Common digraphs¶

All digraphs in Sage can be built through the digraphs object. In order to build a circuit on 15 elements, one can do:

Sage

sage: g = digraphs.Circuit(15)

Python

>>> from sage.all import *
>>> g = digraphs.Circuit(Integer(15))

To get a circulant graph on 10 vertices in which a vertex \(i\) has \(i+2\) and \(i+3\) as outneighbors:

Sage

sage: p = digraphs.Circulant(10,[2,3])

Python

>>> from sage.all import *
>>> p = digraphs.Circulant(Integer(10),[Integer(2),Integer(3)])

More interestingly, one can get the list of all digraphs that Sage knows how to build by typing digraphs. in Sage and then hitting Tab.

`ButterflyGraph()`	Return a \(n\)-dimensional butterfly graph.
`Circuit()`	Return the circuit on \(n\) vertices.
`Circulant()`	Return a circulant digraph on \(n\) vertices from a set of integers.
`Complete()`	Return a complete digraph on \(n\) vertices.
`DeBruijn()`	Return the De Bruijn digraph with parameters \(k,n\).
`GeneralizedDeBruijn()`	Return the generalized de Bruijn digraph of order \(n\) and degree \(d\).
`ImaseItoh()`	Return the digraph of Imase and Itoh of order \(n\) and degree \(d\).
`Kautz()`	Return the Kautz digraph of degree \(d\) and diameter \(D\).
`nauty_directg()`	Return an iterator yielding digraphs using nauty’s `directg` program.
`nauty_posetg()`	Return an iterator yielding Hasse diagrams of posets using nauty’s `genposetg` program.
`Paley()`	Return a Paley digraph on \(q\) vertices.
`Path()`	Return a directed path on \(n\) vertices.
`RandomDirectedAcyclicGraph()`	Return a random (weighted) directed acyclic graph of order \(n\).
`RandomDirectedGNC()`	Return a random growing network with copying (GNC) digraph with \(n\) vertices.
`RandomDirectedGNM()`	Return a random labelled digraph on \(n\) nodes and \(m\) arcs.
`RandomDirectedGNP()`	Return a random digraph on \(n\) nodes.
`RandomDirectedGN()`	Return a random growing network (GN) digraph with \(n\) vertices.
`RandomDirectedGNR()`	Return a random growing network with redirection (GNR) digraph.
`RandomSemiComplete()`	Return a random semi-complete digraph of order \(n\).
`RandomTournament()`	Return a random tournament on \(n\) vertices.
`TransitiveTournament()`	Return a transitive tournament on \(n\) vertices.
`tournaments_nauty()`	Iterator over all tournaments on \(n\) vertices using Nauty.

AUTHORS:

Robert L. Miller (2006)
Emily A. Kirkman (2006)
Michael C. Yurko (2009)
David Coudert (2012)
Janmenjaya Panda (2024)

Functions and methods¶

class sage.graphs.digraph_generators.DiGraphGenerators[source]¶

Bases: object

A class consisting of constructors for several common digraphs, including orderly generation of isomorphism class representatives.

A list of all graphs and graph structures in this database is available via tab completion. Type digraphs. and then hit Tab to see which digraphs are available.

The docstrings include educational information about each named digraph with the hopes that this class can be used as a reference.

The constructors currently in this class include:

Random Directed Graphs:
    - RandomDirectedAcyclicGraph
    - RandomDirectedGN
    - RandomDirectedGNC
    - RandomDirectedGNP
    - RandomDirectedGNM
    - RandomDirectedGNR
    - RandomTournament
    - RandomSemiComplete

Families of Graphs:
    - Complete
    - DeBruijn
    - GeneralizedDeBruijn
    - Kautz
    - Path
    - ImaseItoh
    - RandomTournament
    - TransitiveTournament
    - tournaments_nauty

ORDERLY GENERATION: digraphs(vertices, property=lambda x: True, augment=’edges’, size=None)

Accesses the generator of isomorphism class representatives [McK1998]. Iterates over distinct, exhaustive representatives.

INPUT:

vertices – natural number or None to infinitely generate bigger and bigger digraphs
property – any property to be tested on digraphs before generation
augment – choices:
- 'vertices' – augments by adding a vertex, and edges incident to that vertex. In this case, all digraphs on up to n=vertices are generated. If for any digraph G satisfying the property, every subgraph, obtained from G by deleting one vertex and only edges incident to that vertex, satisfies the property, then this will generate all digraphs with that property. If this does not hold, then all the digraphs generated will satisfy the property, but there will be some missing.
- 'edges' – augments a fixed number of vertices by adding one edge. In this case, all digraphs on exactly n=vertices are generated. If for any graph G satisfying the property, every subgraph, obtained from G by deleting one edge but not the vertices incident to that edge, satisfies the property, then this will generate all digraphs with that property. If this does not hold, then all the digraphs generated will satisfy the property, but there will be some missing.
implementation – which underlying implementation to use (see DiGraph?)
sparse – boolean (default: True); whether to use a sparse or dense data structure. See the documentation of Graph.

EXAMPLES:

Print digraphs on 2 or less vertices:

Sage

sage: for D in digraphs(2, augment='vertices'):
....:     print(D)
Digraph on 0 vertices
Digraph on 1 vertex
Digraph on 2 vertices
Digraph on 2 vertices
Digraph on 2 vertices

Python

>>> from sage.all import *
>>> for D in digraphs(Integer(2), augment='vertices'):
...     print(D)
Digraph on 0 vertices
Digraph on 1 vertex
Digraph on 2 vertices
Digraph on 2 vertices
Digraph on 2 vertices

Print digraphs on 3 vertices:

Sage

sage: for D in digraphs(3):
....:     print(D)
Digraph on 3 vertices
Digraph on 3 vertices
...
Digraph on 3 vertices
Digraph on 3 vertices

Python

>>> from sage.all import *
>>> for D in digraphs(Integer(3)):
...     print(D)
Digraph on 3 vertices
Digraph on 3 vertices
...
Digraph on 3 vertices
Digraph on 3 vertices

Generate all digraphs with 4 vertices and 3 edges:

