# -*- coding: utf-8 -*-
"""
Various small and named graphs, together with some compact generators.
"""
__author__ = """Aric Hagberg (hagberg@lanl.gov)\nPieter Swart (swart@lanl.gov)"""
# Copyright (C) 2004-2018 by
# Aric Hagberg <hagberg@lanl.gov>
# Dan Schult <dschult@colgate.edu>
# Pieter Swart <swart@lanl.gov>
# All rights reserved.
# BSD license.
__all__ = ['make_small_graph',
'LCF_graph',
'bull_graph',
'chvatal_graph',
'cubical_graph',
'desargues_graph',
'diamond_graph',
'dodecahedral_graph',
'frucht_graph',
'heawood_graph',
'hoffman_singleton_graph',
'house_graph',
'house_x_graph',
'icosahedral_graph',
'krackhardt_kite_graph',
'moebius_kantor_graph',
'octahedral_graph',
'pappus_graph',
'petersen_graph',
'sedgewick_maze_graph',
'tetrahedral_graph',
'truncated_cube_graph',
'truncated_tetrahedron_graph',
'tutte_graph']
import networkx as nx
from networkx.generators.classic import empty_graph, cycle_graph, path_graph, complete_graph
from networkx.exception import NetworkXError
#------------------------------------------------------------------------------
# Tools for creating small graphs
#------------------------------------------------------------------------------
def make_small_undirected_graph(graph_description, create_using=None):
"""
Return a small undirected graph described by graph_description.
See make_small_graph.
"""
G = empty_graph(0, create_using)
if G.is_directed():
raise NetworkXError("Directed Graph not supported")
return make_small_graph(graph_description, G)
[docs]def make_small_graph(graph_description, create_using=None):
"""
Return the small graph described by graph_description.
graph_description is a list of the form [ltype,name,n,xlist]
Here ltype is one of "adjacencylist" or "edgelist",
name is the name of the graph and n the number of nodes.
This constructs a graph of n nodes with integer labels 0,..,n-1.
If ltype="adjacencylist" then xlist is an adjacency list
with exactly n entries, in with the j'th entry (which can be empty)
specifies the nodes connected to vertex j.
e.g. the "square" graph C_4 can be obtained by
>>> G=nx.make_small_graph(["adjacencylist","C_4",4,[[2,4],[1,3],[2,4],[1,3]]])
or, since we do not need to add edges twice,
>>> G=nx.make_small_graph(["adjacencylist","C_4",4,[[2,4],[3],[4],[]]])
If ltype="edgelist" then xlist is an edge list
written as [[v1,w2],[v2,w2],...,[vk,wk]],
where vj and wj integers in the range 1,..,n
e.g. the "square" graph C_4 can be obtained by
>>> G=nx.make_small_graph(["edgelist","C_4",4,[[1,2],[3,4],[2,3],[4,1]]])
Use the create_using argument to choose the graph class/type.
"""
ltype = graph_description[0]
name = graph_description[1]
n = graph_description[2]
G = empty_graph(n, create_using)
nodes = G.nodes()
if ltype == "adjacencylist":
adjlist = graph_description[3]
if len(adjlist) != n:
raise NetworkXError("invalid graph_description")
G.add_edges_from([(u - 1, v) for v in nodes for u in adjlist[v]])
elif ltype == "edgelist":
edgelist = graph_description[3]
for e in edgelist:
v1 = e[0] - 1
v2 = e[1] - 1
if v1 < 0 or v1 > n - 1 or v2 < 0 or v2 > n - 1:
raise NetworkXError("invalid graph_description")
else:
G.add_edge(v1, v2)
G.name = name
return G
[docs]def LCF_graph(n, shift_list, repeats, create_using=None):
"""
Return the cubic graph specified in LCF notation.
LCF notation (LCF=Lederberg-Coxeter-Fruchte) is a compressed
notation used in the generation of various cubic Hamiltonian
graphs of high symmetry. See, for example, dodecahedral_graph,
desargues_graph, heawood_graph and pappus_graph below.
n (number of nodes)
The starting graph is the n-cycle with nodes 0,...,n-1.
(The null graph is returned if n < 0.)
shift_list = [s1,s2,..,sk], a list of integer shifts mod n,
repeats
integer specifying the number of times that shifts in shift_list
are successively applied to each v_current in the n-cycle
to generate an edge between v_current and v_current+shift mod n.
For v1 cycling through the n-cycle a total of k*repeats
with shift cycling through shiftlist repeats times connect
v1 with v1+shift mod n
The utility graph $K_{3,3}$
>>> G = nx.LCF_graph(6, [3, -3], 3)
The Heawood graph
>>> G = nx.LCF_graph(14, [5, -5], 7)
See http://mathworld.wolfram.com/LCFNotation.html for a description
and references.
