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# 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()` | Returns a n-dimensional butterfly graph.
`Circuit()` | Returns the circuit on \(n\) vertices.
`Circulant()` | Returns a circulant digraph on \(n\) vertices from a set of integers.
`DeBruijn()` | Returns the De Bruijn digraph with parameters \(k,n\).
`GeneralizedDeBruijn()` | Returns the generalized de Bruijn digraph of order \(n\) and degree \(d\).
`ImaseItoh()` | Returns the digraph of Imase and Itoh of order \(n\) and degree \(d\).
`Kautz()` | Returns the Kautz digraph of degree \(d\) and diameter \(D\).
`Path()` | Returns 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.
`RandomTournament()` | Returns a random tournament on \(n\) vertices.
`TransitiveTournament()` | Returns a transitive tournament on \(n\) vertices.
`tournaments_nauty()` | Returns 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

      Families of Graphs:
          - 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
  - `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(0, 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')

    Returns 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 v[i] != w[i].

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

    INPUT:

    - `vertices` - ‘strings’ (default) or ‘vectors’, specifying whether the vertices are zero-one strings or actually tuples over GF(2).

    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)

    Returns the circuit on \(n\) vertices

    The circuit is an oriented `CycleGraph`

    EXAMPLE:

    A circuit is the smallest strongly connected digraph:

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

  `Circulant`(n, integers)

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

    INPUT:

    - `n` (integer) – number of vertices.
    - `integers` – the list of integers such that there is an edge from \(i\) to \(j\) if and only if `(j-i)%n in integers`.

    EXAMPLE:

        sage: digraphs.Circulant(13,[3,5,7])
        Circulant graph ([3, 5, 7]): Digraph on 13 vertices

    TESTS:

        sage: digraphs.Circulant(13,[3,5,7,"hey"])
        Traceback (most recent call last):
        ...
        ValueError: The list must contain only relative integers.
        sage: digraphs.Circulant(3,[3,5,7,3.4])
        Traceback (most recent call last):
        ...
        ValueError: The list must contain only relative integers.

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

    Returns 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 on 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` – An integer equal to the length of words in the De Bruijn digraph when `vertices == 'strings'`, and also to the diameter of the digraph.

    - `vertices` – ‘strings’ (default) or ‘integers’, specifying whether the vertices are words build upon an alphabet or integers.

    EXAMPLES:

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

    TESTS:

        sage: digraphs.DeBruijn(5,0)
        De Bruijn digraph (k=5, n=0): Looped multi-digraph on 1 vertex
        sage: digraphs.DeBruijn(0,0)
        De Bruijn digraph (k=0, n=0): Looped multi-digraph on 0 vertices

  `GeneralizedDeBruijn`(n, d)

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

    The generalized de Bruijn digraph has been defined in [RPK80] [RPK83]. 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 \(D\).

    INPUTS:

    - `n` – is the number of vertices of the digraph
    - `d` – is the degree of the digraph

    See also

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

    EXAMPLE:

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

    TESTS:

    An exception is raised when the degree is less than one:

        sage: G = digraphs.GeneralizedDeBruijn(2, 0)
        Traceback (most recent call last):
        ...
        ValueError: The generalized de Bruijn digraph is defined for degree at least one.

    An exception is raised when the order of the graph is less than one:

        sage: G = digraphs.GeneralizedDeBruijn(0, 2)
        Traceback (most recent call last):
        ...
        ValueError: The generalized de Bruijn digraph is defined for at least one vertex.

    REFERENCES:

    [RPK80] | S. M. Reddy, D. K. Pradhan, and J. Kuhl. Directed graphs with minimal diameter and maximal connectivity, School Eng., Oakland Univ., Rochester MI, Tech. Rep., July 1980.

    [RPK83] | S. Reddy, P. Raghavan, and J. Kuhl. A Class of Graphs for Processor Interconnection. IEEE International Conference on Parallel Processing, pages 154-157, Los Alamitos, Ca., USA, August 1983.

  `ImaseItoh`(n, d)

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

    The digraph of Imase and Itoh has been defined in [II83]. 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 digraph of Imase and Itoh is isomorphic to the de Bruijn digraph of degree \(d\) and diameter \(D\). When \(n = d^{D-1}(d+1)\), the digraph of Imase and Itoh is isomorphic to the Kautz digraph [Kautz68] of degree \(d\) and diameter \(D\).

    INPUTS:

    - `n` – is the number of vertices of the digraph
    - `d` – is the degree of the digraph

    EXAMPLES:

        sage: II = digraphs.ImaseItoh(8, 2)
        sage: II.is_isomorphic(digraphs.DeBruijn(2, 3), certify = 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: II.is_isomorphic(digraphs.Kautz(2, 3), certify = True)
        (True, {0: '010', 1: '012', 2: '021', 3: '020', 4: '202', 5: '201', 6: '210', 7: '212', 8: '121', 9: '120', 10: '102', 11: '101'})

    TESTS:

    An exception is raised when the degree is less than one:

        sage: G = digraphs.ImaseItoh(2, 0)
        Traceback (most recent call last):
        ...
        ValueError: The digraph of Imase and Itoh is defined for degree at least one.

    An exception is raised when the order of the graph is less than two:

        sage: G = digraphs.ImaseItoh(1, 2)
        Traceback (most recent call last):
        ...
        ValueError: The digraph of Imase and Itoh is defined for at least two vertices.

