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: g = digraphs.Circuit(15)

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

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

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.
Paley() Return a Paley digraph on \(q\) vertices.
Path() Return a directed path on \(n\) vertices.
RandomDirectedGNC() Returns a random GNC (growing network with copying) digraph with \(n\) vertices.
RandomDirectedGNM() Returns a random labelled digraph on \(n\) nodes and \(m\) arcs.
RandomDirectedGNP() Returns a random digraph on \(n\) nodes.
RandomDirectedGN() Returns a random GN (growing network) digraph with \(n\) vertices.
RandomDirectedGNR() Returns a random GNR (growing network with redirection) 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)

Functions and methods

class sage.graphs.digraph_generators.DiGraphGenerators

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 graphs 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:
    - 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. 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 - ignored if implementation is not c_graph

EXAMPLES: Print digraphs on 2 or less vertices.

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

Note that we can also get digraphs with underlying Cython implementation:

sage: for D in digraphs(2, augment='vertices', implementation='c_graph'):
....:     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: for D in digraphs(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: L = digraphs(4, size=3)
sage: len(list(L))
13

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

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

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

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

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

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

REFERENCE:

  • Brendan D. McKay, Isomorph-Free Exhaustive generation. Journal of Algorithms Volume 26, Issue 2, February 1998, pages 306-324.
ButterflyGraph(n, vertices='strings')

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 – integer;
  • vertices – string (default: 'strings'); specifies whether the vertices are zero-one strings (default) or tuples over GF(2) (vertices='vectors')

EXAMPLES:

sage: digraphs.ButterflyGraph(2).edges(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(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))]
Circuit(n)

Return the circuit on \(n\) vertices.

The circuit is an oriented CycleGraph.

EXAMPLES:

A circuit is the smallest strongly connected digraph:

sage: circuit = digraphs.Circuit(15)
sage: len(circuit.strongly_connected_components()) == 1
True
Circulant(n, integers)

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

EXAMPLES:

sage: digraphs.Circulant(13,[3,5,7])
Circulant graph ([3, 5, 7]): Digraph on 13 vertices
Complete(n, loops=False)

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

EXAMPLES:

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
DeBruijn(k, n, vertices='strings')

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

Building a de Bruijn digraph on a different alphabet:

sage: g = digraphs.DeBruijn(['a', 'b'], 2)
sage: g.vertices()
['aa', 'ab', 'ba', 'bb']
sage: g.is_isomorphic(db)
True
sage: g = digraphs.DeBruijn(['AA', 'BB'], 2)
sage: g.vertices()
['AA,AA', 'AA,BB', 'BB,AA', 'BB,BB']
sage: g.is_isomorphic(db)
True
GeneralizedDeBruijn(n, d)

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

EXAMPLES:

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

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: II = digraphs.ImaseItoh(8, 2)
sage: II.is_isomorphic(digraphs.DeBruijn(2, 3), certificate = True)
(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)
sage: b
True
sage: D   # random isomorphism
{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')

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: 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)
sage: K.edges()
[('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)
sage: K.edges()
[('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)

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

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

EXAMPLES:

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

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

A Paley digraph is always self-complementary:

sage: g.complement().is_isomorphic(g)
True
Path(n)

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

INPUT:

  • n – integer; number of vertices in the path

EXAMPLES:

sage: g = digraphs.Path(5)
sage: g.vertices()
[0, 1, 2, 3, 4]
sage: g.size()
4
sage: g.automorphism_group().cardinality()
1
RandomDirectedGN(n, kernel=<function <lambda>>, seed=None)

Returns a random GN (growing network) 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.

INPUT:

  • n - number of vertices.
  • kernel - the attachment kernel.
  • seed - a random.Random seed or a Python int for the random number generator (default: None).