"""
if n <= 0:
return empty_graph(0, create_using)
# start with the n-cycle
G = cycle_graph(n, create_using)
if G.is_directed():
raise NetworkXError("Directed Graph not supported")
G.name = "LCF_graph"
nodes = sorted(list(G))
n_extra_edges = repeats * len(shift_list)
# edges are added n_extra_edges times
# (not all of these need be new)
if n_extra_edges < 1:
return G
for i in range(n_extra_edges):
shift = shift_list[i % len(shift_list)] # cycle through shift_list
v1 = nodes[i % n] # cycle repeatedly through nodes
v2 = nodes[(i + shift) % n]
G.add_edge(v1, v2)
return G
#-------------------------------------------------------------------------------
# Various small and named graphs
#-------------------------------------------------------------------------------
[docs]def bull_graph(create_using=None):
"""Return the Bull graph. """
description = [
"adjacencylist",
"Bull Graph",
5,
[[2, 3], [1, 3, 4], [1, 2, 5], [2], [3]]
]
G = make_small_undirected_graph(description, create_using)
return G
[docs]def chvatal_graph(create_using=None):
"""Return the Chvátal graph."""
description = [
"adjacencylist",
"Chvatal Graph",
12,
[[2, 5, 7, 10], [3, 6, 8], [4, 7, 9], [5, 8, 10],
[6, 9], [11, 12], [11, 12], [9, 12],
[11], [11, 12], [], []]
]
G = make_small_undirected_graph(description, create_using)
return G
[docs]def cubical_graph(create_using=None):
"""Return the 3-regular Platonic Cubical graph."""
description = [
"adjacencylist",
"Platonic Cubical Graph",
8,
[[2, 4, 5], [1, 3, 8], [2, 4, 7], [1, 3, 6],
[1, 6, 8], [4, 5, 7], [3, 6, 8], [2, 5, 7]]
]
G = make_small_undirected_graph(description, create_using)
return G
[docs]def desargues_graph(create_using=None):
""" Return the Desargues graph."""
G = LCF_graph(20, [5, -5, 9, -9], 5, create_using)
G.name = "Desargues Graph"
return G
[docs]def diamond_graph(create_using=None):
"""Return the Diamond graph. """
description = [
"adjacencylist",
"Diamond Graph",
4,
[[2, 3], [1, 3, 4], [1, 2, 4], [2, 3]]
]
G = make_small_undirected_graph(description, create_using)
return G
[docs]def dodecahedral_graph(create_using=None):
""" Return the Platonic Dodecahedral graph. """
G = LCF_graph(20, [10, 7, 4, -4, -7, 10, -4, 7, -7, 4], 2, create_using)
G.name = "Dodecahedral Graph"
return G
[docs]def frucht_graph(create_using=None):
"""Return the Frucht Graph.
The Frucht Graph is the smallest cubical graph whose
automorphism group consists only of the identity element.
"""
G = cycle_graph(7, create_using)
G.add_edges_from([[0, 7], [1, 7], [2, 8], [3, 9], [4, 9], [5, 10], [6, 10],
[7, 11], [8, 11], [8, 9], [10, 11]])
G.name = "Frucht Graph"
return G
[docs]def heawood_graph(create_using=None):
""" Return the Heawood graph, a (3,6) cage. """
G = LCF_graph(14, [5, -5], 7, create_using)
G.name = "Heawood Graph"
return G
[docs]def hoffman_singleton_graph():
'''Return the Hoffman-Singleton Graph.'''
G = nx.Graph()
for i in range(5):
for j in range(5):
G.add_edge(('pentagon', i, j), ('pentagon', i, (j - 1) % 5))
G.add_edge(('pentagon', i, j), ('pentagon', i, (j + 1) % 5))
G.add_edge(('pentagram', i, j), ('pentagram', i, (j - 2) % 5))
G.add_edge(('pentagram', i, j), ('pentagram', i, (j + 2) % 5))
for k in range(5):
G.add_edge(('pentagon', i, j),
('pentagram', k, (i * k + j) % 5))
G = nx.convert_node_labels_to_integers(G)
G.name = 'Hoffman-Singleton Graph'
return G
[docs]def house_graph(create_using=None):
"""Return the House graph (square with triangle on top)."""
description = [
"adjacencylist",
"House Graph",
5,
[[2, 3], [1, 4], [1, 4, 5], [2, 3, 5], [3, 4]]
]
G = make_small_undirected_graph(description, create_using)
return G
[docs]def house_x_graph(create_using=None):
"""Return the House graph with a cross inside the house square."""
description = [
"adjacencylist",
"House-with-X-inside Graph",
5,
[[2, 3, 4], [1, 3, 4], [1, 2, 4, 5], [1, 2, 3, 5], [3, 4]]
]
G = make_small_undirected_graph(description, create_using)
return G
[docs]def icosahedral_graph(create_using=None):
"""Return the Platonic Icosahedral graph."""
description = [
"adjacencylist",
"Platonic Icosahedral Graph",
12,
[[2, 6, 8, 9, 12], [3, 6, 7, 9], [4, 7, 9, 10], [5, 7, 10, 11],
[6, 7, 11, 12], [7, 12], [], [9, 10, 11, 12],
[10], [11], [12], []]
]
G = make_small_undirected_graph(description, create_using)
return G
[docs]def krackhardt_kite_graph(create_using=None):
"""
Return the Krackhardt Kite Social Network.
