r"""
Ribbon Graphs

This file implements objects called *ribbon graphs*. These are graphs 
together with a cyclic ordering of the darts adjacent to each 
vertex. This data allows us to unambiguously "thicken" the ribbon 
graph to an orientable surface with boundary. Also, every orientable
surface with non-empty boundary is the thickening of a ribbon graph.

AUTHORS:

- Pablo Portilla (2016)
"""

#*****************************************************************************
#       Copyright (C) 2016 Pablo Portilla  <p.portilla89@gmail.com>
#
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 2 of the License, or
# (at your option) any later version.
#                  http://www.gnu.org/licenses/
#*****************************************************************************

from sage.structure.sage_object import SageObject
from sage.structure.unique_representation import UniqueRepresentation
from sage.groups.perm_gps.permgroup_element import PermutationGroupElement
from sage.rings.integer_ring import ZZ
from sage.misc.cachefunc import cached_method
from sage.misc.flatten import flatten
from copy import deepcopy

#Auxiliary functions that will be used in the classes.

def _find(l, k):
    r"""
    Return the two coordinates of the element ``k`` in the list of
    lists ``l``.

    INPUT:

    - ``l`` -- a list of lists
    - ``k`` -- a candidate to be in a list in ``l``

    OUTPUT:

    A list with two integers describing the position of the first
    instance of `k`` in ``l``.

    TESTS::

        sage: from sage.geometry.ribbon_graph import _find
        sage: A = [[2,3,4],[4,5,2],[8,7]]
        sage: _find(A,2)
        [0, 0]
        sage: _find(A,7)
        [2, 1]
        sage: _find(A,5)
        [1, 1]
        sage: _find(A,-1)
        Traceback (most recent call last):
        ...
        ValueError: element -1 not found
    """
    for i,lst in enumerate(l):
        if k in lst:
            return [i, lst.index(k)]
    raise ValueError("element {} not found".format(k))

def _clean(l):
    r"""
    Return a list where empty sublists of ``l`` have been removed.

    INPUT:

    - ``l`` -- a list of lists

    OUTPUT:

    - a list which is a copy of ``l`` with all empty sublists removed

    EXAMPLES::

        sage: from sage.geometry.ribbon_graph import _clean
        sage: A = [[1,2],[], [2,1,7],[],[],[1]]
        sage: _clean(A)
        [[1, 2], [2, 1, 7], [1]]
    """
    return [list(elt) for elt in l if elt]


class RibbonGraph(SageObject, UniqueRepresentation):
    r"""
    A ribbon graph codified as two elements of a certain permutation group.

    A comprehensive introduction on the topic can be found in the beginning
    of [GGD2011]_ Chapter 4. More concretely, we will use a variation of what
    is called in the reference "The permutation representation pair of a
    dessin". Note that in that book, ribbon graphs are called "dessins
    d'enfant". For the sake on completeness we reproduce an adapted version
    of that introduction here.

    **Brief introduction**

    Let `\Sigma` be an orientable surface with non-empty boundary and let 
    `\Gamma` be the topological realization of a graph that is embedded in 
    `\Sigma` in such a way that the graph is a strong deformation retract of
    the surface. 

    Let `v(\Gamma)` be the set of vertices of `\Gamma`, suppose that these
    are white vertices. Now we mark black vertices in an interior point
    of each edge. In this way we get a bipartite graph where all the black
    vertices have valency 2 and there is no restriction on the valency
    of the white vertices. We call the edges of this new graph *darts*
    (sometimes they are also called *half eldges* of the original graph).
    Observe that each edge of the original graph is formed by two darts.

    Given a white vertex `v \in v(\Gamma)`, let `d(v)` be the set of darts
    adjacent to `v`. Let `D(\Gamma)` be the set of all the darts of
    `\Gamma` and suppose that we enumerate the set `D(\Gamma)` and that it
    has `n` elements.

    With the orientation of the surface and the embedding of the graph in 
    the surface we can produce two permutations:

    - A permutation that we denote by `\sigma`. This permutation is a
      product of as many cycles as white vertices (that is vertices in
      `\Gamma`). For each vertex consider a small topological circle
      around it in `\Sigma`. This circle intersects each adjacent dart
      once. The circle has an orientation induced by the orientation on
      `\Sigma` and so defines a cycle that sends the number associated
      to one dart to the number associated to the next dart in the
      positive orientation of the circle.

    - A permutation that we denote by `\rho`. This permutation is a
      product of as many `2`-cycles as edges has `\Gamma`. It just tells
      which two darts belong to the same edge.

    .. RUBRIC:: Abstract definition

    Consider a graph `\Gamma` (not a priori embedded in any surface). 
    Now we can again consider one vertex in the interior of each edge 
    splitting each edge in two darts. We label the darts with numbers.

    We say that a ribbon structure on `\Gamma` is a set of two 
    permutations `(\sigma, \rho)`. Where `\sigma` is formed by as many
    disjoint cycles as vertices had `\Gamma`. And each cycle is a 
    cyclic ordering of the darts adjacent to a vertex. The permutation
    `\rho` just tell us which two darts belong to the same edge.

    For any two such permutations there is a way of "thickening" the
    graph to a surface with boundary in such a way that the surface
    retracts (by a strong deformation retract) to the graph and hence
    the graph is embedded in the surface in a such a way that we could
    recover `\sigma` and `\rho`.

    INPUT:

    - ``sigma`` -- a permutation a product of disjoint cycles of any
      length; singletons (vertices of valency 1) need not be specified
    - ``rho`` -- a permutation which is a product of disjoint 2-cycles

    Alternatively, one can pass in 2 integers and this will construct
    a ribbon graph with genus ``sigma`` and ``rho`` boundary components.
    See :func:`~sage.geometry.ribbon_graphs.make_ribbon`.

    One can also construct the bipartite graph modeling the
    corresponding Brieskorn-Pham singularity by passing 2 integers
    and the keyword ``bipartite=True``.
    See :func:`~sage.geometry.ribbon_graphs.bipartite_ribbon_graph`.

