"""
Various small and named graphs, together with some compact generators.
"""
__all__ = [
"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",
]
from functools import wraps
import networkx as nx
from networkx.exception import NetworkXError
from networkx.generators.classic import (
complete_graph,
cycle_graph,
empty_graph,
path_graph,
)
def _raise_on_directed(func):
"""
A decorator which inspects the `create_using` argument and raises a
NetworkX exception when `create_using` is a DiGraph (class or instance) for
graph generators that do not support directed outputs.
`create_using` may be a keyword argument or the first positional argument.
"""
@wraps(func)
def wrapper(*args, **kwargs):
create_using = args[0] if args else kwargs.get("create_using")
if create_using is not None:
G = nx.empty_graph(create_using=create_using)
if G.is_directed():
raise NetworkXError("Directed Graph not supported")
return func(*args, **kwargs)
return wrapper
[docs]
@nx._dispatchable(graphs=None, returns_graph=True)
def LCF_graph(n, shift_list, repeats, create_using=None):
"""
Return the cubic graph specified in LCF notation.
LCF (Lederberg-Coxeter-Fruchte) notation[1]_ 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`.
Nodes are drawn from ``range(n)``. Each node ``n_i`` is connected with
node ``n_i + shift % n`` where ``shift`` is given by cycling through
the input `shift_list` `repeat` s times.
Parameters
----------
n : int
The starting graph is the `n`-cycle with nodes ``0, ..., n-1``.
The null graph is returned if `n` < 1.
shift_list : list
A list of integer shifts mod `n`, ``[s1, s2, .., sk]``
repeats : int
Integer specifying the number of times that shifts in `shift_list`
are successively applied to each current node in the n-cycle
to generate an edge between ``n_current`` and ``n_current + shift mod n``.
Returns
-------
G : Graph
A graph instance created from the specified LCF notation.
Examples
--------
The utility graph $K_{3,3}$
>>> G = nx.LCF_graph(6, [3, -3], 3)
>>> G.edges()
EdgeView([(0, 1), (0, 5), (0, 3), (1, 2), (1, 4), (2, 3), (2, 5), (3, 4), (4, 5)])
The Heawood graph:
>>> G = nx.LCF_graph(14, [5, -5], 7)
>>> nx.is_isomorphic(G, nx.heawood_graph())
True
References
----------
.. [1] https://en.wikipedia.org/wiki/LCF_notation
"""
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(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]
@_raise_on_directed
@nx._dispatchable(graphs=None, returns_graph=True)
def bull_graph(create_using=None):
"""
Returns the Bull Graph
The Bull Graph has 5 nodes and 5 edges. It is a planar undirected
graph in the form of a triangle with two disjoint pendant edges [1]_
The name comes from the triangle and pendant edges representing
respectively the body and legs of a bull.
Parameters
----------
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Returns
-------
G : networkx Graph
A bull graph with 5 nodes
References
----------
.. [1] https://en.wikipedia.org/wiki/Bull_graph.
"""
G = nx.from_dict_of_lists(
{0: [1, 2], 1: [0, 2, 3], 2: [0, 1, 4], 3: [1], 4: [2]},
create_using=create_using,
)
G.name = "Bull Graph"
return G
[docs]
@_raise_on_directed
@nx._dispatchable(graphs=None, returns_graph=True)
def chvatal_graph(create_using=None):
"""
Returns the Chvátal Graph
The Chvátal Graph is an undirected graph with 12 nodes and 24 edges [1]_.
It has 370 distinct (directed) Hamiltonian cycles, giving a unique generalized
LCF notation of order 4, two of order 6 , and 43 of order 1 [2]_.
Parameters
----------
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Returns
-------
G : networkx Graph
The Chvátal graph with 12 nodes and 24 edges
References
----------
.. [1] https://en.wikipedia.org/wiki/Chv%C3%A1tal_graph
.. [2] https://mathworld.wolfram.com/ChvatalGraph.html
"""
G = nx.from_dict_of_lists(
{
0: [1, 4, 6, 9],
1: [2, 5, 7],
2: [3, 6, 8],
3: [4, 7, 9],
4: [5, 8],
5: [10, 11],
6: [10, 11],
7: [8, 11],
8: [10],
9: [10, 11],
},
create_using=create_using,
)
G.name = "Chvatal Graph"
return G
[docs]
@_raise_on_directed
@nx._dispatchable(graphs=None, returns_graph=True)
def cubical_graph(create_using=None):
"""
Returns the 3-regular Platonic Cubical Graph
The skeleton of the cube (the nodes and edges) form a graph, with 8
nodes, and 12 edges. It is a special case of the hypercube graph.