A 10 actor social network introduced by David Krackhardt
to illustrate: degree, betweenness, centrality, closeness, etc.
The traditional labeling is:
Andre=1, Beverley=2, Carol=3, Diane=4,
Ed=5, Fernando=6, Garth=7, Heather=8, Ike=9, Jane=10.
"""
description = [
"adjacencylist",
"Krackhardt Kite Social Network",
10,
[[2, 3, 4, 6], [1, 4, 5, 7], [1, 4, 6], [1, 2, 3, 5, 6, 7], [2, 4, 7],
[1, 3, 4, 7, 8], [2, 4, 5, 6, 8], [6, 7, 9], [8, 10], [9]]
]
G = make_small_undirected_graph(description, create_using)
return G
[docs]def moebius_kantor_graph(create_using=None):
"""Return the Moebius-Kantor graph."""
G = LCF_graph(16, [5, -5], 8, create_using)
G.name = "Moebius-Kantor Graph"
return G
[docs]def octahedral_graph(create_using=None):
"""Return the Platonic Octahedral graph."""
description = [
"adjacencylist",
"Platonic Octahedral Graph",
6,
[[2, 3, 4, 5], [3, 4, 6], [5, 6], [5, 6], [6], []]
]
G = make_small_undirected_graph(description, create_using)
return G
[docs]def pappus_graph():
""" Return the Pappus graph."""
G = LCF_graph(18, [5, 7, -7, 7, -7, -5], 3)
G.name = "Pappus Graph"
return G
[docs]def petersen_graph(create_using=None):
"""Return the Petersen graph."""
description = [
"adjacencylist",
"Petersen Graph",
10,
[[2, 5, 6], [1, 3, 7], [2, 4, 8], [3, 5, 9], [4, 1, 10], [1, 8, 9], [2, 9, 10],
[3, 6, 10], [4, 6, 7], [5, 7, 8]]
]
G = make_small_undirected_graph(description, create_using)
return G
[docs]def sedgewick_maze_graph(create_using=None):
"""
Return a small maze with a cycle.
This is the maze used in Sedgewick,3rd Edition, Part 5, Graph
Algorithms, Chapter 18, e.g. Figure 18.2 and following.
Nodes are numbered 0,..,7
"""
G = empty_graph(0, create_using)
G.add_nodes_from(range(8))
G.add_edges_from([[0, 2], [0, 7], [0, 5]])
G.add_edges_from([[1, 7], [2, 6]])
G.add_edges_from([[3, 4], [3, 5]])
G.add_edges_from([[4, 5], [4, 7], [4, 6]])
G.name = "Sedgewick Maze"
return G
[docs]def tetrahedral_graph(create_using=None):
""" Return the 3-regular Platonic Tetrahedral graph."""
G = complete_graph(4, create_using)
G.name = "Platonic Tetrahedral graph"
return G
[docs]def truncated_cube_graph(create_using=None):
"""Return the skeleton of the truncated cube."""
description = [
"adjacencylist",
"Truncated Cube Graph",
24,
[[2, 3, 5], [12, 15], [4, 5], [7, 9],
[6], [17, 19], [8, 9], [11, 13],
[10], [18, 21], [12, 13], [15],
[14], [22, 23], [16], [20, 24],
[18, 19], [21], [20], [24],
[22], [23], [24], []]
]
G = make_small_undirected_graph(description, create_using)
return G
[docs]def truncated_tetrahedron_graph(create_using=None):
"""Return the skeleton of the truncated Platonic tetrahedron."""
G = path_graph(12, create_using)
# G.add_edges_from([(1,3),(1,10),(2,7),(4,12),(5,12),(6,8),(9,11)])
G.add_edges_from([(0, 2), (0, 9), (1, 6), (3, 11), (4, 11), (5, 7), (8, 10)])
G.name = "Truncated Tetrahedron Graph"
return G
[docs]def tutte_graph(create_using=None):
"""Return the Tutte graph."""
description = [
"adjacencylist",
"Tutte's Graph",
46,
[[2, 3, 4], [5, 27], [11, 12], [19, 20], [6, 34],
[7, 30], [8, 28], [9, 15], [10, 39], [11, 38],
[40], [13, 40], [14, 36], [15, 16], [35],
[17, 23], [18, 45], [19, 44], [46], [21, 46],
[22, 42], [23, 24], [41], [25, 28], [26, 33],
[27, 32], [34], [29], [30, 33], [31],
[32, 34], [33], [], [], [36, 39],
[37], [38, 40], [39], [], [],
[42, 45], [43], [44, 46], [45], [], []]
]
G = make_small_undirected_graph(description, create_using)
return G