r"""
Quiver

A *quiver* is an oriented graph without loops, two-cycles, or multiple
edges. The edges are labelled by pairs `(i,-j)` (with `i` and `j` being
positive integers) such that the matrix `M = (m_{ab})` with
`m_{ab} = i, m_{ba} = -j` for an edge `(i,-j)` between vertices
`a` and `b` is skew-symmetrizable.

.. WARNING:

    This is not the standard definition of a quiver. Normally, in
    cluster algebra theory, a quiver is defined as an oriented graph
    without loops and two-cycles but with multiple edges allowed; the
    edges are unlabelled. This notion of quivers, however, can be seen
    as a particular case of our notion of quivers. Namely, if we have
    a quiver (in the regular sense of this word) with (precisely)
    `i` edges from `a` to `b`, then we represent it by a quiver
    (in our sense of this word) with an edge from `a` to `b` labelled
    by the pair `(i,-i)`.

For the compendium on the cluster algebra and quiver package see ::

    http://arxiv.org/abs/1102.4844.

AUTHORS:

- Gregg Musiker
- Christian Stump

.. seealso:: For mutation types of combinatorial quivers, see :meth:`~sage.combinat.cluster_algebra_quiver.quiver_mutation_type.QuiverMutationType`. Cluster seeds are closely related to :meth:`~sage.combinat.cluster_algebra_quiver.cluster_seed.ClusterSeed`.
"""
#*****************************************************************************
#       Copyright (C) 2011 Gregg Musiker <musiker@math.mit.edu>
#                          Christian Stump <christian.stump@univie.ac.at>
#
#  Distributed under the terms of the GNU General Public License (GPL)
#                  http://www.gnu.org/licenses/
#*****************************************************************************
from __future__ import print_function
from __future__ import absolute_import

from six.moves import range

from sage.structure.sage_object import SageObject
from copy import copy
from sage.misc.all import cached_method
from sage.rings.all import ZZ, CC, infinity
from sage.graphs.all import Graph, DiGraph
from sage.combinat.cluster_algebra_quiver.quiver_mutation_type import QuiverMutationType, QuiverMutationType_Irreducible, QuiverMutationType_Reducible, _edge_list_to_matrix
from sage.combinat.cluster_algebra_quiver.mutation_class import _principal_part, _digraph_mutate, _matrix_to_digraph, _dg_canonical_form, _mutation_class_iter, _digraph_to_dig6, _dig6_to_matrix
from sage.combinat.cluster_algebra_quiver.mutation_type import _connected_mutation_type, _mutation_type_from_data, is_mutation_finite

from sage.misc.decorators import rename_keyword

class ClusterQuiver(SageObject):
    """
    The *quiver* associated to an *exchange matrix*.

    INPUT:

    - ``data`` -- can be any of the following::

        * QuiverMutationType
        * str - a string representing a QuiverMutationType or a common quiver type (see Examples)
        * ClusterQuiver
        * Matrix - a skew-symmetrizable matrix
        * DiGraph - must be the input data for a quiver
        * List of edges - must be the edge list of a digraph for a quiver

    - ``frozen`` -- (default:``None``) sets the number of frozen variables if the input type is a DiGraph, it is ignored otherwise.

    EXAMPLES:

        from a QuiverMutationType::

            sage: Q = ClusterQuiver(['A',5]); Q
            Quiver on 5 vertices of type ['A', 5]

            sage: Q = ClusterQuiver(['B',2]); Q
            Quiver on 2 vertices of type ['B', 2]
            sage: Q2 = ClusterQuiver(['C',2]); Q2
            Quiver on 2 vertices of type ['B', 2]
            sage: MT = Q.mutation_type(); MT.standard_quiver() == Q
            True
            sage: MT = Q2.mutation_type(); MT.standard_quiver() == Q2
            False

            sage: Q = ClusterQuiver(['A',[2,5],1]); Q
            Quiver on 7 vertices of type ['A', [2, 5], 1]

            sage: Q = ClusterQuiver(['A', [5,0],1]); Q
            Quiver on 5 vertices of type ['D', 5]
            sage: Q.is_finite()
            True
            sage: Q.is_acyclic()
            False

            sage: Q = ClusterQuiver(['F', 4, [2,1]]); Q
            Quiver on 6 vertices of type ['F', 4, [1, 2]]
            sage: MT = Q.mutation_type(); MT.standard_quiver() == Q
            False
            sage: dg = Q.digraph(); Q.mutate([2,1,4,0,5,3])
            sage: dg2 = Q.digraph(); dg2.is_isomorphic(dg,edge_labels=True)
            False
            sage: dg2.is_isomorphic(MT.standard_quiver().digraph(),edge_labels=True)
            True

            sage: Q = ClusterQuiver(['G',2, (3,1)]); Q
            Quiver on 4 vertices of type ['G', 2, [1, 3]]
            sage: MT = Q.mutation_type(); MT.standard_quiver() == Q
            False

            sage: Q = ClusterQuiver(['GR',[3,6]]); Q
            Quiver on 4 vertices of type ['D', 4]
            sage: MT = Q.mutation_type(); MT.standard_quiver() == Q
            False

            sage: Q = ClusterQuiver(['GR',[3,7]]); Q
            Quiver on 6 vertices of type ['E', 6]

            sage: Q = ClusterQuiver(['TR',2]); Q
            Quiver on 3 vertices of type ['A', 3]
            sage: MT = Q.mutation_type(); MT.standard_quiver() == Q
            False
            sage: Q.mutate([1,0]); MT.standard_quiver() == Q
            True

            sage: Q = ClusterQuiver(['TR',3]); Q
            Quiver on 6 vertices of type ['D', 6]
            sage: MT = Q.mutation_type(); MT.standard_quiver() == Q
            False

        from a ClusterQuiver::

            sage: Q = ClusterQuiver(['A',[2,5],1]); Q
            Quiver on 7 vertices of type ['A', [2, 5], 1]
            sage: T = ClusterQuiver( Q ); T
            Quiver on 7 vertices of type ['A', [2, 5], 1]

        from a Matrix::

            sage: Q = ClusterQuiver(['A',[2,5],1]); Q
            Quiver on 7 vertices of type ['A', [2, 5], 1]
            sage: T = ClusterQuiver( Q._M ); T
            Quiver on 7 vertices

            sage: Q = ClusterQuiver( matrix([[0,1,-1],[-1,0,1],[1,-1,0],[1,2,3]]) ); Q
            Quiver on 4 vertices with 1 frozen vertex

            sage: Q = ClusterQuiver( matrix([]) ); Q
            Quiver without vertices

        from a DiGraph::

            sage: Q = ClusterQuiver(['A',[2,5],1]); Q
            Quiver on 7 vertices of type ['A', [2, 5], 1]
            sage: T = ClusterQuiver( Q._digraph ); T
            Quiver on 7 vertices

            sage: Q = ClusterQuiver( DiGraph([[1,2],[2,3],[3,4],[4,1]]) ); Q
            Quiver on 4 vertices

        from a List of edges::

            sage: Q = ClusterQuiver(['A',[2,5],1]); Q
            Quiver on 7 vertices of type ['A', [2, 5], 1]
            sage: T = ClusterQuiver( Q._digraph.edges() ); T
            Quiver on 7 vertices

            sage: Q = ClusterQuiver( [[1,2],[2,3],[3,4],[4,1]] ); Q
            Quiver on 4 vertices