    EXAMPLES:

    Consider the ribbon graph consisting of just `1` edge and `2`
    vertices of valency `1`::

        sage: s0 = PermutationGroupElement('(1)(2)')
        sage: r0 = PermutationGroupElement('(1,2)')
        sage: R0 = RibbonGraph(s0,r0); R0
        Ribbon graph of genus 0 and 1 boundary components

    Consider a graph that has `2` vertices of valency `3` (and hence `3`
    edges). That is represented by the following two permutations::

        sage: s1 = PermutationGroupElement('(1,3,5)(2,4,6)')
        sage: r1 = PermutationGroupElement('(1,2)(3,4)(5,6)')
        sage: R1 = RibbonGraph(s1, r1); R1
        Ribbon graph of genus 1 and 1 boundary components

    By drawing the picture in a piece of paper, one can see that its
    thickening has only `1` boundary component. Since the thickening
    is homotopically equivalent to the graph and the graph has Euler
    characteristic `-1`, we find that the thickening has genus `1`::

        sage: R1.number_boundaries()
        1
        sage: R1.genus()
        1

    The following example corresponds to the  complete bipartite graph
    of type `(2,3)`, where we have added one more edge `(8,15)` that
    ends at a vertex of valency `1`. Observe that it is not necessary
    to specify the  vertex `(15)` of valency `1` when we define sigma::

        sage: s2 = PermutationGroupElement('(1,3,5,8)(2,4,6)')
        sage: r2 = PermutationGroupElement('(1,2)(3,4)(5,6)(8,15)')
        sage: R2 = RibbonGraph(s2, r2); R1
        Ribbon graph of genus 1 and 1 boundary components
        sage: R2.sigma()
        (1,3,5,8)(2,4,6)

    This example is constructed by taking the bipartite graph of 
    type `(3,3)`::

        sage: s3 = PermutationGroupElement('(1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)')
        sage: r3 = PermutationGroupElement('(1,16)(2,13)(3,10)(4,17)(5,14)(6,11)(7,18)(8,15)(9,12)')
        sage: R3 = RibbonGraph(s3, r3); R3
        Ribbon graph of genus 1 and 3 boundary components

    The labeling of the darts can omit some numbers::

        sage: s4 = PermutationGroupElement('(3,5,10,12)')
        sage: r4 = PermutationGroupElement('(3,10)(5,12)')
        sage: R4 = RibbonGraph(s4,r4); R4
        Ribbon graph of genus 1 and 1 boundary components

    The next example is the complete bipartite graph of type `(3,3)`, where we
    have added an edge that ends at a vertex of valency 1::

        sage: s5 = PermutationGroupElement('(1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18,19)')
        sage: r5 = PermutationGroupElement('(1,16)(2,13)(3,10)(4,17)(5,14)(6,11)(7,18)(8,15)(9,12)(19,20)')
        sage: R5 = RibbonGraph(s5,r5); R5
        Ribbon graph of genus 1 and 3 boundary components
        sage: C = R5.contract_edge(9); C
        Ribbon graph of genus 1 and 3 boundary components
        sage: C.sigma()
        (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)
        sage: C.rho()
        (1,16)(2,13)(3,10)(4,17)(5,14)(6,11)(7,18)(8,15)(9,12)
        sage: S = R5.reduced(); S
        Ribbon graph of genus 1 and 3 boundary components
        sage: S.sigma()
        (5,6,8,9,14,15,11,12)
        sage: S.rho()
        (5,14)(6,11)(8,15)(9,12)
        sage: R5.boundary()
        [[1, 16, 17, 4, 5, 14, 15, 8, 9, 12, 10, 3],
         [2, 13, 14, 5, 6, 11, 12, 9, 7, 18, 19, 20, 20, 19, 16, 1],
         [3, 10, 11, 6, 4, 17, 18, 7, 8, 15, 13, 2]]
        sage: S.boundary()
        [[5, 14, 15, 8, 9, 12], [6, 11, 12, 9, 14, 5], [8, 15, 11, 6]]
        sage: R5.homology_basis()
        [[[5, 14], [13, 2], [1, 16], [17, 4]],
         [[6, 11], [10, 3], [1, 16], [17, 4]],
         [[8, 15], [13, 2], [1, 16], [18, 7]],
         [[9, 12], [10, 3], [1, 16], [18, 7]]]
        sage: S.homology_basis()
        [[[5, 14]], [[6, 11]], [[8, 15]], [[9, 12]]]

    We construct a ribbon graph corresponding to a genus 0 surface
    with 5 boundary components::

        sage: R = RibbonGraph(0, 5); R
        Ribbon graph of genus 0 and 5 boundary components
        sage: R.sigma()
        (1,9,7,5,3)(2,4,6,8,10)
        sage: R.rho()
        (1,2)(3,4)(5,6)(7,8)(9,10)

    We construct the Brieskorn-Pham singularity of type `(2,3)`::

        sage: B23 = RibbonGraph(2, 3, bipartite=True); B23
        Ribbon graph of genus 1 and 1 boundary components
        sage: B23.sigma()
        (1,2,3)(4,5,6)(7,8)(9,10)(11,12)
        sage: B23.rho()
        (1,8)(2,10)(3,12)(4,7)(5,9)(6,11)
    """
    @staticmethod
    def __classcall_private__(cls, sigma, rho, bipartite=False):
        """
        Normalize input.

        EXAMPLES::

            sage: s1 = PermutationGroupElement('(1,3,5,8)(2,4,6)')
            sage: r1 = PermutationGroupElement('(1,2)(3,4)(5,6)(8,15)')
            sage: R1 = RibbonGraph(s1, r1)
            sage: s2 = PermutationGroupElement('(1,3,5,8)(2,4,6)(15)')
            sage: R2 = RibbonGraph(s2, r1)
            sage: R1 is R2
            True
        """
        if bipartite:
            return bipartite_ribbon_graph(sigma, rho)
        if sigma in ZZ and rho in ZZ:
            return make_ribbon(sigma, rho)
        M = sigma.parent()
        if len(M.domain()) < len(rho.parent().domain()):
            M = rho.parent()
        return super(RibbonGraph, cls).__classcall__(cls, M(sigma), M(rho))

    def __init__(self, sigma, rho):
        r"""
        Initialize ``self``.