    REFERENCE:

    [II83] | (1, 2) M. Imase and M. Itoh. A design for directed graphs with minimum diameter, IEEE Trans. Comput., vol. C-32, pp. 782-784, 1983.

  `Kautz`(k, D, vertices='strings')

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

    The Kautz digraph has been defined in [Kautz68]. The Kautz digraph of degree \(d\) and diameter \(D\) has \(d^{D-1}(d+1)\) vertices. This digraph is build upon a set of vertices equal to the set of words of length \(D\) from an alphabet of \(d+1\) letters such that consecutive letters are differents. 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 digraph of Imase and Itoh [II83] of degree \(d\) and order \(d^{D-1}(d+1)\).

    See also the Wikipedia article on Kautz Graphs.

    INPUTS:

    - `k` – Two possibilities for this parameter :
        - An integer equal to the degree of the digraph to be produced, that is the cardinality minus one of the alphabet to use.
        - 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` – An integer equal to the diameter of the digraph, and also to

        the length of a vertex label when `vertices == 'strings'`.

    - `vertices` – ‘strings’ (default) or ‘integers’, specifying whether

        the vertices are words build upon an alphabet or integers.

    EXAMPLES:

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

        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')]

    TESTS:

    An exception is raised when the degree is less than one:

        sage: G = digraphs.Kautz(0, 2)
        Traceback (most recent call last):
        ...
        ValueError: Kautz digraphs are defined for degree at least one.

        sage: G = digraphs.Kautz(['a'], 2)
        Traceback (most recent call last):
        ...
        ValueError: Kautz digraphs are defined for degree at least one.

    An exception is raised when the diameter of the graph is less than one:

        sage: G = digraphs.Kautz(2, 0)
        Traceback (most recent call last):
        ...
        ValueError: Kautz digraphs are defined for diameter at least one.

    REFERENCE:

    [Kautz68] | (1, 2) W. H. Kautz. Bounds on directed (d, k) graphs. Theory of cellular logic networks and machines, AFCRL-68-0668, SRI Project 7258, Final Rep., pp. 20-28, 1968.

  `Path`(n)

    Returns 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> at 0xed939614>, 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` - for the random number generator

    EXAMPLE:

        sage: D = digraphs.RandomDirectedGN(25)
        sage: D.edges(labels=False)
        [(1, 0), (2, 0), (3, 1), (4, 0), (5, 0), (6, 1), (7, 0), (8, 3), (9, 0), (10, 8), (11, 3), (12, 9), (13, 8), (14, 0), (15, 11), (16, 11), (17, 5), (18, 11), (19, 6), (20, 5), (21, 14), (22, 5), (23, 18), (24, 11)]
        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` - for the random number generator

    EXAMPLE:

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

    EXAMPLE:

        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)]

    TESTS:

        sage: digraphs.RandomDirectedGNM(10,-3)
        Traceback (most recent call last):
        ...
        ValueError: The number of edges must satisfy 0<= m <= n(n-1) when no loops are allowed, and 0<= m <= n^2 otherwise.

        sage: digraphs.RandomDirectedGNM(10,100)
        Traceback (most recent call last):
        ...
        ValueError: The number of edges must satisfy 0<= m <= n(n-1) when no loops are allowed, and 0<= m <= n^2 otherwise.

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

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

    INPUTS:

    - `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] | 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.

    EXAMPLE:

        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` - for the random number generator.

    EXAMPLE:

        sage: D = digraphs.RandomDirectedGNR(25, .2)
        sage: D.edges(labels=False)
        [(1, 0), (2, 0), (2, 1), (3, 0), (4, 0), (4, 1), (5, 0), (5, 1), (5, 2), (6, 0), (6, 1), (7, 0), (7, 1), (7, 4), (8, 0), (9, 0), (9, 8), (10, 0), (10, 1), (10, 2), (10, 5), (11, 0), (11, 8), (11, 9), (12, 0), (12, 8), (12, 9), (13, 0), (13, 1), (14, 0), (14, 8), (14, 9), (14, 12), (15, 0), (15, 8), (15, 9), (15, 12), (16, 0), (16, 1), (16, 4), (16, 7), (17, 0), (17, 8), (17, 9), (17, 12), (18, 0), (18, 8), (19, 0), (19, 1), (19, 4), (19, 7), (20, 0), (20, 1), (20, 4), (20, 7), (20, 16), (21, 0), (21, 8), (22, 0), (22, 1), (22, 4), (22, 7), (22, 19), (23, 0), (23, 8), (23, 9), (23, 12), (23, 14), (24, 0), (24, 8), (24, 9), (24, 12), (24, 15)]
        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.
  `RandomTournament`(n)

    Returns 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 \(i\).

    See Wikipedia article Tournament_(graph_theory)

    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: digraphs.RandomTournament(-1)
        Traceback (most recent call last):
        ...
        ValueError: The number of vertices cannot be strictly negative!

  `TransitiveTournament`(n)

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

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

    See Wikipedia article Tournament_(graph_theory)

    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

    TESTS:

        sage: digraphs.TransitiveTournament(-1)
        Traceback (most recent call last):
        ...
        ValueError: The number of vertices cannot be strictly negative!

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

    Returns 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) – 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/.

    Note

    To use this method you must first install the Nauty spkg.

    EXAMPLES:

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

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