    TESTS::

        sage: Q = ClusterQuiver(DiGraph([[1,1]]))
        Traceback (most recent call last):
        ...
        ValueError: The input DiGraph contains a loop

        sage: Q = ClusterQuiver([[1,1]])
        Traceback (most recent call last):
        ...
        ValueError: The input DiGraph contains a loop

        sage: Q = ClusterQuiver(DiGraph([[1, 0],[0,1]]))
        Traceback (most recent call last):
        ...
        ValueError: The input DiGraph contains two-cycles

        sage: Q = ClusterQuiver('whatever')
        Traceback (most recent call last):
        ...
        ValueError: The input data was not recognized.
    """
    def __init__( self, data, frozen=None ):
        """
        TESTS::

            sage: Q = ClusterQuiver(['A',4])
            sage: TestSuite(Q).run()
        """
        from sage.combinat.cluster_algebra_quiver.cluster_seed import ClusterSeed
        from sage.matrix.matrix import Matrix

        # constructs a quiver from a mutation type
        if type( data ) in [QuiverMutationType_Irreducible,QuiverMutationType_Reducible]:
            if frozen is not None:
                print('The input data is a quiver, therefore the additional parameter frozen is ignored.')

            mutation_type = data
            self.__init__( mutation_type.standard_quiver() )

        # constructs a quiver from string representing a mutation type or a common quiver type (see Examples)
        # NOTE: for now, any string representing a *reducible type* is coerced into the standard quiver, but there is now more flexibility in how to input a connected (irreducible) quiver.
        elif type( data ) in [list,tuple] and ( isinstance(data[0], str) or all(type( comp ) in [list,tuple] and isinstance(comp[0], str) for comp in data) ):
            if frozen is not None:
                print('The input data is a quiver, therefore the additional parameter frozen is ignored.')
            mutation_type = QuiverMutationType( data )

            # The command QuiverMutationType_Irreducible (which is not imported globally) already creates the desired digraph as long as we bypass the mutation type checking of QuiverMutationType and format the input appropriately.  Thus we handle several special cases this way.
            if len(data) == 2 and isinstance(data[0], str):
                if data[0] == 'TR' or data[0] == 'GR' or (data[0] == 'C' and data[1] == 2):
                    if data[1] in ZZ:
                        quiv = ClusterQuiver( QuiverMutationType_Irreducible( data[0], data[1] )._digraph )
                        quiv._mutation_type = mutation_type
                        self.__init__( quiv )
                    elif isinstance(data[1], list):
                        quiv = ClusterQuiver( QuiverMutationType_Irreducible( data[0], tuple(data[1]) )._digraph )
                        quiv._mutation_type = mutation_type
                        self.__init__( quiv )
                else:
                    self.__init__( mutation_type.standard_quiver() )
            elif len(data) == 3 and isinstance(data[0], str):
                if (data[0] == 'F' and data[1] == 4 and data[2] == [2,1])   or (data[0] == 'G' and data[1] == 2 and data[2] == [3,1]):
                    quiv = ClusterQuiver( QuiverMutationType_Irreducible( data[0], data[1], tuple(data[2]) )._digraph )
                    quiv._mutation_type = mutation_type
                    self.__init__( quiv )
                elif (data[0] == 'F' and data[1] == 4 and data[2] == (2,1) )   or (data[0] == 'G' and data[1] == 2 and data[2] == (3,1) ):
                    quiv = ClusterQuiver( QuiverMutationType_Irreducible( data[0], data[1], data[2] )._digraph )
                    quiv._mutation_type = mutation_type
                    self.__init__( quiv )
                elif data[0] == 'A' and isinstance(data[1], list) and data[2] == 1:
                    if len(data[1]) == 2 and min(data[1]) == 0:
                        quiv = ClusterQuiver( QuiverMutationType_Irreducible( data[0], tuple(data[1]), data[2] )._digraph )
                        quiv._mutation_type = mutation_type
                        self.__init__( quiv )
                    else:
                        self.__init__( mutation_type.standard_quiver() )

                elif data[0] == 'A' and isinstance(data[1], tuple) and data[2] == 1:
                    if len(data[1]) == 2 and min(data[1]) == 0:
                        quiv = ClusterQuiver( QuiverMutationType_Irreducible( data[0], data[1], data[2] )._digraph )
                        quiv._mutation_type = mutation_type
                        self.__init__( quiv )
                    else:
                        self.__init__( mutation_type.standard_quiver() )

                else:
                    self.__init__( mutation_type.standard_quiver() )
            else:
                self.__init__( mutation_type.standard_quiver() )

         # constructs a quiver from a cluster seed
        elif isinstance(data, ClusterSeed):
            self.__init__( data.quiver() )

        # constructs a quiver from a quiver
        elif isinstance(data, ClusterQuiver):
            if frozen is not None:
                print('The input data is a quiver, therefore the additional parameter frozen is ignored.')

            self._M = copy(data._M)
            self._M.set_immutable()
            self._n = data._n
            self._m = data._m
            self._digraph = copy( data._digraph )
            self._vertex_dictionary = {}
            self._mutation_type = data._mutation_type
            self._description = data._description

        # constructs a quiver from a matrix
        elif isinstance(data, Matrix):
            if not _principal_part(data).is_skew_symmetrizable( positive=True ):
                raise ValueError('The principal part of the matrix data must be skew-symmetrizable.')
            if frozen is not None:
                print('The input data is a matrix, therefore the additional parameter frozen is ignored.')

            self._M = copy(data).sparse_matrix()
            self._M.set_immutable()
            self._n = n = self._M.ncols()
            self._m = m = self._M.nrows() - self._n
            self._digraph = _matrix_to_digraph( self._M )
            self._vertex_dictionary = {}
            self._mutation_type = None
            if n+m == 0:
                self._description = 'Quiver without vertices'
            elif n+m == 1:
                self._description = 'Quiver on %d vertex' %(n+m)
            else:
                self._description = 'Quiver on %d vertices' %(n+m)