        EXAMPLES::

            sage: s0 = PermutationGroupElement('(1)(2)')
            sage: r0 = PermutationGroupElement('(1,2)')
            sage: R0 = RibbonGraph(s0,r0)
            sage: TestSuite(R0).run()

            sage: s1 = PermutationGroupElement('(1,3,5,8)(2,4,6)')
            sage: r1 = PermutationGroupElement('(1,2)(3,4)(5,6)(8,15)')
            sage: R1 = RibbonGraph(s1, r1)
            sage: TestSuite(R1).run()

            sage: s2 = PermutationGroupElement('(1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)')
            sage: r2 = PermutationGroupElement('(1,16)(2,13)(3,10)(4,17)(5,14)(6,11)(7,18)(8,15)(9,12)')
            sage: R2 = RibbonGraph(s2, r2)
            sage: TestSuite(R2).run()
        """
        self._sigma = sigma
        self._rho = rho

    def _repr_(self):
        r"""
        Return string representation of the two permutations that define
        the ribbon graph.

        EXAMPLES::

            sage: s = PermutationGroupElement('(3,5,10,12)')
            sage: r = PermutationGroupElement('(3,10)(5,12)')
            sage: RibbonGraph(s,r)
            Ribbon graph of genus 1 and 1 boundary components
        """
        return "Ribbon graph of genus {} and {} boundary components".format(self.genus(), self.number_boundaries())

    def sigma(self):
        r"""
        Return the permutation `\sigma` of ``self``.

        EXAMPLES::

            sage: s1 = PermutationGroupElement('(1,3,5,8)(2,4,6)')
            sage: r1 = PermutationGroupElement('(1,2)(3,4)(5,6)(8,15)')
            sage: R = RibbonGraph(s1, r1)
            sage: R.sigma()
            (1,3,5,8)(2,4,6)
        """
        return self._sigma

    def rho(self):
        r"""
        Return the permutation `\rho` of ``self``.

        EXAMPLES::

            sage: s1 = PermutationGroupElement('(1,3,5,8)(2,4,6)')
            sage: r1 = PermutationGroupElement('(1,2)(3,4)(5,6)(8,15)')
            sage: R = RibbonGraph(s1, r1)
            sage: R.rho()
            (1,2)(3,4)(5,6)(8,15)
        """
        return self._rho

    @cached_method
    def number_boundaries(self):
        r"""
        Return number of boundary components of the thickening of the
        ribbon graph.

        EXAMPLES:

        The first example is the ribbon graph corresponding to the torus
        with one hole::

            sage: s1 = PermutationGroupElement('(1,3,5)(2,4,6)')
            sage: r1 = PermutationGroupElement('(1,2)(3,4)(5,6)')
            sage: R1 = RibbonGraph(s1,r1)
            sage: R1.number_boundaries()
            1

        This example is constructed by taking the bipartite graph of 
        type `(3,3)`::

            sage: s2 = PermutationGroupElement('(1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)')
            sage: r2 = PermutationGroupElement('(1,16)(2,13)(3,10)(4,17)(5,14)(6,11)(7,18)(8,15)(9,12)')
            sage: R2 = RibbonGraph(s2,r2)
            sage: R2.number_boundaries()
            3
        """
        # It might seem a bit overkill to call boundary() here but it is
        #   necessary to either call it or do similar computations here.
        #   The function boundary() avoids some patologies with boundaries
        #   formed by just one loop.
        return len(self.boundary())

    def contract_edge(self, k):
        r"""
        Return the ribbon graph resulting from the contraction of
        the ``k``-th edge in ``self``.

        For a ribbon graph `(\sigma, \rho)`, we contract the edge
        corresponding to the `k`-th transposition of `\rho`.

        INPUT:

        - ``k`` -- non-negative integer; the position in `\rho` of the 
          transposition that is going to be contracted

        OUTPUT:

        - a ribbon graph resulting from the contraction of that edge

        EXAMPLES:

        We start again with the one-holed torus ribbon graph::

            sage: s1 = PermutationGroupElement('(1,3,5)(2,4,6)')
            sage: r1 = PermutationGroupElement('(1,2)(3,4)(5,6)')
            sage: R1 = RibbonGraph(s1,r1); R1
            Ribbon graph of genus 1 and 1 boundary components
            sage: S1 = R1.contract_edge(1); S1
            Ribbon graph of genus 1 and 1 boundary components
            sage: S1.sigma()
            (1,6,2,5)
            sage: S1.rho()
            (1,2)(5,6)

        However, this ribbon graphs is formed only by loops and hence
        it cannot be longer reduced, we get an error if we try to
        contract a loop::

            sage: S1.contract_edge(1)
            Traceback (most recent call last):
            ...
            ValueError: the edge is a loop and cannot be contracted

        In this example, we consider a graph that has one edge ``(19,20)``
        such that one of its ends is a vertex of valency 1. This is 
        the vertex ``(20)`` that is not specified when defining `\sigma`.
        We contract precisely this edge and get a ribbon graph with no
        vertices of valency 1::

            sage: s2 = PermutationGroupElement('(1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18,19)')
            sage: r2 = PermutationGroupElement('(1,16)(2,13)(3,10)(4,17)(5,14)(6,11)(7,18)(8,15)(9,12)(19,20)')
            sage: R2 = RibbonGraph(s2,r2); R2
            Ribbon graph of genus 1 and 3 boundary components
            sage: R2.sigma()
            (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18,19)
            sage: R2c = R2.contract_edge(9); R2; R2c.sigma(); R2c.rho()
            Ribbon graph of genus 1 and 3 boundary components
            (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)
            (1,16)(2,13)(3,10)(4,17)(5,14)(6,11)(7,18)(8,15)(9,12)
        """
        #the following two lines convert the list of tuples to list of lists
        aux_sigma = [list(x) for x in self._sigma.cycle_tuples(singletons=True)]
        aux_rho = [list(x) for x in self._rho.cycle_tuples()]
        #The following ''if'' rules out the cases when we would be 
        #contracting a loop (which is not admissible since we would 
        #lose the topological type of the graph).
        if (_find(aux_sigma, aux_rho[k][0])[0] == 
                _find(aux_sigma, aux_rho[k][1])[0]):
            raise ValueError("the edge is a loop and cannot be contracted")
        #We store in auxiliary variables the positions of the vertices
        #that are the ends of the edge to be contracted and we delete
        #from them the darts corresponding to the edge that is going
        #to be contracted. We also delete the contracted edge 
        #from aux_rho
        pos1 = _find(aux_sigma, aux_rho[k][0])
        pos2 = _find(aux_sigma, aux_rho[k][1])
        del aux_sigma[pos1[0]][pos1[1]]
        del aux_sigma[pos2[0]][pos2[1]]
        del aux_rho[k]