It is one of 5 Platonic graphs, each a skeleton of its
Platonic solid [1]_.
Such graphs arise in parallel processing in computers.
Parameters
----------
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Returns
-------
G : networkx Graph
A cubical graph with 8 nodes and 12 edges
References
----------
.. [1] https://en.wikipedia.org/wiki/Cube#Cubical_graph
"""
G = nx.from_dict_of_lists(
{
0: [1, 3, 4],
1: [0, 2, 7],
2: [1, 3, 6],
3: [0, 2, 5],
4: [0, 5, 7],
5: [3, 4, 6],
6: [2, 5, 7],
7: [1, 4, 6],
},
create_using=create_using,
)
G.name = "Platonic Cubical Graph"
return G
[docs]
@nx._dispatchable(graphs=None, returns_graph=True)
def desargues_graph(create_using=None):
"""
Returns the Desargues Graph
The Desargues Graph is a non-planar, distance-transitive cubic graph
with 20 nodes and 30 edges [1]_.
It is a symmetric graph. It can be represented in LCF notation
as [5,-5,9,-9]^5 [2]_.
Parameters
----------
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Returns
-------
G : networkx Graph
Desargues Graph with 20 nodes and 30 edges
References
----------
.. [1] https://en.wikipedia.org/wiki/Desargues_graph
.. [2] https://mathworld.wolfram.com/DesarguesGraph.html
"""
G = LCF_graph(20, [5, -5, 9, -9], 5, create_using)
G.name = "Desargues Graph"
return G
[docs]
@_raise_on_directed
@nx._dispatchable(graphs=None, returns_graph=True)
def diamond_graph(create_using=None):
"""
Returns the Diamond graph
The Diamond Graph is planar undirected graph with 4 nodes and 5 edges.
It is also sometimes known as the double triangle graph or kite graph [1]_.
Parameters
----------
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Returns
-------
G : networkx Graph
Diamond Graph with 4 nodes and 5 edges
References
----------
.. [1] https://mathworld.wolfram.com/DiamondGraph.html
"""
G = nx.from_dict_of_lists(
{0: [1, 2], 1: [0, 2, 3], 2: [0, 1, 3], 3: [1, 2]}, create_using=create_using
)
G.name = "Diamond Graph"
return G
[docs]
@nx._dispatchable(graphs=None, returns_graph=True)
def dodecahedral_graph(create_using=None):
"""
Returns the Platonic Dodecahedral graph.
The dodecahedral graph has 20 nodes and 30 edges. The skeleton of the
dodecahedron forms a graph. It is one of 5 Platonic graphs [1]_.
It can be described in LCF notation as:
``[10, 7, 4, -4, -7, 10, -4, 7, -7, 4]^2`` [2]_.
Parameters
----------
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Returns
-------
G : networkx Graph
Dodecahedral Graph with 20 nodes and 30 edges
References
----------
.. [1] https://en.wikipedia.org/wiki/Regular_dodecahedron#Dodecahedral_graph
.. [2] https://mathworld.wolfram.com/DodecahedralGraph.html
"""
G = LCF_graph(20, [10, 7, 4, -4, -7, 10, -4, 7, -7, 4], 2, create_using)
G.name = "Dodecahedral Graph"
return G
[docs]
@nx._dispatchable(graphs=None, returns_graph=True)
def frucht_graph(create_using=None):
"""
Returns the Frucht Graph.
The Frucht Graph is the smallest cubical graph whose
automorphism group consists only of the identity element [1]_.
It has 12 nodes and 18 edges and no nontrivial symmetries.
It is planar and Hamiltonian [2]_.
Parameters
----------
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Returns
-------
G : networkx Graph
Frucht Graph with 12 nodes and 18 edges
References
----------
.. [1] https://en.wikipedia.org/wiki/Frucht_graph
.. [2] https://mathworld.wolfram.com/FruchtGraph.html
"""
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]
@nx._dispatchable(graphs=None, returns_graph=True)
def heawood_graph(create_using=None):
"""
Returns the Heawood Graph, a (3,6) cage.
The Heawood Graph is an undirected graph with 14 nodes and 21 edges,
named after Percy John Heawood [1]_.
It is cubic symmetric, nonplanar, Hamiltonian, and can be represented
in LCF notation as ``[5,-5]^7`` [2]_.
It is the unique (3,6)-cage: the regular cubic graph of girth 6 with
minimal number of vertices [3]_.