        # constructs a quiver from a digraph
        elif isinstance(data, DiGraph):
            if frozen is None:
                frozen = 0
            elif not ZZ(frozen) == frozen:
                raise ValueError("The optional argument frozen (=%s) must be an integer."%frozen)
            m = self._m = frozen
            n = self._n = data.order() - m
            dg = copy( data )
            edges = data.edges(labels=False)
            if any( (a,a) in edges for a in data.vertices() ):
                raise ValueError("The input DiGraph contains a loop")
            if any( (b,a) in edges for (a,b) in edges ):
                raise ValueError("The input DiGraph contains two-cycles")
            if not set(dg.vertices()) == set(range(n+m)):
                dg.relabel()
            if dg.has_multiple_edges():
                multi_edges = {}
                for v1,v2,label in dg.multiple_edges():
                    if label not in ZZ:
                        raise ValueError("The input DiGraph contains multiple edges labeled by non-integers")
                    elif (v1,v2) in multi_edges:
                        multi_edges[(v1,v2)] += label
                    else:
                        multi_edges[(v1,v2)] = label
                    dg.delete_edge(v1,v2)
                dg.add_edges( [ (v1,v2,multi_edges[(v1,v2)]) for v1,v2 in multi_edges ] )
            for edge in dg.edge_iterator():
                if edge[0] >= n and edge[1] >= n:
                    raise ValueError("The input digraph contains edges within the frozen vertices")
                if edge[2] is None:
                    dg.set_edge_label( edge[0], edge[1], (1,-1) )
                    edge = (edge[0],edge[1],(1,-1))
                elif edge[2] in ZZ:
                    dg.set_edge_label( edge[0], edge[1], (edge[2],-edge[2]) )
                    edge = (edge[0],edge[1],(edge[2],-edge[2]))
                elif isinstance(edge[2], list) and len(edge[2]) != 2:
                    raise ValueError("The input digraph contains an edge with the wrong type of list as a label.")
                elif isinstance(edge[2], list) and len(edge[2]) == 2:
                    dg.set_edge_label( edge[0], edge[1], (edge[2][0], edge[2][1]))
                    edge = (edge[0],edge[1],(edge[2][0],edge[2][1]))
                elif ( edge[0] >= n or edge[1] >= n ) and not edge[2][0] == - edge[2][1]:
                    raise ValueError("The input digraph contains an edge to or from a frozen vertex which is not skew-symmetric.")
                if edge[2][0] < 0:
                    raise ValueError("The input digraph contains an edge of the form (a,-b) with negative a.")

            M = _edge_list_to_matrix( dg.edge_iterator(), n, m )
            if not _principal_part(M).is_skew_symmetrizable( positive=True ):
                raise ValueError("The input digraph must be skew-symmetrizable")
            self._digraph = dg
            self._vertex_dictionary = {}
            self._M = M
            self._M.set_immutable()
            if n+m == 0:
                self._description = 'Quiver without vertices'
            elif n+m == 1:
                self._description = 'Quiver on %d vertex' %(n+m)
            else:
                self._description = 'Quiver on %d vertices' %(n+m)
            self._mutation_type = None

        # if data is a list of edges, the appropriate digraph is constructed.

        elif isinstance(data,list) and all(isinstance(x,(list,tuple)) for x in data ):
            dg = DiGraph( data )
            self.__init__(data=dg, frozen=frozen )

        # otherwise, an error is raised
        else:
            raise ValueError("The input data was not recognized.")

    def __eq__(self, other):
        """
        Returns ``True`` if ``self`` and ``other`` represent the same quiver.

        EXAMPLES::

            sage: Q = ClusterQuiver(['A',5])
            sage: T = Q.mutate( 2, inplace=False )
            sage: Q.__eq__( T )
            False
            sage: T.mutate( 2 )
            sage: Q.__eq__( T )
            True
        """
        return isinstance(other, ClusterQuiver) and self._M == other._M

    def __hash__(self):
        """
        Return a hash of ``self``.

        EXAMPLES::

            sage: Q = ClusterQuiver(['A',5])
            sage: hash(Q)  # indirect doctest
            16
        """
        return hash(self._M)

    def _repr_(self):
        """
        Returns the description of ``self``.

        EXAMPLES::

            sage: Q = ClusterQuiver(['A',5])
            sage: Q._repr_()
            "Quiver on 5 vertices of type ['A', 5]"
        """
        name = self._description
        if self._mutation_type:
            if isinstance(self._mutation_type, str):
                name += ' of ' + self._mutation_type
            else:
                name += ' of type ' + str(self._mutation_type)
        if self._m == 1:
            name += ' with %s frozen vertex'%self._m
        elif self._m > 1:
            name += ' with %s frozen vertices'%self._m
        return name

    def plot(self, circular=True, center=(0, 0), directed=True, mark=None,
             save_pos=False, greens=[]):
        """
        Return the plot of the underlying digraph of ``self``.

        INPUT:

        - ``circular`` -- (default: ``True``) if ``True``, the circular plot
          is chosen, otherwise >>spring<< is used.
        - ``center`` -- (default:(0,0)) sets the center of the circular plot,
          otherwise it is ignored.
        - ``directed`` -- (default: ``True``) if ``True``, the directed
          version is shown, otherwise the undirected.
        - ``mark`` -- (default: ``None``) if set to i, the vertex i is
          highlighted.
        - ``save_pos`` -- (default: ``False``) if ``True``, the positions
          of the vertices are saved.
        - ``greens`` -- (default: []) if set to a list, will display the green
          vertices as green

        EXAMPLES::

            sage: Q = ClusterQuiver(['A',5])
            sage: pl = Q.plot()
            sage: pl = Q.plot(circular=True)
        """
        from sage.plot.colors import rainbow
        from sage.graphs.graph_generators import GraphGenerators
        from sage.all import e, pi, I
        graphs = GraphGenerators()
        # returns positions for graph vertices on two concentric cycles with radius 1 and 2
        def _graphs_concentric_circles(n, m):
            g1 = graphs.CycleGraph(n).get_pos()
            g2 = graphs.CycleGraph(m).get_pos()
            for i in g2:
                z = CC(g2[i])*e**(-pi*I/(2*m))
                g2[i] = (z.real_part(),z.imag_part())
            for i in range(m):
                g1[n+i] = [2*g2[i][0], 2*g2[i][1]]
            return g1

        n, m = self._n, self._m
        colors = rainbow(11)
        color_dict = { colors[0]:[], colors[1]:[], colors[6]:[], colors[5]:[] }
        
