Source code for networkx.algorithms.regular

"""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. Examples -------- >>> G = nx.DiGraph([(1, 2), (2, 3), (3, 4), (4, 1)]) >>> nx.is_regular(G) True """ 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. Examples -------- >>> G = nx.Graph([(1, 2), (2, 3), (3, 4), (4, 1)]) >>> nx.is_k_regular(G, k=3) False """ 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 Examples -------- >>> G = nx.Graph([(1, 2), (2, 3), (3, 4), (4, 1)]) >>> G2 = nx.k_factor(G, k=1) >>> G2.edges() EdgeView([(1, 2), (3, 4)]) 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 is_perfect_matching, max_weight_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