Parameters
----------
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Returns
-------
G : networkx Graph
Heawood Graph with 14 nodes and 21 edges
References
----------
.. [1] https://en.wikipedia.org/wiki/Heawood_graph
.. [2] https://mathworld.wolfram.com/HeawoodGraph.html
.. [3] https://www.win.tue.nl/~aeb/graphs/Heawood.html
"""
G = LCF_graph(14, [5, -5], 7, create_using)
G.name = "Heawood Graph"
return G
[docs]
@nx._dispatchable(graphs=None, returns_graph=True)
def hoffman_singleton_graph():
"""
Returns the Hoffman-Singleton Graph.
The Hoffman–Singleton graph is a symmetrical undirected graph
with 50 nodes and 175 edges.
All indices lie in ``Z % 5``: that is, the integers mod 5 [1]_.
It is the only regular graph of vertex degree 7, diameter 2, and girth 5.
It is the unique (7,5)-cage graph and Moore graph, and contains many
copies of the Petersen graph [2]_.
Returns
-------
G : networkx Graph
Hoffman–Singleton Graph with 50 nodes and 175 edges
Notes
-----
Constructed from pentagon and pentagram as follows: Take five pentagons $P_h$
and five pentagrams $Q_i$ . Join vertex $j$ of $P_h$ to vertex $h·i+j$ of $Q_i$ [3]_.
References
----------
.. [1] https://blogs.ams.org/visualinsight/2016/02/01/hoffman-singleton-graph/
.. [2] https://mathworld.wolfram.com/Hoffman-SingletonGraph.html
.. [3] https://en.wikipedia.org/wiki/Hoffman%E2%80%93Singleton_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]
@_raise_on_directed
@nx._dispatchable(graphs=None, returns_graph=True)
def house_graph(create_using=None):
"""
Returns the House graph (square with triangle on top)
The house graph is a simple undirected graph with
5 nodes and 6 edges [1]_.
Parameters
----------
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Returns
-------
G : networkx Graph
House graph in the form of a square with a triangle on top
References
----------
.. [1] https://mathworld.wolfram.com/HouseGraph.html
"""
G = nx.from_dict_of_lists(
{0: [1, 2], 1: [0, 3], 2: [0, 3, 4], 3: [1, 2, 4], 4: [2, 3]},
create_using=create_using,
)
G.name = "House Graph"
return G
[docs]
@_raise_on_directed
@nx._dispatchable(graphs=None, returns_graph=True)
def house_x_graph(create_using=None):
"""
Returns the House graph with a cross inside the house square.
The House X-graph is the House graph plus the two edges connecting diagonally
opposite vertices of the square base. It is also one of the two graphs
obtained by removing two edges from the pentatope graph [1]_.
Parameters
----------
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Returns
-------
G : networkx Graph
House graph with diagonal vertices connected
References
----------
.. [1] https://mathworld.wolfram.com/HouseGraph.html
"""
G = house_graph(create_using)
G.add_edges_from([(0, 3), (1, 2)])
G.name = "House-with-X-inside Graph"
return G
[docs]
@_raise_on_directed
@nx._dispatchable(graphs=None, returns_graph=True)
def icosahedral_graph(create_using=None):
"""
Returns the Platonic Icosahedral graph.
The icosahedral graph has 12 nodes and 30 edges. It is a Platonic graph
whose nodes have the connectivity of the icosahedron. It is undirected,
regular and Hamiltonian [1]_.
Parameters
----------
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Returns
-------
G : networkx Graph
Icosahedral graph with 12 nodes and 30 edges.
References
----------
.. [1] https://mathworld.wolfram.com/IcosahedralGraph.html
"""
G = nx.from_dict_of_lists(
{
0: [1, 5, 7, 8, 11],
1: [2, 5, 6, 8],
2: [3, 6, 8, 9],
3: [4, 6, 9, 10],
4: [5, 6, 10, 11],
5: [6, 11],
7: [8, 9, 10, 11],
8: [9],
9: [10],
10: [11],
},
create_using=create_using,
)
G.name = "Platonic Icosahedral Graph"
return G
[docs]
@_raise_on_directed
@nx._dispatchable(graphs=None, returns_graph=True)
def krackhardt_kite_graph(create_using=None):
"""
Returns the Krackhardt Kite Social Network.
A 10 actor social network introduced by David Krackhardt
to illustrate different centrality measures [1]_.