        #Now we insert in one of the two vertices, the darts of the other
        #vertex that appears after. We make sure that we don't
        #change the topological type of the thickening of the graph by
        #preserving the cyclic ordering.
        n = len(aux_sigma[pos2[0]])
        for i in range(n):
            aux_sigma[pos1[0]].insert(
                pos1[1] + i,
                aux_sigma[pos2[0]][(pos2[1]+i) % n]
            )
        #Finally we delete the vertex from which we copied all the darts.
        del aux_sigma[pos2[0]]

        #Now we convert this data that is on the form of lists of lists
        #to actual permutations that form a ribbon graph.
        return RibbonGraph(PermutationGroupElement([tuple(x) for x in aux_sigma]), 
                           PermutationGroupElement([tuple(x) for x in aux_rho]))

    def extrude_edge(self, vertex, dart1, dart2):
        r"""
        Return a ribbon graph resulting from extruding an edge from a
        vertex, pulling from it, all darts from ``dart1`` to ``dart2``
        including both.

        INPUT:

        - ``vertex`` -- the position of the vertex in the permutation
          `\sigma`, which must have valency at least 2

        - ``dart1`` -- the position of the first in the
          cycle corresponding to ``vertex``

        - ``dart2`` -- the position of the second dart in the cycle
          corresponding to ``vertex``

        OUTPUT:

        A ribbon graph resulting from extruding a new edge that 
        pulls from ``vertex`` a new vertex that is, now, adjacent
        to all the darts from ``dart1``to ``dart2`` (not including
        ``dart2``) in the cyclic ordering given by the cycle corresponding
        to ``vertex``. Note that ``dart1`` may be equal to ``dart2``
        allowing thus to extrude a contractible edge from a vertex.

        EXAMPLES:

        We try several possibilities in the same graph::

            sage: s1 = PermutationGroupElement('(1,3,5)(2,4,6)')
            sage: r1 = PermutationGroupElement('(1,2)(3,4)(5,6)')
            sage: R1 = RibbonGraph(s1,r1); R1
            Ribbon graph of genus 1 and 1 boundary components
            sage: E1 = R1.extrude_edge(1,1,2); E1
            Ribbon graph of genus 1 and 1 boundary components
            sage: E1.sigma()
            (1,3,5)(2,8,6)(4,7)
            sage: E1.rho()
            (1,2)(3,4)(5,6)(7,8)
            sage: E2 = R1.extrude_edge(1,1,3); E2
            Ribbon graph of genus 1 and 1 boundary components
            sage: E2.sigma()
            (1,3,5)(2,8)(4,6,7)
            sage: E2.rho()
            (1,2)(3,4)(5,6)(7,8)

        We can also extrude a contractible edge from a vertex. This
        new edge will end at a vertex of valency 1::

            sage: E1p = R1.extrude_edge(0,0,0); E1p
            Ribbon graph of genus 1 and 1 boundary components
            sage: E1p.sigma()
            (1,3,5,8)(2,4,6)
            sage: E1p.rho()
            (1,2)(3,4)(5,6)(7,8)

        In the following example we first extrude one edge from a vertex
        of valency 3 generating a new vertex of valency 2. Then we 
        extrude a new edge from this vertex of valency 2::

            sage: s1 = PermutationGroupElement('(1,3,5)(2,4,6)')
            sage: r1 = PermutationGroupElement('(1,2)(3,4)(5,6)')
            sage: R1 = RibbonGraph(s1,r1); R1
            Ribbon graph of genus 1 and 1 boundary components
            sage: E1 = R1.extrude_edge(0,0,1); E1
            Ribbon graph of genus 1 and 1 boundary components
            sage: E1.sigma()
            (1,7)(2,4,6)(3,5,8)
            sage: E1.rho()
            (1,2)(3,4)(5,6)(7,8)
            sage: F1 = E1.extrude_edge(0,0,1); F1
            Ribbon graph of genus 1 and 1 boundary components
            sage: F1.sigma()
            (1,9)(2,4,6)(3,5,8)(7,10)
            sage: F1.rho()
            (1,2)(3,4)(5,6)(7,8)(9,10)
        """
        #We first compute the vertices of valency 1 as in _repr_
        repr_sigma = [list(x) for x in self._sigma.cycle_tuples()]
        repr_rho = [list(x) for x in self._rho.cycle_tuples()]
        darts_rho = flatten(repr_rho)
        darts_sigma = flatten(repr_sigma)
        val_one = [x for x in darts_rho if x not in darts_sigma]
        for val in val_one:
            repr_sigma += [[val]]

        # We find which is the highes value a dart has, in order to 
        #   add new darts that do not conflict with previous ones.
        k = max(darts_rho)

        # We create the new vertex and append it to sigma.
        new_vertex = [repr_sigma[vertex][j] for j in range(dart1, dart2)]
        new_vertex.insert(0, k+1)
        repr_sigma.append(new_vertex)

        # We add the new dart at the vertex from which we are extruding
        #   an edge. Also we delete the darts that have been extruded.
        repr_sigma[vertex].insert(dart1,k+2)
        del repr_sigma[vertex][dart1+1:dart2+1]

        #We update rho
        repr_rho.append([k+1,k+2])

        perm_group = self._sigma.parent()
        return RibbonGraph(PermutationGroupElement([tuple(x) for x in repr_sigma]), 
                           PermutationGroupElement([tuple(x) for x in repr_rho]))

    @cached_method
    def genus(self):
        r"""
        Return the genus of the thickening of ``self``.

