451 lines
14 KiB
Python
451 lines
14 KiB
Python
# -*- coding: utf-8 -*-
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"""
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Various small and named graphs, together with some compact generators.
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"""
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__author__ = """Aric Hagberg (hagberg@lanl.gov)\nPieter Swart (swart@lanl.gov)"""
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# Copyright (C) 2004-2019 by
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# Aric Hagberg <hagberg@lanl.gov>
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# Dan Schult <dschult@colgate.edu>
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# Pieter Swart <swart@lanl.gov>
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# All rights reserved.
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# BSD license.
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__all__ = ['make_small_graph',
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'LCF_graph',
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'bull_graph',
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'chvatal_graph',
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'cubical_graph',
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'desargues_graph',
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'diamond_graph',
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'dodecahedral_graph',
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'frucht_graph',
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'heawood_graph',
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'hoffman_singleton_graph',
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'house_graph',
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'house_x_graph',
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'icosahedral_graph',
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'krackhardt_kite_graph',
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'moebius_kantor_graph',
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'octahedral_graph',
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'pappus_graph',
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'petersen_graph',
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'sedgewick_maze_graph',
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'tetrahedral_graph',
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'truncated_cube_graph',
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'truncated_tetrahedron_graph',
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'tutte_graph']
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import networkx as nx
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from networkx.generators.classic import empty_graph, cycle_graph, path_graph, complete_graph
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from networkx.exception import NetworkXError
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#------------------------------------------------------------------------------
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# Tools for creating small graphs
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#------------------------------------------------------------------------------
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def make_small_undirected_graph(graph_description, create_using=None):
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"""
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Return a small undirected graph described by graph_description.
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See make_small_graph.
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"""
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G = empty_graph(0, create_using)
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if G.is_directed():
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raise NetworkXError("Directed Graph not supported")
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return make_small_graph(graph_description, G)
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def make_small_graph(graph_description, create_using=None):
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"""
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Return the small graph described by graph_description.
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graph_description is a list of the form [ltype,name,n,xlist]
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Here ltype is one of "adjacencylist" or "edgelist",
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name is the name of the graph and n the number of nodes.
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This constructs a graph of n nodes with integer labels 0,..,n-1.
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If ltype="adjacencylist" then xlist is an adjacency list
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with exactly n entries, in with the j'th entry (which can be empty)
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specifies the nodes connected to vertex j.
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e.g. the "square" graph C_4 can be obtained by
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>>> G=nx.make_small_graph(["adjacencylist","C_4",4,[[2,4],[1,3],[2,4],[1,3]]])
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or, since we do not need to add edges twice,
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>>> G=nx.make_small_graph(["adjacencylist","C_4",4,[[2,4],[3],[4],[]]])
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If ltype="edgelist" then xlist is an edge list
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written as [[v1,w2],[v2,w2],...,[vk,wk]],
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where vj and wj integers in the range 1,..,n
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e.g. the "square" graph C_4 can be obtained by
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>>> G=nx.make_small_graph(["edgelist","C_4",4,[[1,2],[3,4],[2,3],[4,1]]])
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Use the create_using argument to choose the graph class/type.
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"""
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ltype = graph_description[0]
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name = graph_description[1]
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n = graph_description[2]
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G = empty_graph(n, create_using)
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nodes = G.nodes()
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if ltype == "adjacencylist":
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adjlist = graph_description[3]
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if len(adjlist) != n:
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raise NetworkXError("invalid graph_description")
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G.add_edges_from([(u - 1, v) for v in nodes for u in adjlist[v]])
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elif ltype == "edgelist":
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edgelist = graph_description[3]
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for e in edgelist:
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v1 = e[0] - 1
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v2 = e[1] - 1
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if v1 < 0 or v1 > n - 1 or v2 < 0 or v2 > n - 1:
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raise NetworkXError("invalid graph_description")
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else:
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G.add_edge(v1, v2)
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G.name = name
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return G
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def LCF_graph(n, shift_list, repeats, create_using=None):
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"""
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Return the cubic graph specified in LCF notation.