Sage

sage: L = digraphs(4, size=3)
sage: len(list(L))
13

Python

>>> from sage.all import *
>>> L = digraphs(Integer(4), size=Integer(3))
>>> len(list(L))
13

Generate all digraphs with 4 vertices and up to 3 edges:

Sage

sage: L = list(digraphs(4, lambda G: G.size() <= 3))
sage: len(L)
20
sage: graphs_list.show_graphs(L)        # long time                             # needs sage.plot

Python

>>> from sage.all import *
>>> L = list(digraphs(Integer(4), lambda G: G.size() <= Integer(3)))
>>> len(L)
20
>>> graphs_list.show_graphs(L)        # long time                             # needs sage.plot

Generate all digraphs with degree at most 2, up to 5 vertices:

Sage

sage: property = lambda G: (max([G.degree(v) for v in G] + [0]) <= 2)
sage: L = list(digraphs(5, property, augment='vertices'))
sage: len(L)
75

Python

>>> from sage.all import *
>>> property = lambda G: (max([G.degree(v) for v in G] + [Integer(0)]) <= Integer(2))
>>> L = list(digraphs(Integer(5), property, augment='vertices'))
>>> len(L)
75

Generate digraphs on the fly (see http://oeis.org/classic/A000273):

Sage

sage: for i in range(5):
....:     print(len(list(digraphs(i))))
1
1
3
16
218

Python

>>> from sage.all import *
>>> for i in range(Integer(5)):
...     print(len(list(digraphs(i))))
1
1
3
16
218

ButterflyGraph(n, vertices='strings')[source]¶

Return a \(n\)-dimensional butterfly graph.

The vertices consist of pairs \((v, i)\), where \(v\) is an \(n\)-dimensional tuple (vector) with binary entries (or a string representation of such) and \(i\) is an integer in \([0..n]\). A directed edge goes from \((v, i)\) to \((w, i + 1)\) if \(v\) and \(w\) are identical except for possibly when \(v[i] \neq w[i]\).

A butterfly graph has \((2^n)(n+1)\) vertices and \(n2^{n+1}\) edges.

INPUT:

n – nonnegative integer; the dimension of the butterfly graph
vertices – string (default: 'strings'); specifies whether the vertices are zero-one strings (default) or tuples over GF(2) (vertices='vectors')

EXAMPLES:

Sage

sage: digraphs.ButterflyGraph(2).edges(sort=True, labels=False)
[(('00', 0), ('00', 1)),
 (('00', 0), ('10', 1)),
 (('00', 1), ('00', 2)),
 (('00', 1), ('01', 2)),
 (('01', 0), ('01', 1)),
 (('01', 0), ('11', 1)),
 (('01', 1), ('00', 2)),
 (('01', 1), ('01', 2)),
 (('10', 0), ('00', 1)),
 (('10', 0), ('10', 1)),
 (('10', 1), ('10', 2)),
 (('10', 1), ('11', 2)),
 (('11', 0), ('01', 1)),
 (('11', 0), ('11', 1)),
 (('11', 1), ('10', 2)),
 (('11', 1), ('11', 2))]
sage: digraphs.ButterflyGraph(2, vertices='vectors').edges(sort=True,       # needs sage.modules sage.rings.finite_rings
....:                                                      labels=False)
[(((0, 0), 0), ((0, 0), 1)),
 (((0, 0), 0), ((1, 0), 1)),
 (((0, 0), 1), ((0, 0), 2)),
 (((0, 0), 1), ((0, 1), 2)),
 (((0, 1), 0), ((0, 1), 1)),
 (((0, 1), 0), ((1, 1), 1)),
 (((0, 1), 1), ((0, 0), 2)),
 (((0, 1), 1), ((0, 1), 2)),
 (((1, 0), 0), ((0, 0), 1)),
 (((1, 0), 0), ((1, 0), 1)),
 (((1, 0), 1), ((1, 0), 2)),
 (((1, 0), 1), ((1, 1), 2)),
 (((1, 1), 0), ((0, 1), 1)),
 (((1, 1), 0), ((1, 1), 1)),
 (((1, 1), 1), ((1, 0), 2)),
 (((1, 1), 1), ((1, 1), 2))]
sage: pos = digraphs.ButterflyGraph(2).get_pos()
sage: pos['11', 0]
(0, 0)

Python

>>> from sage.all import *
>>> digraphs.ButterflyGraph(Integer(2)).edges(sort=True, labels=False)
[(('00', 0), ('00', 1)),
 (('00', 0), ('10', 1)),
 (('00', 1), ('00', 2)),
 (('00', 1), ('01', 2)),
 (('01', 0), ('01', 1)),
 (('01', 0), ('11', 1)),
 (('01', 1), ('00', 2)),
 (('01', 1), ('01', 2)),
 (('10', 0), ('00', 1)),
 (('10', 0), ('10', 1)),
 (('10', 1), ('10', 2)),
 (('10', 1), ('11', 2)),
 (('11', 0), ('01', 1)),
 (('11', 0), ('11', 1)),
 (('11', 1), ('10', 2)),
 (('11', 1), ('11', 2))]
>>> digraphs.ButterflyGraph(Integer(2), vertices='vectors').edges(sort=True,       # needs sage.modules sage.rings.finite_rings
...                                                      labels=False)
[(((0, 0), 0), ((0, 0), 1)),
 (((0, 0), 0), ((1, 0), 1)),
 (((0, 0), 1), ((0, 0), 2)),
 (((0, 0), 1), ((0, 1), 2)),
 (((0, 1), 0), ((0, 1), 1)),
 (((0, 1), 0), ((1, 1), 1)),
 (((0, 1), 1), ((0, 0), 2)),
 (((0, 1), 1), ((0, 1), 2)),
 (((1, 0), 0), ((0, 0), 1)),
 (((1, 0), 0), ((1, 0), 1)),
 (((1, 0), 1), ((1, 0), 2)),
 (((1, 0), 1), ((1, 1), 2)),
 (((1, 1), 0), ((0, 1), 1)),
 (((1, 1), 0), ((1, 1), 1)),
 (((1, 1), 1), ((1, 0), 2)),
 (((1, 1), 1), ((1, 1), 2))]
>>> pos = digraphs.ButterflyGraph(Integer(2)).get_pos()
>>> pos['11', Integer(0)]
(0, 0)

Circuit(n)[source]¶

Return the circuit on \(n\) vertices.