        OUTPUT:

        - ``g`` -- non-negative integer representing the genus of the
          thickening of the ribbon graph

        EXAMPLES::

            sage: s1 = PermutationGroupElement('(1,3,5)(2,4,6)')
            sage: r1 = PermutationGroupElement('(1,2)(3,4)(5,6)')
            sage: R1 = RibbonGraph(s1,r1)
            sage: R1.genus()
            1

            sage: s3=PermutationGroupElement('(1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)')
            sage: r3=PermutationGroupElement('(1,21)(2,17)(3,13)(4,22)(7,23)(5,18)(6,14)(8,19)(9,15)(10,24)(11,20)(12,16)')
            sage: R3 = RibbonGraph(s3,r3); R3.genus()
            3
        """
        #We now use the same procedure as in _repr_ to get the vertices
        #of valency 1 and distinguish them from the extra singletons of
        #the permutation sigma. 
        repr_sigma = [list(x) for x in self._sigma.cycle_tuples()]
        repr_rho = [list(x) for x in self._rho.cycle_tuples()]
        darts_rho = flatten(repr_rho)
        darts_sigma = flatten(repr_sigma)
        val_one = [x for x in darts_rho if x not in darts_sigma]

        #the total number of vertices of sigma is its number of cycles
        #of length >1 plus the number of singletons that are actually
        #vertices of valency 1
        
        vertices = len(self._sigma.cycle_tuples()) + len(val_one)
        edges = len(self._rho.cycle_tuples())
        #formula for the genus using that the thickening is homotopically 
        #equivalent to the graph
        g = (-vertices + edges - self.number_boundaries() + 2) // 2

        return g

    def mu(self):
        r"""
        Return the rank of the first homology group of the thickening
        of the ribbon graph.

        EXAMPLES::

            sage: s1 = PermutationGroupElement('(1,3,5)(2,4,6)')
            sage: r1 = PermutationGroupElement('(1,2)(3,4)(5,6)')
            sage: R1 = RibbonGraph(s1,r1);R1
            Ribbon graph of genus 1 and 1 boundary components
            sage: R1.mu()
            2
        """
        return 2 * self.genus() + self.number_boundaries() - 1

    def boundary(self):
        r"""
        Return the labeled boundaries of ``self``.

        If you cut the thickening of the graph along the graph. you
        get a collection of cylinders (recall that the graph was a
        strong deformation retract of the thickening). In each cylinder
        one of the boundary components has a labelling of its edges
        induced by the labelling of the darts.

        OUTPUT:

        A list of lists. The number of inner lists is the number of 
        boundary components of the surface. Each list in the list
        consists of an ordered tuple of numbers, each number comes
        from the number assigned to the corresponding dart before
        cutting.

        EXAMPLES:

        We start with a ribbon graph whose thickening has one boundary
        component. We compute its labeled boundary, then reduce it and
        compute the labeled boundary of the reduced  ribbon graph::

            sage: s1 = PermutationGroupElement('(1,3,5)(2,4,6)')
            sage: r1 = PermutationGroupElement('(1,2)(3,4)(5,6)')
            sage: R1 = RibbonGraph(s1,r1); R1
            Ribbon graph of genus 1 and 1 boundary components
            sage: R1.boundary()
            [[1, 2, 4, 3, 5, 6, 2, 1, 3, 4, 6, 5]]
            sage: H1 = R1.reduced(); H1
            Ribbon graph of genus 1 and 1 boundary components
            sage: H1.sigma()
            (3,5,4,6)
            sage: H1.rho()
            (3,4)(5,6)
            sage: H1.boundary()
            [[3, 4, 6, 5, 4, 3, 5, 6]]

        We now consider a ribbon graph whose thickening has 3 boundary
        components. Also observe that in one of the labeled boundary
        components, a numbers appears twice in a row. That is because
        the ribbon graph has a vertex of valency 1::

            sage: s2=PermutationGroupElement('(1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18,19)')
            sage: r2=PermutationGroupElement('(1,16)(2,13)(3,10)(4,17)(5,14)(6,11)(7,18)(8,15)(9,12)(19,20)')
            sage: R2 = RibbonGraph(s2,r2)
            sage: R2.number_boundaries()
            3
            sage: R2.boundary()
            [[1, 16, 17, 4, 5, 14, 15, 8, 9, 12, 10, 3],
             [2, 13, 14, 5, 6, 11, 12, 9, 7, 18, 19, 20, 20, 19, 16, 1],
             [3, 10, 11, 6, 4, 17, 18, 7, 8, 15, 13, 2]]
        """
        #initialize and empty list to hold the labels of the boundaries
        bound = []

        #since lists of tuples are not modifiable, we change the data to a
        #list of lists 
        aux_perm = (self._rho * self._sigma).cycle_tuples(singletons=True)

        #the cycles of the permutation rho*sigma are in 1:1 correspondence with 
        #the boundary components of the thickening (see function number_boundaries())
        #but they are not the labeled boundary components.
        #With the next for, we convert the cycles of rho*sigma to actually 
        #the labelling of the edges. Each edge, therefore, should appear twice

        for i,p in enumerate(aux_perm):
            bound = bound + [[]]
            for j in range(len(p)):
                if self._rho(p[j]) != p[j]:
                    bound[i].append(p[j])
                    bound[i].append(self._rho(p[j]))
                else:
                    continue

        #finally the function returns a List of lists. Each list contains
        #a sequence of  numbers and each number corresponds to a half-edge.
        return _clean(bound)

    def reduced(self):
        r"""
        Return a ribbon graph with 1 vertex and `\mu` edges (where `\mu`
        is the first betti number of the graph).

        OUTPUT:

        - a ribbon graph whose `\sigma` permutation has only 1 non-singleton
          cycle and whose `\rho` permutation is a product of `\mu` disjoint
          2-cycles

        EXAMPLES::

            sage: s1 = PermutationGroupElement('(1,3,5)(2,4,6)')
            sage: r1 = PermutationGroupElement('(1,2)(3,4)(5,6)')
            sage: R1 = RibbonGraph(s1,r1); R1
            Ribbon graph of genus 1 and 1 boundary components
            sage: G1 = R1.reduced(); G1
            Ribbon graph of genus 1 and 1 boundary components
            sage: G1.sigma()
            (3,5,4,6)
            sage: G1.rho()
            (3,4)(5,6)

            sage: s2 = PermutationGroupElement('(1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18,19)')
            sage: r2 = PermutationGroupElement('(1,16)(2,13)(3,10)(4,17)(5,14)(6,11)(7,18)(8,15)(9,12)(19,20)')
            sage: R2 = RibbonGraph(s2,r2); R2
            Ribbon graph of genus 1 and 3 boundary components
            sage: G2 = R2.reduced(); G2
            Ribbon graph of genus 1 and 3 boundary components
            sage: G2.sigma()
            (5,6,8,9,14,15,11,12)
            sage: G2.rho()
            (5,14)(6,11)(8,15)(9,12)

            sage: s3 = PermutationGroupElement('(1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)')
            sage: r3 = PermutationGroupElement('(1,21)(2,17)(3,13)(4,22)(7,23)(5,18)(6,14)(8,19)(9,15)(10,24)(11,20)(12,16)')
            sage: R3 = RibbonGraph(s3,r3); R3
            Ribbon graph of genus 3 and 1 boundary components
            sage: G3 = R3.reduced(); G3
            Ribbon graph of genus 3 and 1 boundary components
            sage: G3.sigma()
            (5,6,8,9,11,12,18,19,20,14,15,16)
            sage: G3.rho()
            (5,18)(6,14)(8,19)(9,15)(11,20)(12,16)
        """
        #the following two lines convert the list of tuples to list of lists
        #we have to contract exactly n edges
        aux_ribbon = deepcopy(self)
        aux_rho = [list(x) for x in aux_ribbon._rho.cycle_tuples()]