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LCF notation (LCF=Lederberg-Coxeter-Fruchte) is a compressed
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notation used in the generation of various cubic Hamiltonian
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graphs of high symmetry. See, for example, dodecahedral_graph,
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desargues_graph, heawood_graph and pappus_graph below.
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n (number of nodes)
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The starting graph is the n-cycle with nodes 0,...,n-1.
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(The null graph is returned if n < 0.)
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shift_list = [s1,s2,..,sk], a list of integer shifts mod n,
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repeats
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integer specifying the number of times that shifts in shift_list
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are successively applied to each v_current in the n-cycle
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to generate an edge between v_current and v_current+shift mod n.
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For v1 cycling through the n-cycle a total of k*repeats
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with shift cycling through shiftlist repeats times connect
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v1 with v1+shift mod n
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The utility graph $K_{3,3}$
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>>> G = nx.LCF_graph(6, [3, -3], 3)
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The Heawood graph
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>>> G = nx.LCF_graph(14, [5, -5], 7)
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See http://mathworld.wolfram.com/LCFNotation.html for a description
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and references.
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"""
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if n <= 0:
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return empty_graph(0, create_using)
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# start with the n-cycle
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G = cycle_graph(n, create_using)
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if G.is_directed():
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raise NetworkXError("Directed Graph not supported")
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G.name = "LCF_graph"
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nodes = sorted(list(G))
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n_extra_edges = repeats * len(shift_list)
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# edges are added n_extra_edges times
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# (not all of these need be new)
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if n_extra_edges < 1:
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return G
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for i in range(n_extra_edges):
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shift = shift_list[i % len(shift_list)] # cycle through shift_list
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v1 = nodes[i % n] # cycle repeatedly through nodes
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v2 = nodes[(i + shift) % n]
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G.add_edge(v1, v2)
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return G
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#-------------------------------------------------------------------------------
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# Various small and named graphs
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#-------------------------------------------------------------------------------
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def bull_graph(create_using=None):
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"""Returns the Bull graph. """
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description = [
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"adjacencylist",
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"Bull Graph",
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5,
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[[2, 3], [1, 3, 4], [1, 2, 5], [2], [3]]
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]
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G = make_small_undirected_graph(description, create_using)
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return G
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def chvatal_graph(create_using=None):
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"""Returns the Chvátal graph."""
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description = [
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"adjacencylist",
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"Chvatal Graph",
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12,
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[[2, 5, 7, 10], [3, 6, 8], [4, 7, 9], [5, 8, 10],
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[6, 9], [11, 12], [11, 12], [9, 12],
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[11], [11, 12], [], []]
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]
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G = make_small_undirected_graph(description, create_using)
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return G
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def cubical_graph(create_using=None):
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"""Returns the 3-regular Platonic Cubical graph."""
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description = [
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"adjacencylist",
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"Platonic Cubical Graph",
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8,
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[[2, 4, 5], [1, 3, 8], [2, 4, 7], [1, 3, 6],
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[1, 6, 8], [4, 5, 7], [3, 6, 8], [2, 5, 7]]
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]
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G = make_small_undirected_graph(description, create_using)
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return G
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def desargues_graph(create_using=None):
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""" Return the Desargues graph."""
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G = LCF_graph(20, [5, -5, 9, -9], 5, create_using)
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G.name = "Desargues Graph"
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return G
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def diamond_graph(create_using=None):
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"""Returns the Diamond graph. """
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description = [
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"adjacencylist",
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"Diamond Graph",
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4,
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[[2, 3], [1, 3, 4], [1, 2, 4], [2, 3]]
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]
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G = make_small_undirected_graph(description, create_using)
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return G
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def dodecahedral_graph(create_using=None):
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""" Return the Platonic Dodecahedral graph. """
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G = LCF_graph(20, [10, 7, 4, -4, -7, 10, -4, 7, -7, 4], 2, create_using)
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G.name = "Dodecahedral Graph"
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return G
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def frucht_graph(create_using=None):
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"""Returns the Frucht Graph.