The circuit is an oriented CycleGraph().

EXAMPLES:

A circuit is the smallest strongly connected digraph:

Sage

sage: circuit = digraphs.Circuit(15)
sage: len(circuit.strongly_connected_components()) == 1
True

Python

>>> from sage.all import *
>>> circuit = digraphs.Circuit(Integer(15))
>>> len(circuit.strongly_connected_components()) == Integer(1)
True

Circulant(n, integers)[source]¶

Return a circulant digraph on \(n\) vertices from a set of integers.

A circulant digraph of order \(n\) has an arc from vertex \(i\) to vertex \(i+j \pmod{n}\), for each \(j\) in integers.

INPUT:

n – integer; number of vertices
integers – iterable container (list, set, etc.) of integers such that there is an edge from \(i\) to \(j\) if and only if \((j-i) \pmod{n}\) is an integer

EXAMPLES:

Construct and show the circulant graph [3, 5, 7], a digraph on 13 vertices:

Sage

sage: g = digraphs.Circulant(13, [3, 5, 7])
sage: g.show()                          # long time                             # needs sage.plot

Python

>>> from sage.all import *
>>> g = digraphs.Circulant(Integer(13), [Integer(3), Integer(5), Integer(7)])
>>> g.show()                          # long time                             # needs sage.plot

The Koh-Tindell digraph [LM2024] is the circulant digraph of order 7 with parameters \([1, 5]\). This \(2\)-diregular digraph is vertex-transitive but not arc-transitive. The associated bipartite digraph of the Koh-Tindell digraph is a Pfaffian orientation of the Heawood graph. Construct and show the Koh-Tindell digraph:

Sage

sage: kohTindellDigraph = digraphs.Circulant(7, [1, 5])
sage: kohTindellDigraph.show()          # long time                             # needs sage.plot

Python

>>> from sage.all import *
>>> kohTindellDigraph = digraphs.Circulant(Integer(7), [Integer(1), Integer(5)])
>>> kohTindellDigraph.show()          # long time                             # needs sage.plot

Complete(n, loops=False)[source]¶

Return the complete digraph on \(n\) vertices.

INPUT:

n – integer; number of vertices
loops – boolean (default: False); whether to add loops or not, i.e., edges from \(u\) to itself

See also

RandomSemiComplete()
RandomTournament()

EXAMPLES:

Sage

sage: n = 10
sage: G = digraphs.Complete(n); G
Complete digraph: Digraph on 10 vertices
sage: G.size() == n*(n-1)
True
sage: G = digraphs.Complete(n, loops=True); G
Complete digraph with loops: Looped digraph on 10 vertices
sage: G.size() == n*n
True
sage: digraphs.Complete(-1)
Traceback (most recent call last):
...
ValueError: the number of vertices cannot be strictly negative

Python

>>> from sage.all import *
>>> n = Integer(10)
>>> G = digraphs.Complete(n); G
Complete digraph: Digraph on 10 vertices
>>> G.size() == n*(n-Integer(1))
True
>>> G = digraphs.Complete(n, loops=True); G
Complete digraph with loops: Looped digraph on 10 vertices
>>> G.size() == n*n
True
>>> digraphs.Complete(-Integer(1))
Traceback (most recent call last):
...
ValueError: the number of vertices cannot be strictly negative

DeBruijn(k, n, vertices='strings')[source]¶

Return the De Bruijn digraph with parameters \(k,n\).

The De Bruijn digraph with parameters \(k,n\) is built upon a set of vertices equal to the set of words of length \(n\) from a dictionary of \(k\) letters.

In this digraph, there is an arc \(w_1w_2\) if \(w_2\) can be obtained from \(w_1\) by removing the leftmost letter and adding a new letter at its right end. For more information, see the Wikipedia article De_Bruijn_graph.

INPUT:

k – two possibilities for this parameter :
- An integer equal to the cardinality of the alphabet to use, that is, the degree of the digraph to be produced.
- An iterable object to be used as the set of letters. The degree of the resulting digraph is the cardinality of the set of letters.
n – integer; length of words in the De Bruijn digraph when vertices == 'strings', and also the diameter of the digraph
vertices – string (default: 'strings'); whether the vertices are words over an alphabet (default) or integers (vertices='string')

EXAMPLES:

de Bruijn digraph of degree 2 and diameter 2:

Sage

sage: db = digraphs.DeBruijn(2, 2); db                                      # needs sage.combinat
De Bruijn digraph (k=2, n=2): Looped digraph on 4 vertices
sage: db.order(), db.size()                                                 # needs sage.combinat
(4, 8)
sage: db.diameter()                                                         # needs sage.combinat
2

Python

>>> from sage.all import *
>>> db = digraphs.DeBruijn(Integer(2), Integer(2)); db                                      # needs sage.combinat
De Bruijn digraph (k=2, n=2): Looped digraph on 4 vertices
>>> db.order(), db.size()                                                 # needs sage.combinat
(4, 8)
>>> db.diameter()                                                         # needs sage.combinat
2

Building a de Bruijn digraph on a different alphabet:

Sage

sage: # needs sage.combinat
sage: g = digraphs.DeBruijn(['a', 'b'], 2)
sage: g.vertices(sort=True)
['aa', 'ab', 'ba', 'bb']
sage: g.is_isomorphic(db)
True
sage: g = digraphs.DeBruijn(['AA', 'BB'], 2)
sage: g.vertices(sort=True)
['AA,AA', 'AA,BB', 'BB,AA', 'BB,BB']
sage: g.is_isomorphic(db)
True

Python

>>> from sage.all import *
>>> # needs sage.combinat
>>> g = digraphs.DeBruijn(['a', 'b'], Integer(2))
>>> g.vertices(sort=True)
['aa', 'ab', 'ba', 'bb']
>>> g.is_isomorphic(db)
True
>>> g = digraphs.DeBruijn(['AA', 'BB'], Integer(2))
>>> g.vertices(sort=True)
['AA,AA', 'AA,BB', 'BB,AA', 'BB,BB']
>>> g.is_isomorphic(db)
True

GeneralizedDeBruijn(n, d)[source]¶

Return the generalized de Bruijn digraph of order \(n\) and degree \(d\).

