs=True, returns_graph=True)
def strong_product(G, H):
r"""Returns the strong product of G and H.
The strong product $P$ of the graphs $G$ and $H$ has a node set that
is the Cartesian product of the node sets, $V(P)=V(G) \times V(H)$.
$P$ has an edge $((u,x), (v,y))$ if any of the following conditions
are met:
- $u=v$ and $(x,y)$ is an edge in $H$
- $x=y$ and $(u,v)$ is an edge in $G$
- $(u,v)$ is an edge in $G$ and $(x,y)$ is an edge in $H$
Parameters
----------
G, H: graphs
Networkx graphs.
Returns
-------
P: NetworkX graph
The Cartesian product of G and H. P will be a multi-graph if either G
or H is a multi-graph. Will be a directed if G and H are directed,
and undirected if G and H are undirected.
Raises
------
NetworkXError
If G and H are not both directed or both undirected.
Notes
-----
Node attributes in P are two-tuple of the G and H node attributes.
Missing attributes are assigned None.
Examples
--------
>>> G = nx.Graph()
>>> H = nx.Graph()
>>> G.add_node(0, a1=True)
>>> H.add_node("a", a2="Spam")
>>> P = nx.strong_product(G, H)
>>> list(P)
[(0, 'a')]
Edge attributes and edge keys (for multigraphs) are also copied to the
new product graph
"""
GH = _init_product_graph(G, H)
GH.add_nodes_from(_node_product(G, H))
GH.add_edges_from(_nodes_cross_edges(G, H))
GH.add_edges_from(_edges_cross_nodes(G, H))
GH.add_edges_from(_directed_edges_cross_edges(G, H))
if not GH.is_directed():
GH.add_edges_from(_undirected_edges_cross_edges(G, H))
return GH
[docs]
@not_implemented_for("directed")
@not_implemented_for("multigraph")
@nx._dispatchable(returns_graph=True)
def power(G, k):
"""Returns the specified power of a graph.
The $k$th power of a simple graph $G$, denoted $G^k$, is a
graph on the same set of nodes in which two distinct nodes $u$ and
$v$ are adjacent in $G^k$ if and only if the shortest path
distance between $u$ and $v$ in $G$ is at most $k$.
Parameters
----------
G : graph
A NetworkX simple graph object.
k : positive integer
The power to which to raise the graph `G`.
Returns
-------
NetworkX simple graph
`G` to the power `k`.
Raises
------
ValueError
If the exponent `k` is not positive.
NetworkXNotImplemented
If `G` is not a simple graph.
Examples
--------
The number of edges will never decrease when taking successive
powers:
>>> G = nx.path_graph(4)
>>> list(nx.power(G, 2).edges)
[(0, 1), (0, 2), (1, 2), (1, 3), (2, 3)]
>>> list(nx.power(G, 3).edges)
[(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)]
The `k` th power of a cycle graph on *n* nodes is the complete graph
on *n* nodes, if `k` is at least ``n // 2``:
>>> G = nx.cycle_graph(5)
>>> H = nx.complete_graph(5)
>>> nx.is_isomorphic(nx.power(G, 2), H)
True
>>> G = nx.cycle_graph(8)
>>> H = nx.complete_graph(8)
>>> nx.is_isomorphic(nx.power(G, 4), H)
True
References
----------
.. [1] J. A. Bondy, U. S. R. Murty, *Graph Theory*. Springer, 2008.
Notes
-----
This definition of "power graph" comes from Exercise 3.1.6 of
*Graph Theory* by Bondy and Murty [1]_.
"""
if k <= 0:
raise ValueError("k must be a positive integer")
H = nx.Graph()
H.add_nodes_from(G)
# update BFS code to ignore self loops.
for n in G:
seen = {} # level (number of hops) when seen in BFS
level = 1 # the current level
nextlevel = G[n]
while nextlevel:
thislevel = nextlevel # advance to next level
nextlevel = {} # and start a new list (fringe)
for v in thislevel:
if v == n: # avoid self loop
continue
if v not in seen:
seen[v] = level # set the level of vertex v
nextlevel.update(G[v]) # add neighbors of v
if k <= level:
break
level += 1
H.add_edges_from((n, nbr) for nbr in seen)
return H
[docs]
@not_implemented_for("multigraph")
@nx._dispatchable(graphs=_G_H, returns_graph=True)
def rooted_product(G, H, root):
"""Return the rooted product of graphs G and H rooted at root in H.
A new graph is constructed representing the rooted product of
the inputted graphs, G and H, with a root in H.
A rooted product duplicates H for each nodes in G with the root
of H corresponding to the node in G. Nodes are renamed as the direct
product of G and H. The result is a subgraph of the cartesian product.
Parameters
----------
G,H : graph
A NetworkX graph
root : node
A node in H
Returns
-------
R : The rooted product of G and H with a specified root in H
Notes
-----
The nodes of R are the Cartesian Product of the nodes of G and H.
The nodes of G and H are not relabeled.
"""
if root not in H:
raise nx.NodeNotFound("root must be a vertex in H")
R = nx.Graph()
R.add_nodes_from(product(G, H))
R.add_edges_from(((e[0], root), (e[1], root)) for e in G.edges())
R.add_edges_from(((g, e[0]), (g, e[1])) for g in G for e in H.edges())
return R
[docs]
@not_implemented_for("directed")
@not_implemented_for("multigraph")
@nx._dispatchable(graphs=_G_H, returns_graph=True)
def corona_product(G, H):
r"""Returns the Corona product of G and H.