        if directed:
            dg = DiGraph( self._digraph )
        else:
            dg = Graph( self._digraph )
        for edge in dg.edges():
            v1,v2,(a,b) = edge
            if v1 < n and v2 < n:
                if (a,b) == (1,-1):
                    color_dict[ colors[0] ].append((v1,v2))
                else:
                    color_dict[ colors[6] ].append((v1,v2))
            else:
                if (a,b) == (1,-1):
                    color_dict[ colors[1] ].append((v1,v2))
                else:
                    color_dict[ colors[5] ].append((v1,v2))
            if a == -b:
                if a == 1:
                    dg.set_edge_label( v1,v2,'' )
                else:
                    dg.set_edge_label( v1,v2,a )
        if mark is not None:
            if mark < n:
                partition = (list(range(mark)) + list(range(mark+1,n)),
                             list(range(n,n+m)),[mark])
            elif mark > n:
                partition = (list(range(n)),list(range(n,mark))+list(range(mark+1,n+m)),[mark])
            else:
                raise ValueError("The given mark is not a vertex of self.")
        else:
            nr = list(range(n))
            mr = list(range(n, n + m))
            for i in greens:
                if i < n:
                    nr.remove(i)
                else:
                    mr.remove(i)
            partition = (nr,mr,greens)
            
        # fix labels
        for i in range(2):
            for p in list(enumerate(partition[i])):
                key = p[0]
                part = p[1]
                if part in self._vertex_dictionary:
                    partition[0][key]= self._vertex_dictionary[part]
        
        vertex_color_dict = {}
        vertex_color_dict[ colors[0] ] = partition[0]
        vertex_color_dict[ colors[6] ] = partition[1]
        vertex_color_dict[ colors[4] ] = partition[2]

        options = {
            'graph_border' : True,
            'edge_colors': color_dict,
            'vertex_colors': vertex_color_dict,
            'edge_labels' : True,
            'vertex_labels': True,
        }
        if circular:
            pp = _graphs_concentric_circles( n, m )
            for v in pp:
                pp[v] = (pp[v][0]+center[0],pp[v][1]+center[1])
            options[ 'pos' ] = pp
        return dg.plot( **options )

    def show(self, fig_size=1, circular=False, directed=True, mark=None, save_pos=False, greens=[]):
        """
        Show the plot of the underlying digraph of ``self``.

        INPUT:

        - ``fig_size`` -- (default: 1) factor by which the size of the plot
          is multiplied.
        - ``circular`` -- (default: False) if True, the circular plot is
          chosen, otherwise >>spring<< is used.
        - ``directed`` -- (default: True) if True, the directed version is
          shown, otherwise the undirected.
        - ``mark`` -- (default: None) if set to i, the vertex i is highlighted.
        - ``save_pos`` -- (default:False) if True, the positions of the
          vertices are saved.
        - ``greens`` -- (default:[]) if set to a list, will display the green
          vertices as green

        TESTS::

            sage: Q = ClusterQuiver(['A',5])
            sage: Q.show() # long time
        """
        n, m = self._n, self._m
        plot = self.plot( circular=circular, directed=directed, mark=mark, save_pos=save_pos, greens=greens)
        if circular:
            plot.show( figsize=[fig_size*3*(n+m)/4+1,fig_size*3*(n+m)/4+1] )
        else:
            plot.show( figsize=[fig_size*n+1,fig_size*n+1] )

    def interact(self, fig_size=1, circular=True):
        """
        Only in notebook mode. Starts an interactive window for cluster seed mutations.

        INPUT:

        - ``fig_size`` -- (default: 1) factor by which the size of the plot is multiplied.
        - ``circular`` -- (default: False) if True, the circular plot is chosen, otherwise >>spring<< is used.

        TESTS::

            sage: Q = ClusterQuiver(['A',4])
            sage: Q.interact() # long time
            'The interactive mode only runs in the Sage notebook.'
        """
        from sage.plot.plot import EMBEDDED_MODE
        from sagenb.notebook.interact import interact, selector
        from sage.misc.all import html,latex

        if not EMBEDDED_MODE:
            return "The interactive mode only runs in the Sage notebook."
        else:
            seq = []
            sft = [True]
            sss = [True]
            ssm = [True]
            ssl = [True]
            @interact
            def player(k=selector(values=list(range(self._n)),nrows = 1,label='Mutate at: '), show_seq=("Mutation sequence:", True), show_matrix=("B-Matrix:", True), show_lastmutation=("Show last mutation:", True) ):
                ft,ss,sm,sl = sft.pop(), sss.pop(), ssm.pop(), ssl.pop()
                if ft:
                    self.show(fig_size=fig_size, circular=circular)
                elif show_seq is not ss or show_matrix is not sm or show_lastmutation is not sl:
                    if seq and show_lastmutation:
                        self.show(fig_size=fig_size, circular=circular, mark=seq[len(seq)-1])
                    else:
                        self.show(fig_size=fig_size, circular=circular )
                else:
                    self.mutate(k)
                    seq.append(k)
                    if not show_lastmutation:
                        self.show(fig_size=fig_size, circular=circular)
                    else:
                        self.show(fig_size=fig_size, circular=circular,mark=k)
                sft.append(False)
                sss.append(show_seq)
                ssm.append(show_matrix)
                ssl.append(show_lastmutation)
                if show_seq: html( "Mutation sequence: $" + str( [ seq[i] for i in range(len(seq)) ] ).strip('[]') + "$" )
                if show_matrix:
                    html( "B-Matrix:" )
                    m = self._M
                    #m = matrix(range(1,self._n+1),sparse=True).stack(m)
                    m = latex(m)
                    m = m.split('(')[1].split('\\right')[0]
                    html( "$ $" )
                    html( "$\\begin{align*} " + m + "\\end{align*}$" )
                    #html( "$" + m + "$" )
                    html( "$ $" )

    def save_image(self,filename,circular=False):
        """
        Saves the plot of the underlying digraph of ``self``.

        INPUT:

        - ``filename`` -- the filename the image is saved to.
        - ``circular`` -- (default: False) if True, the circular plot is chosen, otherwise >>spring<< is used.

        EXAMPLES::

            sage: Q = ClusterQuiver(['F',4,[1,2]])
            sage: Q.save_image(os.path.join(SAGE_TMP, 'sage.png'))
        """
        graph_plot = self.plot( circular=circular )
        graph_plot.save( filename=filename )

    def qmu_save(self,filename=None):
        """
        Saves a .qmu file of ``self`` that can then be opened in Bernhard Keller's Quiver Applet.

        INPUT:

        - ``filename`` -- the filename the image is saved to.

        If a filename is not specified, the default name is from_sage.qmu in the current sage directory.