Parameters
----------
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Returns
-------
G : networkx Graph
Krackhardt Kite graph with 10 nodes and 18 edges
Notes
-----
The traditional labeling is:
Andre=1, Beverley=2, Carol=3, Diane=4,
Ed=5, Fernando=6, Garth=7, Heather=8, Ike=9, Jane=10.
References
----------
.. [1] Krackhardt, David. "Assessing the Political Landscape: Structure,
Cognition, and Power in Organizations". Administrative Science Quarterly.
35 (2): 342–369. doi:10.2307/2393394. JSTOR 2393394. June 1990.
"""
G = nx.from_dict_of_lists(
{
0: [1, 2, 3, 5],
1: [0, 3, 4, 6],
2: [0, 3, 5],
3: [0, 1, 2, 4, 5, 6],
4: [1, 3, 6],
5: [0, 2, 3, 6, 7],
6: [1, 3, 4, 5, 7],
7: [5, 6, 8],
8: [7, 9],
9: [8],
},
create_using=create_using,
)
G.name = "Krackhardt Kite Social Network"
return G
[docs]
@nx._dispatchable(graphs=None, returns_graph=True)
def moebius_kantor_graph(create_using=None):
"""
Returns the Moebius-Kantor graph.
The Möbius-Kantor graph is the cubic symmetric graph on 16 nodes.
Its LCF notation is [5,-5]^8, and it is isomorphic to the generalized
Petersen graph [1]_.
Parameters
----------
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Returns
-------
G : networkx Graph
Moebius-Kantor graph
References
----------
.. [1] https://en.wikipedia.org/wiki/M%C3%B6bius%E2%80%93Kantor_graph
"""
G = LCF_graph(16, [5, -5], 8, create_using)
G.name = "Moebius-Kantor Graph"
return G
[docs]
@_raise_on_directed
@nx._dispatchable(graphs=None, returns_graph=True)
def octahedral_graph(create_using=None):
"""
Returns the Platonic Octahedral graph.
The octahedral graph is the 6-node 12-edge Platonic graph having the
connectivity of the octahedron [1]_. If 6 couples go to a party,
and each person shakes hands with every person except his or her partner,
then this graph describes the set of handshakes that take place;
for this reason it is also called the cocktail party graph [2]_.
Parameters
----------
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Returns
-------
G : networkx Graph
Octahedral graph
References
----------
.. [1] https://mathworld.wolfram.com/OctahedralGraph.html
.. [2] https://en.wikipedia.org/wiki/Tur%C3%A1n_graph#Special_cases
"""
G = nx.from_dict_of_lists(
{0: [1, 2, 3, 4], 1: [2, 3, 5], 2: [4, 5], 3: [4, 5], 4: [5]},
create_using=create_using,
)
G.name = "Platonic Octahedral Graph"
return G
[docs]
@nx._dispatchable(graphs=None, returns_graph=True)
def pappus_graph():
"""
Returns the Pappus graph.
The Pappus graph is a cubic symmetric distance-regular graph with 18 nodes
and 27 edges. It is Hamiltonian and can be represented in LCF notation as
[5,7,-7,7,-7,-5]^3 [1]_.
Returns
-------
G : networkx Graph
Pappus graph
References
----------
.. [1] https://en.wikipedia.org/wiki/Pappus_graph
"""
G = LCF_graph(18, [5, 7, -7, 7, -7, -5], 3)
G.name = "Pappus Graph"
return G
[docs]
@_raise_on_directed
@nx._dispatchable(graphs=None, returns_graph=True)
def petersen_graph(create_using=None):
"""
Returns the Petersen graph.
The Peterson graph is a cubic, undirected graph with 10 nodes and 15 edges [1]_.
Julius Petersen constructed the graph as the smallest counterexample
against the claim that a connected bridgeless cubic graph
has an edge colouring with three colours [2]_.
Parameters
----------
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Returns
-------
G : networkx Graph
Petersen graph
References
----------
.. [1] https://en.wikipedia.org/wiki/Petersen_graph
.. [2] https://www.win.tue.nl/~aeb/drg/graphs/Petersen.html
"""
G = nx.from_dict_of_lists(
{
0: [1, 4, 5],
1: [0, 2, 6],
2: [1, 3, 7],
3: [2, 4, 8],
4: [3, 0, 9],
5: [0, 7, 8],
6: [1, 8, 9],
7: [2, 5, 9],
8: [3, 5, 6],
9: [4, 6, 7],
},
create_using=create_using,
)
G.name = "Petersen Graph"
return G
[docs]
@nx._dispatchable(graphs=None, returns_graph=True)
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 [1]_.