The corona product of $G$ and $H$ is the graph $C = G \circ H$ obtained by
taking one copy of $G$, called the center graph, $|V(G)|$ copies of $H$,
called the outer graph, and making the $i$-th vertex of $G$ adjacent to
every vertex of the $i$-th copy of $H$, where $1 ≤ i ≤ |V(G)|$.
Parameters
----------
G, H: NetworkX graphs
The graphs to take the carona product of.
`G` is the center graph and `H` is the outer graph
Returns
-------
C: NetworkX graph
The Corona product of G and H.
Raises
------
NetworkXError
If G and H are not both directed or both undirected.
Examples
--------
>>> G = nx.cycle_graph(4)
>>> H = nx.path_graph(2)
>>> C = nx.corona_product(G, H)
>>> list(C)
[0, 1, 2, 3, (0, 0), (0, 1), (1, 0), (1, 1), (2, 0), (2, 1), (3, 0), (3, 1)]
>>> print(C)
Graph with 12 nodes and 16 edges
References
----------
[1] M. Tavakoli, F. Rahbarnia, and A. R. Ashrafi,
"Studying the corona product of graphs under some graph invariants,"
Transactions on Combinatorics, vol. 3, no. 3, pp. 43–49, Sep. 2014,
doi: 10.22108/toc.2014.5542.
[2] A. Faraji, "Corona Product in Graph Theory," Ali Faraji, May 11, 2021.
https://blog.alifaraji.ir/math/graph-theory/corona-product.html (accessed Dec. 07, 2021).
"""
GH = _init_product_graph(G, H)
GH.add_nodes_from(G)
GH.add_edges_from(G.edges)
for G_node in G:
# copy nodes of H in GH, call it H_i
GH.add_nodes_from((G_node, v) for v in H)
# copy edges of H_i based on H
GH.add_edges_from(
((G_node, e0), (G_node, e1), d) for e0, e1, d in H.edges.data()
)
# creating new edges between H_i and a G's node
GH.add_edges_from((G_node, (G_node, H_node)) for H_node in H)
return GH
[docs]
@nx._dispatchable(
graphs=_G_H, preserve_edge_attrs=True, preserve_node_attrs=True, returns_graph=True
)
def modular_product(G, H):
r"""Returns the Modular product of G and H.
The modular product of `G` and `H` is the graph $M = G \nabla H$,
consisting of the node set $V(M) = V(G) \times V(H)$ that is the Cartesian
product of the node sets of `G` and `H`. Further, M contains an edge ((u, v), (x, y)):
- if u is adjacent to x in `G` and v is adjacent to y in `H`, or
- if u is not adjacent to x in `G` and v is not adjacent to y in `H`.
More formally::
E(M) = {((u, v), (x, y)) | ((u, x) in E(G) and (v, y) in E(H)) or
((u, x) not in E(G) and (v, y) not in E(H))}
Parameters
----------
G, H: NetworkX graphs
The graphs to take the modular product of.
Returns
-------
M: NetworkX graph
The Modular product of `G` and `H`.
Raises
------
NetworkXNotImplemented
If `G` is not a simple graph.
Examples
--------
>>> G = nx.cycle_graph(4)
>>> H = nx.path_graph(2)
>>> M = nx.modular_product(G, H)
>>> list(M)
[(0, 0), (0, 1), (1, 0), (1, 1), (2, 0), (2, 1), (3, 0), (3, 1)]
>>> print(M)
Graph with 8 nodes and 8 edges
Notes
-----
The *modular product* is defined in [1]_ and was first
introduced as the *weak modular product*.
The modular product reduces the problem of counting isomorphic subgraphs
in `G` and `H` to the problem of counting cliques in M. The subgraphs of
`G` and `H` that are induced by the nodes of a clique in M are
isomorphic [2]_ [3]_.
References
----------
.. [1] R. Hammack, W. Imrich, and S. Klavžar,
"Handbook of Product Graphs", CRC Press, 2011.
.. [2] H. G. Barrow and R. M. Burstall,
"Subgraph isomorphism, matching relational structures and maximal
cliques", Information Processing Letters, vol. 4, issue 4, pp. 83-84,
1976, https://doi.org/10.1016/0020-0190(76)90049-1.
.. [3] V. G. Vizing, "Reduction of the problem of isomorphism and isomorphic
entrance to the task of finding the nondensity of a graph." Proc. Third
All-Union Conference on Problems of Theoretical Cybernetics. 1974.
"""
if G.is_directed() or H.is_directed():
raise nx.NetworkXNotImplemented(
"Modular product not implemented for directed graphs"
)
if G.is_multigraph() or H.is_multigraph():
raise nx.NetworkXNotImplemented(
"Modular product not implemented for multigraphs"
)
GH = _init_product_graph(G, H)
GH.add_nodes_from(_node_product(G, H))
for u, v, c in G.edges(data=True):
for x, y, d in H.edges(data=True):
GH.add_edge((u, x), (v, y), **_dict_product(c, d))
GH.add_edge((v, x), (u, y), **_dict_product(c, d))
G = nx.complement(G)
H = nx.complement(H)
for u, v, c in G.edges(data=True):
for x, y, d in H.edges(data=True):
GH.add_edge((u, x), (v, y), **_dict_product(c, d))
GH.add_edge((v, x), (u, y), **_dict_product(c, d))
return GH