        EXAMPLES::

            sage: Q = ClusterQuiver(['F',4,[1,2]])
            sage: Q.qmu_save(os.path.join(SAGE_TMP, 'sage.qmu'))

        Make sure we can save quivers with `m != n` frozen variables, see :trac:`14851`::

            sage: S=ClusterSeed(['A',3])
            sage: T1=S.principal_extension()
            sage: Q=T1.quiver()
            sage: Q.qmu_save(os.path.join(SAGE_TMP, 'sage.qmu'))
        """
        M = self.b_matrix()
        if self.m() > 0:
            from sage.matrix.constructor import Matrix
            from sage.matrix.constructor import block_matrix
            M1 = M.matrix_from_rows(list(range(self.n())))
            M2 = M.matrix_from_rows(list(range(self.n(),self.n()+self.m())))
            M3 = Matrix(self.m(),self.m())
            M = block_matrix([[M1,-M2.transpose()],[M2,M3]])
        dg = self.digraph()
        dg.plot(save_pos=True)
        PP = dg.get_pos()
        m = M.ncols()
        if filename is None:
            filename = 'from_sage.qmu'
        try:
            self._default_filename = filename
        except AttributeError:
            pass
        if filename[-4:] != '.qmu':
            filename = filename + '.qmu'
        myfile = open(filename, 'w')
        myfile.write('//Number of points'); myfile.write('\n')
        myfile.write(str(m)); myfile.write('\n')
        myfile.write('//Vertex radius'); myfile.write('\n')
        myfile.write(str(9)); myfile.write('\n')
        myfile.write('//Labels shown'); myfile.write('\n')
        myfile.write(str(1)); myfile.write('\n')
        myfile.write('//Matrix'); myfile.write('\n')
        myfile.write(str(m)); myfile.write(' '); myfile.write(str(m)); myfile.write('\n')
        for i in range(m):
            for j in range(m):
                myfile.write(str(M[i,j])); myfile.write(' ')
            myfile.write('\n')
        myfile.write('//Points'); myfile.write('\n')
        for i in range(m):
            myfile.write(str(9)); myfile.write(' '); myfile.write(str(100*PP[i][0])); myfile.write(' ');
            myfile.write(str(100*PP[i][1]));
            if i > self.n()-1:
                myfile.write(' '); myfile.write(str(1))
            myfile.write('\n')
        myfile.write('//Historycounter'); myfile.write('\n')
        myfile.write(str(-1)); myfile.write('\n')
        myfile.write('//History'); myfile.write('\n'); myfile.write('\n')
        myfile.write('//Cluster is null');
        myfile.close()

    def b_matrix(self):
        """
        Returns the b-matrix of ``self``.

        EXAMPLES::

            sage: ClusterQuiver(['A',4]).b_matrix()
            [ 0  1  0  0]
            [-1  0 -1  0]
            [ 0  1  0  1]
            [ 0  0 -1  0]

            sage: ClusterQuiver(['B',4]).b_matrix()
            [ 0  1  0  0]
            [-1  0 -1  0]
            [ 0  1  0  1]
            [ 0  0 -2  0]

            sage: ClusterQuiver(['D',4]).b_matrix()
            [ 0  1  0  0]
            [-1  0 -1 -1]
            [ 0  1  0  0]
            [ 0  1  0  0]

            sage: ClusterQuiver(QuiverMutationType([['A',2],['B',2]])).b_matrix()
            [ 0  1  0  0]
            [-1  0  0  0]
            [ 0  0  0  1]
            [ 0  0 -2  0]
        """
        return copy(self._M)

    def digraph(self):
        """
        Returns the underlying digraph of ``self``.

        EXAMPLES::

            sage: ClusterQuiver(['A',1]).digraph()
            Digraph on 1 vertex
            sage: ClusterQuiver(['A',1]).digraph().vertices()
            [0]
            sage: ClusterQuiver(['A',1]).digraph().edges()
            []

            sage: ClusterQuiver(['A',4]).digraph()
            Digraph on 4 vertices
            sage: ClusterQuiver(['A',4]).digraph().edges()
            [(0, 1, (1, -1)), (2, 1, (1, -1)), (2, 3, (1, -1))]

            sage: ClusterQuiver(['B',4]).digraph()
            Digraph on 4 vertices
            sage: ClusterQuiver(['A',4]).digraph().edges()
            [(0, 1, (1, -1)), (2, 1, (1, -1)), (2, 3, (1, -1))]

            sage: ClusterQuiver(QuiverMutationType([['A',2],['B',2]])).digraph()
            Digraph on 4 vertices

            sage: ClusterQuiver(QuiverMutationType([['A',2],['B',2]])).digraph().edges()
            [(0, 1, (1, -1)), (2, 3, (1, -2))]
        """
        return copy( self._digraph )

    def mutation_type(self):
        """
        Returns the mutation type of ``self``.

        Returns the mutation_type of each connected component of self if it can be determined,
        otherwise, the mutation type of this component is set to be unknown.

        The mutation types of the components are ordered by vertex labels.

        If you do many type recognitions, you should consider to save
        exceptional mutation types using
        `meth`:sage.combinat.cluster_algebra_quiver.quiver_mutation_type.save_exceptional_data

        WARNING:

        - All finite types can be detected,
        - All affine types can be detected, EXCEPT affine type D (the algorithm is not yet implemented)
        - All exceptional types can be detected.

        EXAMPLES::

            sage: ClusterQuiver(['A',4]).mutation_type()
            ['A', 4]
            sage: ClusterQuiver(['A',(3,1),1]).mutation_type()
            ['A', [1, 3], 1]
            sage: ClusterQuiver(['C',2]).mutation_type()
            ['B', 2]
            sage: ClusterQuiver(['B',4,1]).mutation_type()
            ['BD', 4, 1]

        finite types::

            sage: Q = ClusterQuiver(['A',5])
            sage: Q._mutation_type = None
            sage: Q.mutation_type()
            ['A', 5]

            sage: Q = ClusterQuiver([(0,1),(1,2),(2,3),(3,4)])
            sage: Q.mutation_type()
            ['A', 5]

        affine types::

            sage: Q = ClusterQuiver(['E',8,[1,1]]); Q
            Quiver on 10 vertices of type ['E', 8, [1, 1]]
            sage: Q._mutation_type = None; Q
            Quiver on 10 vertices
            sage: Q.mutation_type() # long time
            ['E', 8, [1, 1]]

        the not yet working affine type D (unless user has saved small classical quiver data)::

            sage: Q = ClusterQuiver(['D',4,1])
            sage: Q._mutation_type = None
            sage: Q.mutation_type() # todo: not implemented
            ['D', 4, 1]

        the exceptional types::