The generalized de Bruijn digraph was defined in [RPK1980] [RPK1983]. It has vertex set \(V=\{0, 1,..., n-1\}\) and there is an arc from vertex \(u \in V\) to all vertices \(v \in V\) such that \(v \equiv (u*d + a) \mod{n}\) with \(0 \leq a < d\).

When \(n = d^{D}\), the generalized de Bruijn digraph is isomorphic to the de Bruijn digraph of degree \(d\) and diameter \(D\).

INPUT:

n – integer; number of vertices of the digraph (must be at least one)
d – integer; degree of the digraph (must be at least one)

See also

sage.graphs.generic_graph.GenericGraph.is_circulant() – checks whether a (di)graph is circulant, and/or returns all possible sets of parameters.

EXAMPLES:

Sage

sage: GB = digraphs.GeneralizedDeBruijn(8, 2)
sage: GB.is_isomorphic(digraphs.DeBruijn(2, 3), certificate=True)           # needs sage.combinat
(True, {0: '000', 1: '001', 2: '010', 3: '011',
        4: '100', 5: '101', 6: '110', 7: '111'})

Python

>>> from sage.all import *
>>> GB = digraphs.GeneralizedDeBruijn(Integer(8), Integer(2))
>>> GB.is_isomorphic(digraphs.DeBruijn(Integer(2), Integer(3)), certificate=True)           # needs sage.combinat
(True, {0: '000', 1: '001', 2: '010', 3: '011',
        4: '100', 5: '101', 6: '110', 7: '111'})

ImaseItoh(n, d)[source]¶

Return the Imase-Itoh digraph of order \(n\) and degree \(d\).

The Imase-Itoh digraph was defined in [II1983]. It has vertex set \(V=\{0, 1,..., n-1\}\) and there is an arc from vertex \(u \in V\) to all vertices \(v \in V\) such that \(v \equiv (-u*d-a-1) \mod{n}\) with \(0 \leq a < d\).

When \(n = d^{D}\), the Imase-Itoh digraph is isomorphic to the de Bruijn digraph of degree \(d\) and diameter \(D\). When \(n = d^{D-1}(d+1)\), the Imase-Itoh digraph is isomorphic to the Kautz digraph [Kau1968] of degree \(d\) and diameter \(D\).

INPUT:

n – integer; number of vertices of the digraph (must be greater than or equal to two)
d – integer; degree of the digraph (must be greater than or equal to one)

EXAMPLES:

Sage

sage: II = digraphs.ImaseItoh(8, 2)
sage: II.is_isomorphic(digraphs.DeBruijn(2, 3), certificate=True)           # needs sage.combinat
(True, {0: '010', 1: '011', 2: '000', 3: '001',
        4: '110', 5: '111', 6: '100', 7: '101'})

sage: II = digraphs.ImaseItoh(12, 2)
sage: b,D = II.is_isomorphic(digraphs.Kautz(2, 3), certificate=True)        # needs sage.combinat
sage: b                                                                     # needs sage.combinat
True
sage: D   # random isomorphism                                              # needs sage.combinat
{0: '202', 1: '201', 2: '210', 3: '212', 4: '121',
 5: '120', 6: '102', 7: '101', 8: '010', 9: '012',
 10: '021', 11: '020'}

Python

>>> from sage.all import *
>>> II = digraphs.ImaseItoh(Integer(8), Integer(2))
>>> II.is_isomorphic(digraphs.DeBruijn(Integer(2), Integer(3)), certificate=True)           # needs sage.combinat
(True, {0: '010', 1: '011', 2: '000', 3: '001',
        4: '110', 5: '111', 6: '100', 7: '101'})

>>> II = digraphs.ImaseItoh(Integer(12), Integer(2))
>>> b,D = II.is_isomorphic(digraphs.Kautz(Integer(2), Integer(3)), certificate=True)        # needs sage.combinat
>>> b                                                                     # needs sage.combinat
True
>>> D   # random isomorphism                                              # needs sage.combinat
{0: '202', 1: '201', 2: '210', 3: '212', 4: '121',
 5: '120', 6: '102', 7: '101', 8: '010', 9: '012',
 10: '021', 11: '020'}

Kautz(k, D, vertices='strings')[source]¶

Return the Kautz digraph of degree \(d\) and diameter \(D\).

The Kautz digraph has been defined in [Kau1968]. The Kautz digraph of degree \(d\) and diameter \(D\) has \(d^{D-1}(d+1)\) vertices. This digraph is built from a set of vertices equal to the set of words of length \(D\) over an alphabet of \(d+1\) letters such that consecutive letters are different. There is an arc from vertex \(u\) to vertex \(v\) if \(v\) can be obtained from \(u\) by removing the leftmost letter and adding a new letter, distinct from the rightmost letter of \(u\), at the right end.

The Kautz digraph of degree \(d\) and diameter \(D\) is isomorphic to the Imase-Itoh digraph [II1983] of degree \(d\) and order \(d^{D-1}(d+1)\).

See the Wikipedia article Kautz_graph for more information.