        #Observe that in the end we will have `\mu` edges, so we
        #know exactly how many steps we will iterate
        while len(aux_rho) > self.mu():
            aux_sigma = [list(x) for x in aux_ribbon._sigma.cycle_tuples(singletons=True)]
            aux_rho = [list(x) for x in aux_ribbon._rho.cycle_tuples()]
            for j in range(len(aux_rho)):
                if (_find(aux_sigma, aux_rho[j][0])[0] != 
                        _find(aux_sigma, aux_rho[j][1])[0]):
                    aux_ribbon = aux_ribbon.contract_edge(j)
                    aux_rho = [list(x) for 
                    x in aux_ribbon._rho.cycle_tuples()]
                    break
        #finally we change the data to a list of tuples and return the
        #information as a ribbon graph. 
        return aux_ribbon
    
    #the next function computes a basis of homology, it uses
    #the previous function.

    def make_generic(self):
        r"""
        Return a ribbon graph equivalent to ``self`` but where every
        vertex has valency 3.

        OUTPUT:

        - a ribbon graph that is equivalent to ``self`` but is generic
          in the sense that all vertices have valency 3

        EXAMPLES::

            sage: R = RibbonGraph(1,3); R
            Ribbon graph of genus 1 and 3 boundary components
            sage: R.sigma()
            (1,2,3,9,7)(4,8,10,5,6)
            sage: R.rho()
            (1,4)(2,5)(3,6)(7,8)(9,10)
            sage: G = R.make_generic(); G
            Ribbon graph of genus 1 and 3 boundary components
            sage: G.sigma()
            (2,3,11)(5,6,13)(7,8,15)(9,16,17)(10,14,19)(12,18,21)(20,22)
            sage: G.rho()
            (2,5)(3,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)
            sage: R.genus() == G.genus() and R.number_boundaries() == G.number_boundaries()
            True

            sage: R = RibbonGraph(5,4); R
            Ribbon graph of genus 5 and 4 boundary components
            sage: R.sigma()
            (1,2,3,4,5,6,7,8,9,10,11,27,25,23)(12,24,26,28,13,14,15,16,17,18,19,20,21,22)
            sage: R.rho()
            (1,12)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(23,24)(25,26)(27,28)
            sage: G = R.reduced(); G
            Ribbon graph of genus 5 and 4 boundary components
            sage: G.sigma()
            (2,3,4,5,6,7,8,9,10,11,27,25,23,24,26,28,13,14,15,16,17,18,19,20,21,22)
            sage: G.rho()
            (2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(23,24)(25,26)(27,28)
            sage: G.genus() == R.genus() and G.number_boundaries() == R.number_boundaries()
            True

            sage: R = RibbonGraph(0,6); R
            Ribbon graph of genus 0 and 6 boundary components
            sage: R.sigma()
            (1,11,9,7,5,3)(2,4,6,8,10,12)
            sage: R.rho()
            (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)
            sage: G = R.reduced(); G
            Ribbon graph of genus 0 and 6 boundary components
            sage: G.sigma()
            (3,4,6,8,10,12,11,9,7,5)
            sage: G.rho()
            (3,4)(5,6)(7,8)(9,10)(11,12)
            sage: G.genus() == R.genus() and G.number_boundaries() == R.number_boundaries()
            True
        """
        aux_ribbon = self.reduced()

        for i in range(2*aux_ribbon.mu() - 2):
            aux_ribbon = aux_ribbon.extrude_edge(i,0,2)

        return aux_ribbon

    def homology_basis(self):
        r"""
        Return an oriented basis of the first homology group of the 
        graph.

        OUTPUT:

        - A 2-dimensional array of ordered edges in the graph (given by pairs).
          The length of the first dimension is `\mu`. Each row corresponds
          to an element of the basis and is a circle contained in the graph.

        EXAMPLES::

            sage: R = RibbonGraph(0,6); R
            Ribbon graph of genus 0 and 6 boundary components
            sage: R.mu()
            5
            sage: R.homology_basis()
            [[[3, 4], [2, 1]],
             [[5, 6], [2, 1]],
             [[7, 8], [2, 1]],
             [[9, 10], [2, 1]],
             [[11, 12], [2, 1]]]

            sage: R = RibbonGraph(1,1); R
            Ribbon graph of genus 1 and 1 boundary components
            sage: R.mu()
            2
            sage: R.homology_basis()
            [[[2, 5], [4, 1]], [[3, 6], [4, 1]]]
            sage: H = R.reduced(); H
            Ribbon graph of genus 1 and 1 boundary components
            sage: H.sigma()
            (2,3,5,6)
            sage: H.rho()
            (2,5)(3,6)
            sage: H.homology_basis()
            [[[2, 5]], [[3, 6]]]

