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