            sage: Q = ClusterQuiver(['X',6])
            sage: Q._mutation_type = None
            sage: Q.mutation_type() # long time
            ['X', 6]

        examples from page 8 of Keller's article "Cluster algebras, quiver representations
        and triangulated categories" (arXiv:0807.1960)::

            sage: dg = DiGraph(); dg.add_edges([(9,0),(9,4),(4,6),(6,7),(7,8),(8,3),(3,5),(5,6),(8,1),(2,3)])
            sage: ClusterQuiver( dg ).mutation_type() # long time
            ['E', 8, [1, 1]]

            sage: dg = DiGraph( { 0:[3], 1:[0,4], 2:[0,6], 3:[1,2,7], 4:[3,8], 5:[2], 6:[3,5], 7:[4,6], 8:[7] } )
            sage: ClusterQuiver( dg ).mutation_type() # long time
            ['E', 8, 1]

            sage: dg = DiGraph( { 0:[3,9], 1:[0,4], 2:[0,6], 3:[1,2,7], 4:[3,8], 5:[2], 6:[3,5], 7:[4,6], 8:[7], 9:[1] } )
            sage: ClusterQuiver( dg ).mutation_type() # long time
            ['E', 8, [1, 1]]

        infinite types::

            sage: Q = ClusterQuiver(['GR',[4,9]])
            sage: Q._mutation_type = None
            sage: Q.mutation_type()
            'undetermined infinite mutation type'

        reducible types::

            sage: Q = ClusterQuiver([['A', 3], ['B', 3]])
            sage: Q._mutation_type = None
            sage: Q.mutation_type()
            [ ['A', 3], ['B', 3] ]

            sage: Q = ClusterQuiver([['A', 3], ['T', [4,4,4]]])
            sage: Q._mutation_type = None
            sage: Q.mutation_type()
            [['A', 3], 'undetermined infinite mutation type']

            sage: Q = ClusterQuiver([['A', 3], ['B', 3], ['T', [4,4,4]]])
            sage: Q._mutation_type = None
            sage: Q.mutation_type()
            [['A', 3], ['B', 3], 'undetermined infinite mutation type']

            sage: Q = ClusterQuiver([[0,1,2],[1,2,2],[2,0,2],[3,4,1],[4,5,1]])
            sage: Q.mutation_type()
            ['undetermined finite mutation type', ['A', 3]]

        TESTS::

            sage: Q = ClusterQuiver(matrix([[0, 3], [-1, 0], [1, 0], [0, 1]]))
            sage: Q.mutation_type()
            ['G', 2]
            sage: Q = ClusterQuiver(matrix([[0, -1, -1, 1, 0], [1, 0, 1, 0, 1], [1, -1, 0, -1, 0], [-1, 0, 1, 0, 1], [0, -1, 0, -1, 0], [0, 1, 0, -1, -1], [0, 1, -1, 0, 0]]))
            sage: Q.mutation_type()
            'undetermined infinite mutation type'
        """
        # checking if the mutation type is known already
        if self._mutation_type is None:
            # checking mutation type only for the principal part
            if self._m > 0:
                dg = self._digraph.subgraph(list(range(self._n)))
            else:
                dg = self._digraph

            # checking the type for each connected component
            mutation_type = []
            connected_components = sorted(dg.connected_components())
            for component in connected_components:
                # constructing the digraph for this component
                dg_component = dg.subgraph( component )
                dg_component.relabel()
                # turning dg_component into a canonical form
                iso, orbits = _dg_canonical_form( dg_component, dg_component.num_verts(), 0 )
                # turning dg_component into a canonical form
                dig6 = _digraph_to_dig6( dg_component, hashable=True )
                # and getting the corresponding matrix
                M = _dig6_to_matrix(dig6)

                # checking if this quiver is mutation infinite
                is_finite, path = is_mutation_finite(M)
                if is_finite is False:
                    mut_type_part = 'undetermined infinite mutation type'
                else:
                    # checking if this quiver is in the database
                    mut_type_part = _mutation_type_from_data( dg_component.order(), dig6, compute_if_necessary=False )
                    # checking if the algorithm can determine the mutation type
                    if mut_type_part == 'unknown':
                        mut_type_part = _connected_mutation_type(dg_component)
                    # checking if this quiver is of exceptional type by computing the exceptional mutation classes
                    if mut_type_part == 'unknown':
                        mut_type_part = _mutation_type_from_data(dg_component.order(), dig6, compute_if_necessary=True)
                    if mut_type_part == 'unknown':
                        mut_type_part = 'undetermined finite mutation type'
                mutation_type.append( mut_type_part )

            # the empty quiver case
            if len( mutation_type ) == 0:
                Warning('Quiver has no vertices')
                mutation_type = None
            # the connected quiver case
            elif len( mutation_type ) == 1:
                mutation_type = mutation_type[0]
            # the reducible quiver case
            elif len( mutation_type ) > 1:
                if any( isinstance(mut_type_part, str) for mut_type_part in mutation_type ):
                    pass
                else:
                    mutation_type = QuiverMutationType( mutation_type )
            self._mutation_type = mutation_type
        return self._mutation_type

    def n(self):
        """
        Returns the number of free vertices of ``self``.

        EXAMPLES::

            sage: ClusterQuiver(['A',4]).n()
            4
            sage: ClusterQuiver(['A',(3,1),1]).n()
            4
            sage: ClusterQuiver(['B',4]).n()
            4
            sage: ClusterQuiver(['B',4,1]).n()
            5
        """
        return self._n

    def m(self):
        """
        Returns the number of frozen vertices of ``self``.

        EXAMPLES::

            sage: Q = ClusterQuiver(['A',4])
            sage: Q.m()
            0

            sage: T = ClusterQuiver( Q.digraph().edges(), frozen=1 )
            sage: T.n()
            3
            sage: T.m()
            1
        """
        return self._m

    @rename_keyword(deprecation=21111, certify='certificate')
    def canonical_label( self, certificate=False ):
        """
        Returns the canonical labelling of ``self``, see
        :meth:`sage.graphs.graph.GenericGraph.canonical_label`.

        INPUT:

        - ``certificate`` -- (default: False) if True, the dictionary from ``self.vertices()`` to the vertices of the returned quiver is returned as well.

