"""Functions for computing and verifying regular graphs."""
import networkx as nx
from networkx.utils import not_implemented_for
__all__ = ["is_regular", "is_k_regular", "k_factor"]
[docs]def is_regular(G):
"""Determines whether the graph ``G`` is a regular graph.
A regular graph is a graph where each vertex has the same degree. A
regular digraph is a graph where the indegree and outdegree of each
vertex are equal.
Parameters
----------
G : NetworkX graph
Returns
-------
bool
Whether the given graph or digraph is regular.
"""
n1 = nx.utils.arbitrary_element(G)
if not G.is_directed():
d1 = G.degree(n1)
return all(d1 == d for _, d in G.degree)
else:
d_in = G.in_degree(n1)
in_regular = all(d_in == d for _, d in G.in_degree)
d_out = G.out_degree(n1)
out_regular = all(d_out == d for _, d in G.out_degree)
return in_regular and out_regular
[docs]@not_implemented_for("directed")
def is_k_regular(G, k):
"""Determines whether the graph ``G`` is a k-regular graph.
A k-regular graph is a graph where each vertex has degree k.
Parameters
----------
G : NetworkX graph
Returns
-------
bool
Whether the given graph is k-regular.
"""
return all(d == k for n, d in G.degree)
[docs]@not_implemented_for("directed")
@not_implemented_for("multigraph")
def k_factor(G, k, matching_weight="weight"):
"""Compute a k-factor of G
A k-factor of a graph is a spanning k-regular subgraph.
A spanning k-regular subgraph of G is a subgraph that contains
each vertex of G and a subset of the edges of G such that each
vertex has degree k.
Parameters
----------
G : NetworkX graph
Undirected graph
matching_weight: string, optional (default='weight')
Edge data key corresponding to the edge weight.
Used for finding the max-weighted perfect matching.
If key not found, uses 1 as weight.
Returns
-------
G2 : NetworkX graph
A k-factor of G
References
----------
.. [1] "An algorithm for computing simple k-factors.",
Meijer, Henk, Yurai Núñez-Rodríguez, and David Rappaport,
Information processing letters, 2009.
"""
from networkx.algorithms.matching import max_weight_matching
from networkx.algorithms.matching import is_perfect_matching
class LargeKGadget:
def __init__(self, k, degree, node, g):
self.original = node
self.g = g
self.k = k
self.degree = degree
self.outer_vertices = [(node, x) for x in range(degree)]
self.core_vertices = [(node, x + degree) for x in range(degree - k)]
def replace_node(self):
adj_view = self.g[self.original]
neighbors = list(adj_view.keys())
edge_attrs = list(adj_view.values())
for (outer, neighbor, edge_attrs) in zip(
self.outer_vertices, neighbors, edge_attrs
):
self.g.add_edge(outer, neighbor, **edge_attrs)
for core in self.core_vertices:
for outer in self.outer_vertices:
self.g.add_edge(core, outer)
self.g.remove_node(self.original)
def restore_node(self):
self.g.add_node(self.original)
for outer in self.outer_vertices:
adj_view = self.g[outer]
for neighbor, edge_attrs in list(adj_view.items()):
if neighbor not in self.core_vertices:
self.g.add_edge(self.original, neighbor, **edge_attrs)
break
g.remove_nodes_from(self.outer_vertices)
g.remove_nodes_from(self.core_vertices)
class SmallKGadget:
def __init__(self, k, degree, node, g):
self.original = node
self.k = k
self.degree = degree
self.g = g
self.outer_vertices = [(node, x) for x in range(degree)]
self.inner_vertices = [(node, x + degree) for x in range(degree)]
self.core_vertices = [(node, x + 2 * degree) for x in range(k)]
def replace_node(self):
adj_view = self.g[self.original]
for (outer, inner, (neighbor, edge_attrs)) in zip(
self.outer_vertices, self.inner_vertices, list(adj_view.items())
):
self.g.add_edge(outer, inner)
self.g.add_edge(outer, neighbor, **edge_attrs)
for core in self.core_vertices:
for inner in self.inner_vertices:
self.g.add_edge(core, inner)
self.g.remove_node(self.original)
def restore_node(self):
self.g.add_node(self.original)
for outer in self.outer_vertices:
adj_view = self.g[outer]
for neighbor, edge_attrs in adj_view.items():
if neighbor not in self.core_vertices:
self.g.add_edge(self.original, neighbor, **edge_attrs)
break
self.g.remove_nodes_from(self.outer_vertices)
self.g.remove_nodes_from(self.inner_vertices)
self.g.remove_nodes_from(self.core_vertices)
# Step 1
if any(d < k for _, d in G.degree):
raise nx.NetworkXUnfeasible("Graph contains a vertex with degree less than k")
g = G.copy()
# Step 2
gadgets = []
for node, degree in list(g.degree):
if k < degree / 2.0:
gadget = SmallKGadget(k, degree, node, g)
else:
gadget = LargeKGadget(k, degree, node, g)
gadget.replace_node()
gadgets.append(gadget)
# Step 3
matching = max_weight_matching(g, maxcardinality=True, weight=matching_weight)
# Step 4
if not is_perfect_matching(g, matching):
raise nx.NetworkXUnfeasible(
"Cannot find k-factor because no perfect matching exists"
)
for edge in g.edges():
if edge not in matching and (edge[1], edge[0]) not in matching:
g.remove_edge(edge[0], edge[1])
for gadget in gadgets:
gadget.restore_node()
return g