            sage: s3 = PermutationGroupElement('(1,2,3,4,5,6,7,8,9,10,11,27,25,23)(12,24,26,28,13,14,15,16,17,18,19,20,21,22)')
            sage: r3 = PermutationGroupElement('(1,12)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(23,24)(25,26)(27,28)')
            sage: R3 = RibbonGraph(s3,r3); R3
            Ribbon graph of genus 5 and 4 boundary components
            sage: R3.mu()
            13
            sage: R3.homology_basis()
            [[[2, 13], [12, 1]],
             [[3, 14], [12, 1]],
             [[4, 15], [12, 1]],
             [[5, 16], [12, 1]],
             [[6, 17], [12, 1]],
             [[7, 18], [12, 1]],
             [[8, 19], [12, 1]],
             [[9, 20], [12, 1]],
             [[10, 21], [12, 1]],
             [[11, 22], [12, 1]],
             [[23, 24], [12, 1]],
             [[25, 26], [12, 1]],
             [[27, 28], [12, 1]]]
            sage: H3 = R3.reduced(); H3
            Ribbon graph of genus 5 and 4 boundary components
            sage: H3.sigma()
            (2,3,4,5,6,7,8,9,10,11,27,25,23,24,26,28,13,14,15,16,17,18,19,20,21,22)
            sage: H3.rho()
            (2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(23,24)(25,26)(27,28)
            sage: H3.homology_basis()
            [[[2, 13]],
             [[3, 14]],
             [[4, 15]],
             [[5, 16]],
             [[6, 17]],
             [[7, 18]],
             [[8, 19]],
             [[9, 20]],
             [[10, 21]],
             [[11, 22]],
             [[23, 24]],
             [[25, 26]],
             [[27, 28]]]
        """
        aux_sigma = [list(x) for  x in self._sigma.cycle_tuples(singletons=True)]

        basis = [[list(x)] for x in self.reduced()._rho.cycle_tuples()]

        #Now we define center as the set of edges that were contracted 
        #in reduced() this set is contractible and can be define as the 
        #complement of reduced_rho in rho

        center = [list(x) for x in self._rho.cycle_tuples() 
                  if (x not in self.reduced()._rho.cycle_tuples())]

        #We define an auxiliary list 'vertices' that will contain the
        #vertices (cycles of sigma) corresponding to each half edge. 

        vertices = []

        for i in range(len(basis)):
            vertices = vertices + [[]]
            basis[i].extend(deepcopy(center))

            for j in range (len(basis[i])):
                vertices[i].append(_find(aux_sigma, basis[i][j][0])[0])
                vertices[i].append(_find(aux_sigma, basis[i][j][1])[0])
            k = 0

            while k < len(vertices[i]):
                if vertices[i].count(vertices[i][k]) == 1:
                    m = k // 2
                    del basis[i][m]
                    del vertices[i][2*m:2*m+2]
                    k = 0
                else:
                    k+=1

        for i in range(len(basis)):
            for j in range(1, len(basis[i])):
                n = [t for t, n in enumerate(vertices[i])
                     if n == vertices[i][2*j-1]][1]

                ind = n // 2

                if j != ind:
                    basis[i][j], basis[i][ind] = basis[i][ind], basis[i][j]

                    vertices[i][2*j], vertices[i][2*ind] = \
                        vertices[i][2*ind], vertices[i][2*j]

                    vertices[i][2*j+1], vertices[i][2*ind+1] = \
                        vertices[i][2*ind+1], vertices[i][2*j+1]

                if (vertices[i][2*j-1] != vertices[i][2*j]):
                    vertices[i][2*j], vertices[i][2*j+1] = \
                        vertices[i][2*j+1], vertices[i][2*j]

                    basis[i][j][0], basis[i][j][1] = \
                        basis[i][j][1], basis[i][j][0]

        #the variable basis is a LIST of Lists of lists. Each List 
        #corresponds to an element of the basis and each list in a List
        #is just a 2-tuple which corresponds to an ''ordered'' edge of rho.

        return basis

    def normalize(self):
        r"""
        Return an equivalent graph such that the enumeration of its darts
        exhausts all numbers from 1 to the number of darts.

        OUTPUT:

        - a ribbon graph equivalent to ``self`` such that the enumeration
          of its darts exhausts all numbers from 1 to the number of darts.

        EXAMPLES::

            sage: s0 = PermutationGroupElement('(1,22,3,4,5,6,7,15)(8,16,9,10,11,12,13,14)')
            sage: r0 = PermutationGroupElement('(1,8)(22,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,16)')
            sage: R0 = RibbonGraph(s0,r0); R0
            Ribbon graph of genus 3 and 2 boundary components
            sage: RN0 = R0.normalize(); RN0; RN0.sigma(); RN0.rho()
            Ribbon graph of genus 3 and 2 boundary components
            (1,16,2,3,4,5,6,14)(7,15,8,9,10,11,12,13)
            (1,7)(2,9)(3,10)(4,11)(5,12)(6,13)(8,16)(14,15)

            sage: s1 = PermutationGroupElement('(5,10,12)(30,34,78)')
            sage: r1 = PermutationGroupElement('(5,30)(10,34)(12,78)')
            sage: R1 = RibbonGraph(s1,r1); R1
            Ribbon graph of genus 1 and 1 boundary components
            sage: RN1 = R1.normalize(); RN1; RN1.sigma(); RN1.rho()
            Ribbon graph of genus 1 and 1 boundary components
            (1,2,3)(4,5,6)
            (1,4)(2,5)(3,6)

        """
        #First we compute the vertices of valency 1 and store them in val_one.
        aux_sigma = [list(x) for x in self._sigma.cycle_tuples()]
        aux_rho = [list(x) for x in self._rho.cycle_tuples()]
        darts_rho = flatten(aux_rho)
        darts_sigma = flatten(aux_sigma)
        val_one = [x for x in darts_rho if x not in darts_sigma]
 
        #We add them to aux_sigma
        for i in range(len(val_one)):
            aux_sigma += [[val_one[i]]]
        #Now we proceed to normalize the numbers enumerating the darts.
        #We do this by checking if every number from 1 to len(darts_rho)
        #is actually in darts_rho.
        for i in range(len(darts_rho)):
            found = i+1 in darts_rho
            #if a value is not in darts_rho, we take the next number that appears
            #and change it to the new value.
            if not found:
                aux_val = min(x for x in darts_rho if x > i+1)
                pos_darts = darts_rho.index(aux_val)
                pos_rho = _find(aux_rho,aux_val)
                pos_sigma = _find(aux_sigma,aux_val)

                #Now we set the found positions to the new normalized value
                darts_rho[pos_darts]=i+1
                aux_sigma[pos_sigma[0]][pos_sigma[1]]=i+1
                aux_rho[pos_rho[0]][pos_rho[1]]=i+1

        return RibbonGraph(
                        PermutationGroupElement([tuple(x) for x in aux_sigma]), 
                        PermutationGroupElement([tuple(x) for x in aux_rho])
                        )

def make_ribbon(g, r):
    r"""
    Return a ribbon graph whose thickening has genus ``g`` and ``r``
    boundary components.