INPUT:

k – two possibilities for this parameter. In either case the degree must be at least one:
- An integer equal to the degree of the digraph to be produced, that is, the cardinality of the alphabet to be used minus one.
- An iterable object to be used as the set of letters. The degree of the resulting digraph is the cardinality of the set of letters minus one.
D – integer; diameter of the digraph, and length of a vertex label when vertices == 'strings' (must be at least one)
vertices – string (default: 'strings'); whether the vertices are words over an alphabet (default) or integers (vertices='strings')

EXAMPLES:

Sage

sage: # needs sage.combinat
sage: K = digraphs.Kautz(2, 3)
sage: b, D = K.is_isomorphic(digraphs.ImaseItoh(12, 2), certificate=True)
sage: b
True
sage: D  # random isomorphism
{'010': 8, '012': 9, '020': 11, '021': 10, '101': 7,  '102': 6,
 '120': 5, '121': 4, '201': 1, '202': 0, '210': 2, '212': 3}

sage: K = digraphs.Kautz([1,'a','B'], 2)                                    # needs sage.combinat
sage: K.edges(sort=True)                                                    # needs sage.combinat
[('1B', 'B1', '1'), ('1B', 'Ba', 'a'), ('1a', 'a1', '1'),
 ('1a', 'aB', 'B'), ('B1', '1B', 'B'), ('B1', '1a', 'a'),
 ('Ba', 'a1', '1'), ('Ba', 'aB', 'B'), ('a1', '1B', 'B'),
 ('a1', '1a', 'a'), ('aB', 'B1', '1'), ('aB', 'Ba', 'a')]

sage: K = digraphs.Kautz([1,'aA','BB'], 2)                                  # needs sage.combinat
sage: K.edges(sort=True)                                                    # needs sage.combinat
[('1,BB', 'BB,1', '1'), ('1,BB', 'BB,aA', 'aA'),
 ('1,aA', 'aA,1', '1'), ('1,aA', 'aA,BB', 'BB'),
 ('BB,1', '1,BB', 'BB'), ('BB,1', '1,aA', 'aA'),
 ('BB,aA', 'aA,1', '1'), ('BB,aA', 'aA,BB', 'BB'),
 ('aA,1', '1,BB', 'BB'), ('aA,1', '1,aA', 'aA'),
 ('aA,BB', 'BB,1', '1'), ('aA,BB', 'BB,aA', 'aA')]

Python

>>> from sage.all import *
>>> # needs sage.combinat
>>> K = digraphs.Kautz(Integer(2), Integer(3))
>>> b, D = K.is_isomorphic(digraphs.ImaseItoh(Integer(12), Integer(2)), certificate=True)
>>> b
True
>>> D  # random isomorphism
{'010': 8, '012': 9, '020': 11, '021': 10, '101': 7,  '102': 6,
 '120': 5, '121': 4, '201': 1, '202': 0, '210': 2, '212': 3}

>>> K = digraphs.Kautz([Integer(1),'a','B'], Integer(2))                                    # needs sage.combinat
>>> K.edges(sort=True)                                                    # needs sage.combinat
[('1B', 'B1', '1'), ('1B', 'Ba', 'a'), ('1a', 'a1', '1'),
 ('1a', 'aB', 'B'), ('B1', '1B', 'B'), ('B1', '1a', 'a'),
 ('Ba', 'a1', '1'), ('Ba', 'aB', 'B'), ('a1', '1B', 'B'),
 ('a1', '1a', 'a'), ('aB', 'B1', '1'), ('aB', 'Ba', 'a')]

>>> K = digraphs.Kautz([Integer(1),'aA','BB'], Integer(2))                                  # needs sage.combinat
>>> K.edges(sort=True)                                                    # needs sage.combinat
[('1,BB', 'BB,1', '1'), ('1,BB', 'BB,aA', 'aA'),
 ('1,aA', 'aA,1', '1'), ('1,aA', 'aA,BB', 'BB'),
 ('BB,1', '1,BB', 'BB'), ('BB,1', '1,aA', 'aA'),
 ('BB,aA', 'aA,1', '1'), ('BB,aA', 'aA,BB', 'BB'),
 ('aA,1', '1,BB', 'BB'), ('aA,1', '1,aA', 'aA'),
 ('aA,BB', 'BB,1', '1'), ('aA,BB', 'BB,aA', 'aA')]

Paley(q)[source]¶

Return a Paley digraph on \(q\) vertices.

Parameter \(q\) must be the power of a prime number and congruent to 3 mod 4.

See also

EXAMPLES:

A Paley digraph has \(n * (n-1) / 2\) edges, its underlying graph is a clique, and so it is a tournament:

Sage

sage: g = digraphs.Paley(7); g                                              # needs sage.rings.finite_rings
Paley digraph with parameter 7: Digraph on 7 vertices
sage: g.size() == g.order() * (g.order() - 1) / 2                           # needs sage.rings.finite_rings
True
sage: g.to_undirected().is_clique()                                         # needs sage.rings.finite_rings
True

Python

>>> from sage.all import *
>>> g = digraphs.Paley(Integer(7)); g                                              # needs sage.rings.finite_rings
Paley digraph with parameter 7: Digraph on 7 vertices
>>> g.size() == g.order() * (g.order() - Integer(1)) / Integer(2)                           # needs sage.rings.finite_rings
True
>>> g.to_undirected().is_clique()                                         # needs sage.rings.finite_rings
True

A Paley digraph is always self-complementary:

Sage

sage: g.complement().is_isomorphic(g)                                       # needs sage.rings.finite_rings
True

Python

>>> from sage.all import *
>>> g.complement().is_isomorphic(g)                                       # needs sage.rings.finite_rings
True

Path(n)[source]¶

Return a directed path on \(n\) vertices.

INPUT:

n – integer; number of vertices in the path

EXAMPLES:

Sage

sage: g = digraphs.Path(5)
sage: g.vertices(sort=True)
[0, 1, 2, 3, 4]
sage: g.size()
4
sage: g.automorphism_group().cardinality()                                  # needs sage.groups
1

Python

>>> from sage.all import *
>>> g = digraphs.Path(Integer(5))
>>> g.vertices(sort=True)
[0, 1, 2, 3, 4]
>>> g.size()
4
>>> g.automorphism_group().cardinality()                                  # needs sage.groups
1

RandomDirectedAcyclicGraph(n, p, weight_max=None)[source]¶

Return a random (weighted) directed acyclic graph of order \(n\).

The method starts with the sink vertex and adds vertices one at a time. A vertex is connected only to previously defined vertices, and the probability of each possible connection is given by the probability \(p\). The weight of an edge is a random integer between 1 and weight_max.

INPUT:

n – number of nodes of the graph
p – probability of an edge
weight_max – (default: None) by default, the returned DAG is unweighted. When weight_max is set to a positive integer, edges are assigned a random integer weight between 1 and weight_max.

