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"""
Module to simplify the specification of user-defined equality functions for
node and edge attributes during isomorphism checks.
During the construction of an isomorphism, the algorithm considers two
candidate nodes n1 in G1 and n2 in G2. The graphs G1 and G2 are then
compared with respect to properties involving n1 and n2, and if the outcome
is good, then the candidate nodes are considered isomorphic. NetworkX
provides a simple mechanism for users to extend the comparisons to include
node and edge attributes.
Node attributes are handled by the node_match keyword. When considering
n1 and n2, the algorithm passes their node attribute dictionaries to
node_match, and if it returns False, then n1 and n2 cannot be
considered to be isomorphic.
Edge attributes are handled by the edge_match keyword. When considering
n1 and n2, the algorithm must verify that outgoing edges from n1 are
commensurate with the outgoing edges for n2. If the graph is directed,
then a similar check is also performed for incoming edges.
Focusing only on outgoing edges, we consider pairs of nodes (n1, v1) from
G1 and (n2, v2) from G2. For graphs and digraphs, there is only one edge
between (n1, v1) and only one edge between (n2, v2). Those edge attribute
dictionaries are passed to edge_match, and if it returns False, then
n1 and n2 cannot be considered isomorphic. For multigraphs and
multidigraphs, there can be multiple edges between (n1, v1) and also
multiple edges between (n2, v2). Now, there must exist an isomorphism
from "all the edges between (n1, v1)" to "all the edges between (n2, v2)".
So, all of the edge attribute dictionaries are passed to edge_match, and
it must determine if there is an isomorphism between the two sets of edges.
"""
from . import isomorphvf2 as vf2
__all__ = ["GraphMatcher", "DiGraphMatcher", "MultiGraphMatcher", "MultiDiGraphMatcher"]
def _semantic_feasibility(self, G1_node, G2_node):
"""Returns True if mapping G1_node to G2_node is semantically feasible."""
# Make sure the nodes match
if self.node_match is not None:
nm = self.node_match(self.G1.nodes[G1_node], self.G2.nodes[G2_node])
if not nm:
return False
# Make sure the edges match
if self.edge_match is not None:
# Cached lookups
G1nbrs = self.G1_adj[G1_node]
G2nbrs = self.G2_adj[G2_node]
core_1 = self.core_1
edge_match = self.edge_match
for neighbor in G1nbrs:
# G1_node is not in core_1, so we must handle R_self separately
if neighbor == G1_node:
if G2_node in G2nbrs and not edge_match(
G1nbrs[G1_node], G2nbrs[G2_node]
):
return False
elif neighbor in core_1:
G2_nbr = core_1[neighbor]
if G2_nbr in G2nbrs and not edge_match(
G1nbrs[neighbor], G2nbrs[G2_nbr]
):
return False
# syntactic check has already verified that neighbors are symmetric
return True
class GraphMatcher(vf2.GraphMatcher):
"""VF2 isomorphism checker for undirected graphs."""
def __init__(self, G1, G2, node_match=None, edge_match=None):
"""Initialize graph matcher.
Parameters
----------
G1, G2: graph
The graphs to be tested.
node_match: callable
A function that returns True iff node n1 in G1 and n2 in G2
should be considered equal during the isomorphism test. The
function will be called like::
node_match(G1.nodes[n1], G2.nodes[n2])
That is, the function will receive the node attribute dictionaries
of the nodes under consideration. If None, then no attributes are
considered when testing for an isomorphism.
edge_match: callable
A function that returns True iff the edge attribute dictionary for
the pair of nodes (u1, v1) in G1 and (u2, v2) in G2 should be
considered equal during the isomorphism test. The function will be
called like::
edge_match(G1[u1][v1], G2[u2][v2])
That is, the function will receive the edge attribute dictionaries
of the edges under consideration. If None, then no attributes are
considered when testing for an isomorphism.
"""
vf2.GraphMatcher.__init__(self, G1, G2)
self.node_match = node_match
self.edge_match = edge_match
# These will be modified during checks to minimize code repeat.
self.G1_adj = self.G1.adj
self.G2_adj = self.G2.adj
semantic_feasibility = _semantic_feasibility
class DiGraphMatcher(vf2.DiGraphMatcher):
"""VF2 isomorphism checker for directed graphs."""
def __init__(self, G1, G2, node_match=None, edge_match=None):
"""Initialize graph matcher.
Parameters
----------
G1, G2 : graph
The graphs to be tested.
node_match : callable
A function that returns True iff node n1 in G1 and n2 in G2
should be considered equal during the isomorphism test. The
function will be called like::
node_match(G1.nodes[n1], G2.nodes[n2])
That is, the function will receive the node attribute dictionaries
of the nodes under consideration. If None, then no attributes are
considered when testing for an isomorphism.
edge_match : callable
A function that returns True iff the edge attribute dictionary for
the pair of nodes (u1, v1) in G1 and (u2, v2) in G2 should be
considered equal during the isomorphism test. The function will be
called like::
edge_match(G1[u1][v1], G2[u2][v2])
That is, the function will receive the edge attribute dictionaries
of the edges under consideration. If None, then no attributes are
considered when testing for an isomorphism.
"""
vf2.DiGraphMatcher.__init__(self, G1, G2)
self.node_match = node_match
self.edge_match = edge_match
# These will be modified during checks to minimize code repeat.
self.G1_adj = self.G1.adj
self.G2_adj = self.G2.adj
def semantic_feasibility(self, G1_node, G2_node):
"""Returns True if mapping G1_node to G2_node is semantically feasible."""
# Test node_match and also test edge_match on successors
feasible = _semantic_feasibility(self, G1_node, G2_node)
if not feasible:
return False
# Test edge_match on predecessors
self.G1_adj = self.G1.pred
self.G2_adj = self.G2.pred
feasible = _semantic_feasibility(self, G1_node, G2_node)
self.G1_adj = self.G1.adj
self.G2_adj = self.G2.adj
return feasible
# The "semantics" of edge_match are different for multi(di)graphs, but
# the implementation is the same. So, technically we do not need to
# provide "multi" versions, but we do so to match NetworkX's base classes.
class MultiGraphMatcher(GraphMatcher):
"""VF2 isomorphism checker for undirected multigraphs. """
pass
class MultiDiGraphMatcher(DiGraphMatcher):
"""VF2 isomorphism checker for directed multigraphs. """
pass
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