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The Frucht Graph is the smallest cubical graph whose
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automorphism group consists only of the identity element.
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"""
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G = cycle_graph(7, create_using)
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G.add_edges_from([[0, 7], [1, 7], [2, 8], [3, 9], [4, 9], [5, 10], [6, 10],
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[7, 11], [8, 11], [8, 9], [10, 11]])
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G.name = "Frucht Graph"
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return G
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def heawood_graph(create_using=None):
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""" Return the Heawood graph, a (3,6) cage. """
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G = LCF_graph(14, [5, -5], 7, create_using)
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G.name = "Heawood Graph"
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return G
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def hoffman_singleton_graph():
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'''Return the Hoffman-Singleton Graph.'''
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G = nx.Graph()
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for i in range(5):
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for j in range(5):
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G.add_edge(('pentagon', i, j), ('pentagon', i, (j - 1) % 5))
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G.add_edge(('pentagon', i, j), ('pentagon', i, (j + 1) % 5))
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G.add_edge(('pentagram', i, j), ('pentagram', i, (j - 2) % 5))
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G.add_edge(('pentagram', i, j), ('pentagram', i, (j + 2) % 5))
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for k in range(5):
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G.add_edge(('pentagon', i, j),
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('pentagram', k, (i * k + j) % 5))
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G = nx.convert_node_labels_to_integers(G)
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G.name = 'Hoffman-Singleton Graph'
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return G
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def house_graph(create_using=None):
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"""Returns the House graph (square with triangle on top)."""
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description = [
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"adjacencylist",
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"House Graph",
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5,
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[[2, 3], [1, 4], [1, 4, 5], [2, 3, 5], [3, 4]]
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]
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G = make_small_undirected_graph(description, create_using)
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return G
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def house_x_graph(create_using=None):
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"""Returns the House graph with a cross inside the house square."""
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description = [
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"adjacencylist",
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"House-with-X-inside Graph",
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5,
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[[2, 3, 4], [1, 3, 4], [1, 2, 4, 5], [1, 2, 3, 5], [3, 4]]
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]
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G = make_small_undirected_graph(description, create_using)
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return G
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def icosahedral_graph(create_using=None):
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"""Returns the Platonic Icosahedral graph."""
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description = [
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"adjacencylist",
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"Platonic Icosahedral Graph",
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12,
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[[2, 6, 8, 9, 12], [3, 6, 7, 9], [4, 7, 9, 10], [5, 7, 10, 11],
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[6, 7, 11, 12], [7, 12], [], [9, 10, 11, 12],
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[10], [11], [12], []]
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]
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G = make_small_undirected_graph(description, create_using)
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return G
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def krackhardt_kite_graph(create_using=None):
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"""
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Return the Krackhardt Kite Social Network.
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A 10 actor social network introduced by David Krackhardt
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to illustrate: degree, betweenness, centrality, closeness, etc.
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The traditional labeling is:
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Andre=1, Beverley=2, Carol=3, Diane=4,
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Ed=5, Fernando=6, Garth=7, Heather=8, Ike=9, Jane=10.
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"""
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description = [
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"adjacencylist",
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"Krackhardt Kite Social Network",
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10,
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[[2, 3, 4, 6], [1, 4, 5, 7], [1, 4, 6], [1, 2, 3, 5, 6, 7], [2, 4, 7],
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[1, 3, 4, 7, 8], [2, 4, 5, 6, 8], [6, 7, 9], [8, 10], [9]]
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]
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G = make_small_undirected_graph(description, create_using)
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return G
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def moebius_kantor_graph(create_using=None):
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"""Returns the Moebius-Kantor graph."""