EXAMPLES:

Sage

sage: D = digraphs.RandomDirectedAcyclicGraph(5, .5); D
RandomDAG(5, 0.500000000000000): Digraph on 5 vertices
sage: D.is_directed_acyclic()
True
sage: D = digraphs.RandomDirectedAcyclicGraph(5, .5, weight_max=3); D
RandomWeightedDAG(5, 0.500000000000000, 3): Digraph on 5 vertices
sage: D.is_directed_acyclic()
True

Python

>>> from sage.all import *
>>> D = digraphs.RandomDirectedAcyclicGraph(Integer(5), RealNumber('.5')); D
RandomDAG(5, 0.500000000000000): Digraph on 5 vertices
>>> D.is_directed_acyclic()
True
>>> D = digraphs.RandomDirectedAcyclicGraph(Integer(5), RealNumber('.5'), weight_max=Integer(3)); D
RandomWeightedDAG(5, 0.500000000000000, 3): Digraph on 5 vertices
>>> D.is_directed_acyclic()
True

RandomDirectedGN(n, kernel=None, seed=None)[source]¶

Return a random growing network (GN) digraph with \(n\) vertices.

The digraph is constructed by adding vertices with a link to one previously added vertex. The vertex to link to is chosen with a preferential attachment model, i.e. probability is proportional to degree. The default attachment kernel is a linear function of degree. The digraph is always a tree, so in particular it is a directed acyclic graph. See [KR2001b] for more details.

INPUT:

n – integer; number of vertices
kernel – the attachment kernel (default: identity function)
seed – a random.Random seed or a Python int for the random number generator (default: None)

EXAMPLES:

Sage

sage: # needs networkx
sage: D = digraphs.RandomDirectedGN(25)
sage: D.num_verts()
25
sage: D.num_edges()
24
sage: D.is_connected()
True
sage: D.parent() is DiGraph
True
sage: D.show()                      # long time

Python

>>> from sage.all import *
>>> # needs networkx
>>> D = digraphs.RandomDirectedGN(Integer(25))
>>> D.num_verts()
25
>>> D.num_edges()
24
>>> D.is_connected()
True
>>> D.parent() is DiGraph
True
>>> D.show()                      # long time

RandomDirectedGNC(n, seed=None)[source]¶

Return a random growing network with copying (GNC) digraph with \(n\) vertices.

The digraph is constructed by adding vertices with a link to one previously added vertex. The vertex to link to is chosen with a preferential attachment model, i.e. probability is proportional to degree. The new vertex is also linked to all of the previously added vertex’s successors. See [KR2005] for more details.

INPUT:

n – integer; number of vertices
seed – a random.Random seed or a Python int for the random number generator (default: None)

EXAMPLES:

Sage

sage: # needs networkx
sage: D = digraphs.RandomDirectedGNC(25)
sage: D.is_directed_acyclic()
True
sage: D.topological_sort()
[24, 23, ..., 1, 0]
sage: D.show()                      # long time

Python

>>> from sage.all import *
>>> # needs networkx
>>> D = digraphs.RandomDirectedGNC(Integer(25))
>>> D.is_directed_acyclic()
True
>>> D.topological_sort()
[24, 23, ..., 1, 0]
>>> D.show()                      # long time

RandomDirectedGNM(n, m, loops=False)[source]¶

Return a random labelled digraph on \(n\) nodes and \(m\) arcs.

INPUT:

n – integer; number of vertices
m – integer; number of edges
loops – boolean (default: False); whether to allow loops

PLOTTING: When plotting, this graph will use the default spring-layout algorithm, unless a position dictionary is specified.

EXAMPLES:

Sage

sage: D = digraphs.RandomDirectedGNM(10, 5)
sage: D.num_verts()
10
sage: D.num_edges()
5

Python

>>> from sage.all import *
>>> D = digraphs.RandomDirectedGNM(Integer(10), Integer(5))
>>> D.num_verts()
10
>>> D.num_edges()
5

With loops:

Sage

sage: D = digraphs.RandomDirectedGNM(10, 100, loops = True)
sage: D.num_verts()
10
sage: D.loops()
[(0, 0, None), (1, 1, None), (2, 2, None), (3, 3, None),
 (4, 4, None), (5, 5, None), (6, 6, None), (7, 7, None),
 (8, 8, None), (9, 9, None)]

Python

>>> from sage.all import *
>>> D = digraphs.RandomDirectedGNM(Integer(10), Integer(100), loops = True)
>>> D.num_verts()
10
>>> D.loops()
[(0, 0, None), (1, 1, None), (2, 2, None), (3, 3, None),
 (4, 4, None), (5, 5, None), (6, 6, None), (7, 7, None),
 (8, 8, None), (9, 9, None)]

RandomDirectedGNP(n, p, loops=False, seed=None)[source]¶

Return a random digraph on \(n\) nodes.

Each edge is inserted independently with probability \(p\). See [ER1959] and [Gil1959] for more details.

INPUT:

n – integer; number of nodes of the digraph
p – float; probability of an edge
loops – boolean (default: False); whether the random digraph may have loops
seed – integer (default: None); seed for random number generator

PLOTTING: When plotting, this graph will use the default spring-layout algorithm, unless a position dictionary is specified.

EXAMPLES:

Sage

sage: D = digraphs.RandomDirectedGNP(10, .2)
sage: D.num_verts()
10
sage: D.parent() is DiGraph
True

Python

>>> from sage.all import *
>>> D = digraphs.RandomDirectedGNP(Integer(10), RealNumber('.2'))
>>> D.num_verts()
10
>>> D.parent() is DiGraph
True

RandomDirectedGNR(n, p, seed=None)[source]¶

Return a random growing network with redirection (GNR) digraph with \(n\) vertices and redirection probability \(p\).

The digraph is constructed by adding vertices with a link to one previously added vertex. The vertex to link to is chosen uniformly. With probability p, the arc is instead redirected to the successor vertex. The digraph is always a tree. See [KR2001b] for more details.