        EXAMPLES::

            sage: Q = ClusterQuiver(['A',4]); Q.digraph().edges()
            [(0, 1, (1, -1)), (2, 1, (1, -1)), (2, 3, (1, -1))]

            sage: T = Q.canonical_label(); T.digraph().edges()
            [(0, 3, (1, -1)), (1, 2, (1, -1)), (1, 3, (1, -1))]

            sage: T,iso = Q.canonical_label(certificate=True); T.digraph().edges(); iso
            [(0, 3, (1, -1)), (1, 2, (1, -1)), (1, 3, (1, -1))]
            {0: 0, 1: 3, 2: 1, 3: 2}

            sage: Q = ClusterQuiver(QuiverMutationType([['B',2],['A',1]])); Q
            Quiver on 3 vertices of type [ ['B', 2], ['A', 1] ]

            sage: Q.canonical_label()
            Quiver on 3 vertices of type [ ['A', 1], ['B', 2] ]

            sage: Q.canonical_label(certificate=True)
            (Quiver on 3 vertices of type [ ['A', 1], ['B', 2] ], {0: 1, 1: 2, 2: 0})

        TESTS::

            sage: Q = ClusterQuiver(['A',4])
            sage: _,iso = Q.canonical_label(certify=True); iso
            doctest:...: DeprecationWarning: use the option 'certificate' instead of 'certify'
            See http://trac.sagemath.org/21111 for details.
            {0: 0, 1: 3, 2: 1, 3: 2}
        """
        n = self._n
        m = self._m

        # computing the canonical form respecting the frozen variables
        dg = copy( self._digraph )
        iso, orbits = _dg_canonical_form( dg, n, m )
        Q = ClusterQuiver( dg )
        # getting the new ordering for the mutation type if necessary
        if self._mutation_type:
            if dg.is_connected():
                Q._mutation_type = self._mutation_type
            else:
                CC = sorted( self._digraph.connected_components() )
                CC_new = sorted( zip( [ sorted( [ iso[i] for i in L ] ) for L in CC ], list(range(len(CC))) ) )
                comp_iso = [ L[1] for L in CC_new ]
                Q._mutation_type = []
                for i in range( len( CC_new ) ):
                    Q._mutation_type.append( copy( self._mutation_type.irreducible_components()[ comp_iso[i] ] ) )
                Q._mutation_type = QuiverMutationType( Q._mutation_type )
        if certificate:
            return Q, iso
        else:
            return Q

    def is_acyclic(self):
        """
        Returns true if ``self`` is acyclic.

        EXAMPLES::

            sage: ClusterQuiver(['A',4]).is_acyclic()
            True

            sage: ClusterQuiver(['A',[2,1],1]).is_acyclic()
            True

            sage: ClusterQuiver([[0,1],[1,2],[2,0]]).is_acyclic()
            False
        """
        return self._digraph.is_directed_acyclic()

    def is_bipartite(self,return_bipartition=False):
        """
        Returns true if ``self`` is bipartite.

        EXAMPLES::

            sage: ClusterQuiver(['A',[3,3],1]).is_bipartite()
            True

            sage: ClusterQuiver(['A',[4,3],1]).is_bipartite()
            False
        """
        dg = copy( self._digraph )
        dg.delete_vertices(list(range(self._n, self._n + self._m)))
        innie = dg.in_degree()
        outie = dg.out_degree()
        is_bip = sum( [ innie[i]*outie[i] for i in range(len(innie)) ] ) == 0
        if not is_bip:
            return False
        else:
            if not return_bipartition:
                return True
            else:
                g = dg.to_undirected()
                return g.bipartite_sets()

    def exchangeable_part(self):
        """
        Returns the restriction to the principal part (i.e. exchangeable part) of ``self``, the subquiver obtained by deleting the frozen vertices of ``self``.

        EXAMPLES::

            sage: Q = ClusterQuiver(['A',4])
            sage: T = ClusterQuiver( Q.digraph().edges(), frozen=1 )
            sage: T.digraph().edges()
            [(0, 1, (1, -1)), (2, 1, (1, -1)), (2, 3, (1, -1))]

            sage: T.exchangeable_part().digraph().edges()
            [(0, 1, (1, -1)), (2, 1, (1, -1))]

            sage: Q2 = Q.principal_extension()
            sage: Q3 = Q2.principal_extension()
            sage: Q2.exchangeable_part() == Q3.exchangeable_part()
            True
        """
        dg = DiGraph( self._digraph )
        dg.delete_vertices(list(range(self._n, self._n + self._m)))
        Q = ClusterQuiver( dg )
        Q._mutation_type = self._mutation_type
        return Q

    def principal_extension(self, inplace=False):
        """
        Returns the principal extension of ``self``, adding n frozen vertices to any previously frozen vertices. I.e., the quiver obtained by adding an outgoing edge to every mutable vertex of ``self``.

        EXAMPLES::

            sage: Q = ClusterQuiver(['A',2]); Q
            Quiver on 2 vertices of type ['A', 2]
            sage: T = Q.principal_extension(); T
            Quiver on 4 vertices of type ['A', 2] with 2 frozen vertices
            sage: T2 = T.principal_extension(); T2
            Quiver on 6 vertices of type ['A', 2] with 4 frozen vertices
            sage: Q.digraph().edges()
            [(0, 1, (1, -1))]
            sage: T.digraph().edges()
            [(0, 1, (1, -1)), (2, 0, (1, -1)), (3, 1, (1, -1))]
            sage: T2.digraph().edges()
            [(0, 1, (1, -1)), (2, 0, (1, -1)), (3, 1, (1, -1)), (4, 0, (1, -1)), (5, 1, (1, -1))]
        """
        dg = DiGraph( self._digraph )
        dg.add_edges( [(self._n+self._m+i,i) for i in range(self._n)] )
        Q = ClusterQuiver( dg, frozen=self._m+self._n )
        Q._mutation_type = self._mutation_type
        if inplace:
            self.__init__(Q)
        else:
            return Q


    def first_sink(self):
        r"""
        Return the first vertex of ``self`` that is a sink

        EXAMPLES::

            sage: Q = ClusterQuiver(['A',5]);
            sage: Q.mutate([1,2,4,3,2]);
            sage: Q.first_sink()
            0
        """
        sinks = self.digraph().sinks()

        if len(sinks) > 0:
            return sinks[0]
        return None


    def sinks(self):
        r"""
        Return all vertices of ``self`` that are sinks

        EXAMPLES::

            sage: Q = ClusterQuiver(['A',5]);
            sage: Q.mutate([1,2,4,3,2]);
            sage: Q.sinks()
            [0, 2]

            sage: Q = ClusterQuiver(['A',5])
            sage: Q.mutate([2,1,3,4,2])
            sage: Q.sinks()
            [3]
        """
        return self.digraph().sinks()

    def first_source(self):
        r"""
        Return the first vertex of ``self`` that is a source

        EXAMPLES::

            sage: Q = ClusterQuiver(['A',5])
            sage: Q.mutate([2,1,3,4,2])
            sage: Q.first_source()
            1
        """
        sources = self.digraph().sources()

        if len(sources) > 0:
            return sources[0]
        return None

    def sources(self):
        r"""
        Returns all vertices of ``self`` that are sources

        EXAMPLES::

            sage: Q = ClusterQuiver(['A',5]);
            sage: Q.mutate([1,2,4,3,2]);
            sage: Q.sources()
            []

            sage: Q = ClusterQuiver(['A',5])
            sage: Q.mutate([2,1,3,4,2])
            sage: Q.sources()
            [1]
        """
        return self.digraph().sources()

    def mutate(self, data, inplace=True):
        """
        Mutates ``self`` at a sequence of vertices.