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G = LCF_graph(16, [5, -5], 8, create_using)
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G.name = "Moebius-Kantor Graph"
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return G
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def octahedral_graph(create_using=None):
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"""Returns the Platonic Octahedral graph."""
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description = [
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"adjacencylist",
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"Platonic Octahedral Graph",
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6,
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[[2, 3, 4, 5], [3, 4, 6], [5, 6], [5, 6], [6], []]
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]
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G = make_small_undirected_graph(description, create_using)
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return G
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def pappus_graph():
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""" Return the Pappus graph."""
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G = LCF_graph(18, [5, 7, -7, 7, -7, -5], 3)
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G.name = "Pappus Graph"
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return G
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def petersen_graph(create_using=None):
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"""Returns the Petersen graph."""
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description = [
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"adjacencylist",
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"Petersen Graph",
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10,
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[[2, 5, 6], [1, 3, 7], [2, 4, 8], [3, 5, 9], [4, 1, 10], [1, 8, 9], [2, 9, 10],
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[3, 6, 10], [4, 6, 7], [5, 7, 8]]
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]
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G = make_small_undirected_graph(description, create_using)
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return G
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def sedgewick_maze_graph(create_using=None):
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"""
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Return a small maze with a cycle.
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This is the maze used in Sedgewick,3rd Edition, Part 5, Graph
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Algorithms, Chapter 18, e.g. Figure 18.2 and following.
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Nodes are numbered 0,..,7
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"""
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G = empty_graph(0, create_using)
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G.add_nodes_from(range(8))
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G.add_edges_from([[0, 2], [0, 7], [0, 5]])
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G.add_edges_from([[1, 7], [2, 6]])
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G.add_edges_from([[3, 4], [3, 5]])
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G.add_edges_from([[4, 5], [4, 7], [4, 6]])
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G.name = "Sedgewick Maze"
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return G
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def tetrahedral_graph(create_using=None):
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""" Return the 3-regular Platonic Tetrahedral graph."""
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G = complete_graph(4, create_using)
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G.name = "Platonic Tetrahedral graph"
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return G
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def truncated_cube_graph(create_using=None):
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"""Returns the skeleton of the truncated cube."""
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description = [
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"adjacencylist",
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"Truncated Cube Graph",
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24,
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[[2, 3, 5], [12, 15], [4, 5], [7, 9],
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[6], [17, 19], [8, 9], [11, 13],
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[10], [18, 21], [12, 13], [15],
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[14], [22, 23], [16], [20, 24],
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[18, 19], [21], [20], [24],
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[22], [23], [24], []]
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]
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G = make_small_undirected_graph(description, create_using)
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return G
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def truncated_tetrahedron_graph(create_using=None):
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"""Returns the skeleton of the truncated Platonic tetrahedron."""
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G = path_graph(12, create_using)
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# G.add_edges_from([(1,3),(1,10),(2,7),(4,12),(5,12),(6,8),(9,11)])
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G.add_edges_from([(0, 2), (0, 9), (1, 6), (3, 11), (4, 11), (5, 7), (8, 10)])
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G.name = "Truncated Tetrahedron Graph"
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return G
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def tutte_graph(create_using=None):
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"""Returns the Tutte graph."""
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description = [
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"adjacencylist",
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"Tutte's Graph",
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46,
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[[2, 3, 4], [5, 27], [11, 12], [19, 20], [6, 34],
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[7, 30], [8, 28], [9, 15], [10, 39], [11, 38],
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[40], [13, 40], [14, 36], [15, 16], [35],
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[17, 23], [18, 45], [19, 44], [46], [21, 46],
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[22, 42], [23, 24], [41], [25, 28], [26, 33],
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[27, 32], [34], [29], [30, 33], [31],
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[32, 34], [33], [], [], [36, 39],
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[37], [38, 40], [39], [], [],
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[42, 45], [43], [44, 46], [45], [], []]
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]
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G = make_small_undirected_graph(description, create_using)
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return G
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