Nodes are numbered 0,..,7
Parameters
----------
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Returns
-------
G : networkx Graph
Small maze with a cycle
References
----------
.. [1] Figure 18.2, Chapter 18, Graph Algorithms (3rd Ed), Sedgewick
"""
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]
@nx._dispatchable(graphs=None, returns_graph=True)
def tetrahedral_graph(create_using=None):
"""
Returns the 3-regular Platonic Tetrahedral graph.
Tetrahedral graph has 4 nodes and 6 edges. It is a
special case of the complete graph, K4, and wheel graph, W4.
It is one of the 5 platonic graphs [1]_.
Parameters
----------
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Returns
-------
G : networkx Graph
Tetrahedral Graph
References
----------
.. [1] https://en.wikipedia.org/wiki/Tetrahedron#Tetrahedral_graph
"""
G = complete_graph(4, create_using)
G.name = "Platonic Tetrahedral Graph"
return G
[docs]
@_raise_on_directed
@nx._dispatchable(graphs=None, returns_graph=True)
def truncated_cube_graph(create_using=None):
"""
Returns the skeleton of the truncated cube.
The truncated cube is an Archimedean solid with 14 regular
faces (6 octagonal and 8 triangular), 36 edges and 24 nodes [1]_.
The truncated cube is created by truncating (cutting off) the tips
of the cube one third of the way into each edge [2]_.
Parameters
----------
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Returns
-------
G : networkx Graph
Skeleton of the truncated cube
References
----------
.. [1] https://en.wikipedia.org/wiki/Truncated_cube
.. [2] https://www.coolmath.com/reference/polyhedra-truncated-cube
"""
G = nx.from_dict_of_lists(
{
0: [1, 2, 4],
1: [11, 14],
2: [3, 4],
3: [6, 8],
4: [5],
5: [16, 18],
6: [7, 8],
7: [10, 12],
8: [9],
9: [17, 20],
10: [11, 12],
11: [14],
12: [13],
13: [21, 22],
14: [15],
15: [19, 23],
16: [17, 18],
17: [20],
18: [19],
19: [23],
20: [21],
21: [22],
22: [23],
},
create_using=create_using,
)
G.name = "Truncated Cube Graph"
return G
[docs]
@nx._dispatchable(graphs=None, returns_graph=True)
def truncated_tetrahedron_graph(create_using=None):
"""
Returns the skeleton of the truncated Platonic tetrahedron.
The truncated tetrahedron is an Archimedean solid with 4 regular hexagonal faces,
4 equilateral triangle faces, 12 nodes and 18 edges. It can be constructed by truncating
all 4 vertices of a regular tetrahedron at one third of the original edge length [1]_.
Parameters
----------
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Returns
-------
G : networkx Graph
Skeleton of the truncated tetrahedron
References
----------
.. [1] https://en.wikipedia.org/wiki/Truncated_tetrahedron
"""
G = path_graph(12, create_using)
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]
@_raise_on_directed
@nx._dispatchable(graphs=None, returns_graph=True)
def tutte_graph(create_using=None):
"""
Returns the Tutte graph.
The Tutte graph is a cubic polyhedral, non-Hamiltonian graph. It has
46 nodes and 69 edges.
It is a counterexample to Tait's conjecture that every 3-regular polyhedron
has a Hamiltonian cycle.
It can be realized geometrically from a tetrahedron by multiply truncating
three of its vertices [1]_.
Parameters
----------
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Returns
-------
G : networkx Graph
Tutte graph
References
----------
.. [1] https://en.wikipedia.org/wiki/Tutte_graph
"""
G = nx.from_dict_of_lists(
{
0: [1, 2, 3],
1: [4, 26],
2: [10, 11],
3: [18, 19],
4: [5, 33],
5: [6, 29],
6: [7, 27],
7: [8, 14],
8: [9, 38],
9: [10, 37],
10: [39],
11: [12, 39],
12: [13, 35],
13: [14, 15],
14: [34],
15: [16, 22],
16: [17, 44],
17: [18, 43],
18: [45],
19: [20, 45],
20: [21, 41],
21: [22, 23],
22: [40],
23: [24, 27],
24: [25, 32],
25: [26, 31],
26: [33],
27: [28],
28: [29, 32],
29: [30],
30: [31, 33],
31: [32],
34: [35, 38],
35: [36],
36: [37, 39],
37: [38],
40: [41, 44],
41: [42],
42: [43, 45],
43: [44],
},
create_using=create_using,
)
G.name = "Tutte's Graph"
return G