INPUT:

n – integer; number of vertices
p – redirection probability
seed – a random.Random seed or a Python int for the random number generator (default: None)

EXAMPLES:

Sage

sage: # needs networkx
sage: D = digraphs.RandomDirectedGNR(25, .2)
sage: D.is_directed_acyclic()
True
sage: D.to_undirected().is_tree()
True
sage: D.show()                      # long time                             # needs sage.plot

Python

>>> from sage.all import *
>>> # needs networkx
>>> D = digraphs.RandomDirectedGNR(Integer(25), RealNumber('.2'))
>>> D.is_directed_acyclic()
True
>>> D.to_undirected().is_tree()
True
>>> D.show()                      # long time                             # needs sage.plot

RandomSemiComplete(n)[source]¶

Return a random semi-complete digraph on \(n\) vertices.

A directed graph \(G=(V,E)\) is semi-complete if for any pair of vertices \(u\) and \(v\), there is at least one arc between them.

To generate randomly a semi-complete digraph, we have to ensure, for any pair of distinct vertices \(u\) and \(v\), that with probability \(1/3\) we have only arc \(uv\), with probability \(1/3\) we have only arc \(vu\), and with probability \(1/3\) we have both arc \(uv\) and arc \(vu\). We do so by selecting a random integer \(coin\) in \([1,3]\). When \(coin==1\) we select only arc \(uv\), when \(coin==3\) we select only arc \(vu\), and when \(coin==2\) we select both arcs. In other words, we select arc \(uv\) when \(coin\leq 2\) and arc \(vu\) when \(coin\geq 2\).

INPUT:

n – integer; the number of nodes

See also

Complete()
RandomTournament()

EXAMPLES:

Sage

sage: SC = digraphs.RandomSemiComplete(10); SC
Random Semi-Complete digraph: Digraph on 10 vertices
sage: SC.size() >= binomial(10, 2)                                          # needs sage.symbolic
True
sage: digraphs.RandomSemiComplete(-1)
Traceback (most recent call last):
...
ValueError: the number of vertices cannot be strictly negative

Python

>>> from sage.all import *
>>> SC = digraphs.RandomSemiComplete(Integer(10)); SC
Random Semi-Complete digraph: Digraph on 10 vertices
>>> SC.size() >= binomial(Integer(10), Integer(2))                                          # needs sage.symbolic
True
>>> digraphs.RandomSemiComplete(-Integer(1))
Traceback (most recent call last):
...
ValueError: the number of vertices cannot be strictly negative

RandomTournament(n)[source]¶

Return a random tournament on \(n\) vertices.

For every pair of vertices, the tournament has an edge from \(i\) to \(j\) with probability \(1/2\), otherwise it has an edge from \(j\) to \(i\).

INPUT:

n – integer; number of vertices

EXAMPLES:

Sage

sage: T = digraphs.RandomTournament(10); T
Random Tournament: Digraph on 10 vertices
sage: T.size() == binomial(10, 2)                                           # needs sage.symbolic
True
sage: T.is_tournament()
True
sage: digraphs.RandomTournament(-1)
Traceback (most recent call last):
...
ValueError: the number of vertices cannot be strictly negative

Python

>>> from sage.all import *
>>> T = digraphs.RandomTournament(Integer(10)); T
Random Tournament: Digraph on 10 vertices
>>> T.size() == binomial(Integer(10), Integer(2))                                           # needs sage.symbolic
True
>>> T.is_tournament()
True
>>> digraphs.RandomTournament(-Integer(1))
Traceback (most recent call last):
...
ValueError: the number of vertices cannot be strictly negative

See also

StronglyRegular(n)[source]¶

Return a Strongly Regular digraph with \(n\) vertices.

The adjacency matrix of the graph is constructed from a skew Hadamard matrix of order \(n+1\). These graphs were first constructed in [Duv1988].

INPUT:

n – integer; the number of vertices of the digraph

See also

EXAMPLES:

A Strongly Regular digraph satisfies the condition \(AJ = JA = kJ\) where \(A\) is the adjacency matrix:

Sage

sage: # needs sage.combinat sage.modules
sage: g = digraphs.StronglyRegular(7); g
Strongly regular digraph: Digraph on 7 vertices
sage: A = g.adjacency_matrix()*ones_matrix(7)
sage: B = ones_matrix(7)*g.adjacency_matrix()
sage: A == B == A[0, 0]*ones_matrix(7)
True

Python

>>> from sage.all import *
>>> # needs sage.combinat sage.modules
>>> g = digraphs.StronglyRegular(Integer(7)); g
Strongly regular digraph: Digraph on 7 vertices
>>> A = g.adjacency_matrix()*ones_matrix(Integer(7))
>>> B = ones_matrix(Integer(7))*g.adjacency_matrix()
>>> A == B == A[Integer(0), Integer(0)]*ones_matrix(Integer(7))
True

TransitiveTournament(n)[source]¶

Return a transitive tournament on \(n\) vertices.

In this tournament there is an edge from \(i\) to \(j\) if \(i<j\).

See the Wikipedia article Tournament_(graph_theory) for more information.

INPUT:

n – integer; number of vertices in the tournament

EXAMPLES:

Sage

sage: g = digraphs.TransitiveTournament(5)
sage: g.vertices(sort=True)
[0, 1, 2, 3, 4]
sage: g.size()
10
sage: g.automorphism_group().cardinality()                                  # needs sage.groups
1

Python

>>> from sage.all import *
>>> g = digraphs.TransitiveTournament(Integer(5))
>>> g.vertices(sort=True)
[0, 1, 2, 3, 4]
>>> g.size()
10
>>> g.automorphism_group().cardinality()                                  # needs sage.groups
1

See also

nauty_directg(graphs, options='', debug=False)[source]¶

Return an iterator yielding digraphs using nauty’s directg program.

Description from directg --help: Read undirected graphs and orient their edges in all possible ways. Edges can be oriented in either or both directions (3 possibilities). Isomorphic directed graphs derived from the same input are suppressed. If the input graphs are non-isomorphic then the output graphs are also.