EXAMPLES:

sage: D = digraphs.RandomDirectedGN(25)
sage: D.edges(labels=False)
[(1, 0), (2, 0), (3, 2), (4, 2), (5, 4), (6, 3), (7, 0), (8, 4), (9, 4), (10, 3), (11, 4), (12, 4), (13, 3), (14, 4), (15, 4), (16, 0), (17, 2), (18, 4), (19, 6), (20, 14), (21, 4), (22, 0), (23, 22), (24, 14)]  # 32-bit
[(1, 0), (2, 1), (3, 0), (4, 2), (5, 0), (6, 2), (7, 3), (8, 2), (9, 3), (10, 4), (11, 5), (12, 9), (13, 2), (14, 2), (15, 5), (16, 2), (17, 15), (18, 1), (19, 5), (20, 2), (21, 5), (22, 1), (23, 5), (24, 14)]   # 64-bit
sage: D.show()  # long time

REFERENCE:

  • [1] Krapivsky, P.L. and Redner, S. Organization of Growing Random Networks, Phys. Rev. E vol. 63 (2001), p. 066123.
RandomDirectedGNC(n, seed=None)

Returns a random GNC (growing network with copying) 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.

INPUT:

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

EXAMPLES:

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

REFERENCE:

  • [1] Krapivsky, P.L. and Redner, S. Network Growth by Copying, Phys. Rev. E vol. 71 (2005), p. 036118.
RandomDirectedGNM(n, m, loops=False)

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

INPUT:

  • n (integer) – number of vertices.
  • m (integer) – number of edges.
  • loops (boolean) – whether to allow loops (set to False by default).

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

EXAMPLES:

sage: D = digraphs.RandomDirectedGNM(10, 5)
sage: D.num_verts()
10
sage: D.edges(labels=False)
[(0, 3), (1, 5), (5, 1), (7, 0), (8, 5)]

With loops:

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

Returns a random digraph on \(n\) nodes. Each edge is inserted independently with probability \(p\).

INPUT:

  • n – number of nodes of the digraph
  • p – probability of an edge
  • loops – is a boolean set to True if the random digraph may have loops, and False (default) otherwise.
  • seed – integer seed for random number generator (default=None).

REFERENCES:

[1]P. Erdos and A. Renyi, On Random Graphs, Publ. Math. 6, 290 (1959).
[2]E. N. Gilbert, Random Graphs, Ann. Math. Stat., 30, 1141 (1959).

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

EXAMPLES:

sage: set_random_seed(0)
sage: D = digraphs.RandomDirectedGNP(10, .2)
sage: D.num_verts()
10
sage: D.edges(labels=False)
[(1, 0), (1, 2), (3, 6), (3, 7), (4, 5), (4, 7), (4, 8), (5, 2), (6, 0), (7, 2), (8, 1), (8, 9), (9, 4)]
RandomDirectedGNR(n, p, seed=None)

Returns a random GNR (growing network with redirection) 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.

INPUT:

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

EXAMPLES:

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

REFERENCE:

  • [1] Krapivsky, P.L. and Redner, S. Organization of Growing Random Networks, Phys. Rev. E vol. 63 (2001), p. 066123.
RandomSemiComplete(n)

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

EXAMPLES:

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

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

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: g = digraphs.TransitiveTournament(5)
sage: g.vertices()
[0, 1, 2, 3, 4]
sage: g.size()
10
sage: g.automorphism_group().cardinality()
1
nauty_directg(graphs, options='', debug=False)

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

    -e# | -e#:#  specify a value or range of the total number of arcs
    -o     orient each edge in only one direction, never both
    -f#  Use only the subgroup that fixes the first # vertices setwise
    -V  only output graphs with nontrivial groups (including exchange of
          isolated vertices).  The -f option is respected.
    -s#/#  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: 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

See also

tournaments_nauty(n, min_out_degree=None, max_out_degree=None, strongly_connected=False, debug=False, options='')

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 genbg’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 genbg. See its documentation for more information : http://cs.anu.edu.au/~bdm/nauty/.

EXAMPLES:

sage: for g in digraphs.tournaments_nauty(4):
....:     print(g.edges(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]