    INPUT:

    - ``g`` -- non-negative integer representing the genus of the
      thickening

    - ``r`` -- positive integer representing the number of boundary
      components of the thickening

    OUTPUT:

    - a ribbon graph that has 2 vertices (two non-trivial cycles 
      in its sigma permutation) of valency `2g + r` and it has 
      `2g + r` edges (and hence `4g + 2r` darts)

    EXAMPLES::

        sage: from sage.geometry.ribbon_graph import make_ribbon
        sage: R = make_ribbon(0,1); R
        Ribbon graph of genus 0 and 1 boundary components
        sage: R.sigma()
        ()
        sage: R.rho()
        (1,2)

        sage: R = make_ribbon(0,5); R
        Ribbon graph of genus 0 and 5 boundary components
        sage: R.sigma()
        (1,9,7,5,3)(2,4,6,8,10)
        sage: R.rho()
        (1,2)(3,4)(5,6)(7,8)(9,10)

        sage: R = make_ribbon(1,1); R
        Ribbon graph of genus 1 and 1 boundary components
        sage: R.sigma()
        (1,2,3)(4,5,6)
        sage: R.rho()
        (1,4)(2,5)(3,6)

        sage: R = make_ribbon(7,3); R
        Ribbon graph of genus 7 and 3 boundary components
        sage: R.sigma()
        (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,33,31)(16,32,34,17,18,19,20,21,22,23,24,25,26,27,28,29,30)
        sage: R.rho()
        (1,16)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30)(31,32)(33,34)

    """
    #Initialize the two vertices of sigma and the edge joining them
    repr_sigma = [[1],[2*g+2]]
    repr_rho = [[1,2*g+2]]

    #We first generate the surface of genus g and 1 boundary component.
    #This is done by considering the usual planar representation of
    #a surface as a polygon of 4*g+2 edges with identifications. (see
    #any topology  book on the classification of surfaces)
    for i in range(2*g):
        repr_sigma[0].append(i+2)
        repr_sigma[1].append(i+(2*g+2)+1)
        repr_rho += [[i+2,i+(2*g+2)+1]]

    #finally we add an edge for each additional boundary component. 
    max_dart = 4*g+2
    for j in range(r-1):
        repr_sigma[0].insert(0, max_dart+2*(j+1)-1)
        repr_sigma[1].insert(j+1, max_dart+2*(j+1))
        repr_rho += [[max_dart+2*(j+1)-1, max_dart+2*(j+1)]]

    return RibbonGraph(PermutationGroupElement([tuple(x) for x in repr_sigma]), 
                       PermutationGroupElement([tuple(x) for x in repr_rho]))

def bipartite_ribbon_graph(p, q):
    r"""
    Return the bipartite graph modeling the corresponding
    Brieskorn-Pham singularity.

    Take two parallel lines in the plane, and consider `p` points in
    one of them and `q` points in the other. Join with a line each
    point from the first set with every point with the second set.
    The resulting is a planar projection of the complete bipartite
    graph of type `(p,q)`. If you consider the cyclic ordering at
    each vertex induced by the positive orientation of the plane,
    the result is a ribbon graph whose associated orientable surface
    with boundary is homeomorphic to the Milnor fiber of the
    Brieskorn-Pham singularity `x^p + y^q`. It satisfies that it has
    `\gcd(p,q)` number of  boundary components and genus
    `(pq - p - q - \gcd(p,q) - 2) / 2`.

    INPUT:

    - ``p`` -- a positive integer
    - ``q`` -- a positive integer

    EXAMPLES::

        sage: B23 = RibbonGraph(2,3,bipartite=True); B23; B23.sigma(); B23.rho()
        Ribbon graph of genus 1 and 1 boundary components
        (1,2,3)(4,5,6)(7,8)(9,10)(11,12)
        (1,8)(2,10)(3,12)(4,7)(5,9)(6,11)

        sage: B32 = RibbonGraph(3,2,bipartite=True); B32; B32.sigma(); B32.rho()
        Ribbon graph of genus 1 and 1 boundary components
        (1,2)(3,4)(5,6)(7,8,9)(10,11,12)
        (1,9)(2,12)(3,8)(4,11)(5,7)(6,10)

        sage: B33 = RibbonGraph(3,3,bipartite=True); B33; B33.sigma(); B33.rho()
        Ribbon graph of genus 1 and 3 boundary components
        (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)
        (1,12)(2,15)(3,18)(4,11)(5,14)(6,17)(7,10)(8,13)(9,16)

        sage: B24 = RibbonGraph(2,4,bipartite=True); B24; B24.sigma(); B24.rho()
        Ribbon graph of genus 1 and 2 boundary components
        (1,2,3,4)(5,6,7,8)(9,10)(11,12)(13,14)(15,16)
        (1,10)(2,12)(3,14)(4,16)(5,9)(6,11)(7,13)(8,15)

        sage: B47 = RibbonGraph(4,7, bipartite=True); B47; B47.sigma(); B47.rho()
        Ribbon graph of genus 9 and 1 boundary components
        (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21)(22,23,24,25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)
        (1,32)(2,36)(3,40)(4,44)(5,48)(6,52)(7,56)(8,31)(9,35)(10,39)(11,43)(12,47)(13,51)(14,55)(15,30)(16,34)(17,38)(18,42)(19,46)(20,50)(21,54)(22,29)(23,33)(24,37)(25,41)(26,45)(27,49)(28,53)

    """
    sigma = []
    rho = []
    for i in range(p):
        aux_tuple = [i*q + j + 1  for j in range(q)]
        sigma += [aux_tuple]
    for i in range(q):
        aux_tuple = [p*q + i*p + j +1  for j in range(p)]
        sigma += [aux_tuple]
    for i in range(p*q):
        if (i+1) % q == 0:
            k = q
        elif (i+1) % q != 0:
            k = (i+1) % q
        t = 0
        if (i+1) %  q != 0:
            t = 1
        aux_edge = [i+1, p*q + k*p - ((i+1 + t*q)/q).floor() +1]
        rho += [aux_edge]
    return RibbonGraph(
                       PermutationGroupElement([tuple(x) for x in sigma]), 
                       PermutationGroupElement([tuple(x) for x in rho])
                       )