INPUT:

graphs – a Graph or an iterable containing Graph. The graph6 string of these graphs is used as an input for directg.

options – string passed to directg as if it was run at a system command line. Available options from directg --help:

-e<int> | -e<int>:<int>  specify a value or range of the total number of arcs
-o       orient each edge in only one direction, never both
-a       only make acyclic orientations (implies -o)
-f<int>  Use only the subgroup that fixes the first <int> vertices setwise
-V       only output graphs with nontrivial groups (including exchange of
         isolated vertices).  The -f option is respected.
-s<int>/<int>  Make only a fraction of the orientations: The first integer is
         the part number (first is 0) and the second is the number of
         parts. Splitting is done per input graph independently.

debug – boolean (default: False); if True directg standard error and standard output are displayed

EXAMPLES:

Sage

sage: gen = graphs.nauty_geng("-c 3")
sage: dgs = list(digraphs.nauty_directg(gen))
sage: len(dgs)
13
sage: dgs[0]
Digraph on 3 vertices
sage: dgs[0]._bit_vector()
'001001000'
sage: len(list(digraphs.nauty_directg(graphs.PetersenGraph(), options='-o')))
324

Python

>>> from sage.all import *
>>> gen = graphs.nauty_geng("-c 3")
>>> dgs = list(digraphs.nauty_directg(gen))
>>> len(dgs)
13
>>> dgs[Integer(0)]
Digraph on 3 vertices
>>> dgs[Integer(0)]._bit_vector()
'001001000'
>>> len(list(digraphs.nauty_directg(graphs.PetersenGraph(), options='-o')))
324

Generate non-isomorphic acyclic orientations:

Sage

sage: K = graphs.CompleteGraph(4)
sage: all(d.is_directed_acyclic() for d in digraphs.nauty_directg(K, options='-a'))
True
sage: sum(1 for _ in digraphs.nauty_directg(K, options='-a'))
1
sage: S = graphs.StarGraph(4)
sage: all(d.is_directed_acyclic() for d in digraphs.nauty_directg(S, options='-a'))
True
sage: sum(1 for _ in digraphs.nauty_directg(S, options='-a'))
5

Python

>>> from sage.all import *
>>> K = graphs.CompleteGraph(Integer(4))
>>> all(d.is_directed_acyclic() for d in digraphs.nauty_directg(K, options='-a'))
True
>>> sum(Integer(1) for _ in digraphs.nauty_directg(K, options='-a'))
1
>>> S = graphs.StarGraph(Integer(4))
>>> all(d.is_directed_acyclic() for d in digraphs.nauty_directg(S, options='-a'))
True
>>> sum(Integer(1) for _ in digraphs.nauty_directg(S, options='-a'))
5

See also

nauty_posetg(options='', debug=False)[source]¶

Return a generator which creates all posets using nauty.

Here a poset is seen through its Hasse diagram, which is an acyclic and transitively reduced digraph.

INPUT:

options – string (default: ""); a string passed to genposetg as if it was run at a system command line. At a minimum, you must pass the number of vertices you desire and a choice between o and t for the output order.
debug – boolean (default: False); if True the first line of genposetg’s output to standard error is captured and the first call to the generator’s next() function will return this line as a string. A line leading with “>A” indicates a successful initiation of the program with some information on the arguments, while a line beginning with “>E” indicates an error with the input.

The possible options, obtained as output of genposetg --help:

n: the number of vertices, between 0 and 16
o: digraph6 output in arbitrary order
t: digraph6 output in topological order

EXAMPLES:

Sage

sage: gen = digraphs.nauty_posetg("5 o")
sage: len(list(gen))
63

Python

>>> from sage.all import *
>>> gen = digraphs.nauty_posetg("5 o")
>>> len(list(gen))
63

This coincides with OEIS sequence A000112.

tournaments_nauty(n, min_out_degree=None, max_out_degree=None, strongly_connected=False, debug=False, options='')[source]¶

Iterator over all tournaments on \(n\) vertices using Nauty.

INPUT:

n – integer; number of vertices
min_out_degree, max_out_degree – integers; if set to None (default), then the min/max out-degree is not constrained
debug – boolean (default: False); if True the first line of gentourng’s output to standard error is captured and the first call to the generator’s next() function will return this line as a string. A line leading with “>A” indicates a successful initiation of the program with some information on the arguments, while a line beginning with “>E” indicates an error with the input.
options – string; anything else that should be forwarded as input to Nauty’s gentourng. See its documentation for more information : https://pallini.di.uniroma1.it.

EXAMPLES:

Sage

sage: for g in digraphs.tournaments_nauty(4):
....:     print(g.edges(sort=True, labels = False))
[(1, 0), (2, 0), (2, 1), (3, 0), (3, 1), (3, 2)]
[(1, 0), (1, 3), (2, 0), (2, 1), (3, 0), (3, 2)]
[(0, 2), (1, 0), (2, 1), (3, 0), (3, 1), (3, 2)]
[(0, 2), (0, 3), (1, 0), (2, 1), (3, 1), (3, 2)]
sage: tournaments = digraphs.tournaments_nauty
sage: [len(list(tournaments(x))) for x in range(1,8)]
[1, 1, 2, 4, 12, 56, 456]
sage: [len(list(tournaments(x, strongly_connected = True))) for x in range(1,9)]
[1, 0, 1, 1, 6, 35, 353, 6008]

Python

>>> from sage.all import *
>>> for g in digraphs.tournaments_nauty(Integer(4)):
...     print(g.edges(sort=True, labels = False))
[(1, 0), (2, 0), (2, 1), (3, 0), (3, 1), (3, 2)]
[(1, 0), (1, 3), (2, 0), (2, 1), (3, 0), (3, 2)]
[(0, 2), (1, 0), (2, 1), (3, 0), (3, 1), (3, 2)]
[(0, 2), (0, 3), (1, 0), (2, 1), (3, 1), (3, 2)]
>>> tournaments = digraphs.tournaments_nauty
>>> [len(list(tournaments(x))) for x in range(Integer(1),Integer(8))]
[1, 1, 2, 4, 12, 56, 456]
>>> [len(list(tournaments(x, strongly_connected = True))) for x in range(Integer(1),Integer(9))]
[1, 0, 1, 1, 6, 35, 353, 6008]