        INPUT:

        - ``sequence`` -- a vertex of ``self``, an iterator of vertices of ``self``,
          a function which takes in the ClusterQuiver and returns a vertex or an iterator of vertices,
          or a string of the parameter wanting to be called on ClusterQuiver that will return a vertex or 
          an iterator of vertices.
        - ``inplace`` -- (default: True) if False, the result is returned, otherwise ``self`` is modified.

        EXAMPLES::

            sage: Q = ClusterQuiver(['A',4]); Q.b_matrix()
            [ 0  1  0  0]
            [-1  0 -1  0]
            [ 0  1  0  1]
            [ 0  0 -1  0]

            sage: Q.mutate(0); Q.b_matrix()
            [ 0 -1  0  0]
            [ 1  0 -1  0]
            [ 0  1  0  1]
            [ 0  0 -1  0]

            sage: T = Q.mutate(0, inplace=False); T
            Quiver on 4 vertices of type ['A', 4]

            sage: Q.mutate(0)
            sage: Q == T
            True

            sage: Q.mutate([0,1,0])
            sage: Q.b_matrix()
            [ 0 -1  1  0]
            [ 1  0  0  0]
            [-1  0  0  1]
            [ 0  0 -1  0]

            sage: Q = ClusterQuiver(QuiverMutationType([['A',1],['A',3]]))
            sage: Q.b_matrix()
            [ 0  0  0  0]
            [ 0  0  1  0]
            [ 0 -1  0 -1]
            [ 0  0  1  0]

            sage: T = Q.mutate(0,inplace=False)
            sage: Q == T
            True
            
            sage: Q = ClusterQuiver(['A',3]); Q.b_matrix()
            [ 0  1  0]
            [-1  0 -1]
            [ 0  1  0]
            sage: Q.mutate('first_sink'); Q.b_matrix()
            [ 0 -1  0]
            [ 1  0  1]
            [ 0 -1  0]
            sage: Q.mutate('first_source'); Q.b_matrix()
            [ 0  1  0]
            [-1  0 -1]
            [ 0  1  0]



        TESTS::

            sage: Q = ClusterQuiver(['A',4]); Q.mutate(0,1)
            Traceback (most recent call last):
            ...
            ValueError: The second parameter must be boolean.  To mutate at a sequence of length 2, input it as a list.

            sage: Q = ClusterQuiver(['A',4]); Q.mutate(0,0)
            Traceback (most recent call last):
            ...
            ValueError: The second parameter must be boolean.  To mutate at a sequence of length 2, input it as a list.
        """

        n = self._n
        m = self._m
        dg = self._digraph
        V = list(range(n))

        # If we get a string, execute as a function
        if isinstance(data, str):
            data = getattr(self, data)()

        # If we get a function, execute it
        if hasattr(data, '__call__'):
            # function should return either integer or sequence
            data = data(self)

        if data is None:
            raise ValueError('Not mutating: No vertices given.')


        if data in V:
            seq = [data]
        else:
            seq = data
        if isinstance(seq, tuple):
            seq = list( seq )
        if not isinstance(seq, list):
            raise ValueError('The quiver can only be mutated at a vertex or at a sequence of vertices')
        if not isinstance(inplace, bool):
            raise ValueError('The second parameter must be boolean.  To mutate at a sequence of length 2, input it as a list.')
        if any( v not in V for v in seq ):
            v = filter( lambda v: v not in V, seq )[0]
            raise ValueError('The quiver cannot be mutated at the vertex %s'%v)

        for v in seq:
            dg = _digraph_mutate( dg, v, n, m )
        M = _edge_list_to_matrix( dg.edge_iterator(), n, m )
        if inplace:
            self._M = M
            self._M.set_immutable()
            self._digraph = dg
        else:
            Q = ClusterQuiver( M )
            Q._mutation_type = self._mutation_type
            return Q

    def mutation_sequence(self, sequence, show_sequence=False, fig_size=1.2 ):
        """
        Returns a list containing the sequence of quivers obtained from ``self`` by a sequence of mutations on vertices.

        INPUT:

        - ``sequence`` -- a list or tuple of vertices of ``self``.
        - ``show_sequence`` -- (default: False) if True, a png containing the mutation sequence is shown.
        - ``fig_size`` -- (default: 1.2) factor by which the size of the sequence is expanded.

        EXAMPLES::

            sage: Q = ClusterQuiver(['A',4])
            sage: seq = Q.mutation_sequence([0,1]); seq
            [Quiver on 4 vertices of type ['A', 4], Quiver on 4 vertices of type ['A', 4], Quiver on 4 vertices of type ['A', 4]]
            sage: [T.b_matrix() for T in seq]
            [
            [ 0  1  0  0]  [ 0 -1  0  0]  [ 0  1 -1  0]
            [-1  0 -1  0]  [ 1  0 -1  0]  [-1  0  1  0]
            [ 0  1  0  1]  [ 0  1  0  1]  [ 1 -1  0  1]
            [ 0  0 -1  0], [ 0  0 -1  0], [ 0  0 -1  0]
            ]
        """
        from sage.plot.plot import Graphics
        from sage.plot.text import text
        n = self._n
        m = self._m
        if m == 0:
            width_factor = 3
            fig_size = fig_size*2*n/3
        else:
            width_factor = 6
            fig_size = fig_size*4*n/3
        V = list(range(n))

        if isinstance(sequence, tuple):
            sequence = list( sequence )
        if not isinstance(sequence, list):
            raise ValueError('The quiver can only be mutated at a vertex or at a sequence of vertices')
        if any( v not in V for v in sequence ):
            v = filter( lambda v: v not in V, sequence )[0]
            raise ValueError('The quiver can only be mutated at the vertex %s'%v )

        quiver = copy( self )
        quiver_sequence = []
        quiver_sequence.append( copy( quiver ) )

        for v in sequence:
            quiver.mutate( v )
            quiver_sequence.append( copy( quiver ) )

        if show_sequence:
            def _plot_arrow( v, k, center=(0,0) ):
                return text("$\longleftrightarrow$",(center[0],center[1]), fontsize=25) + text("$\mu_"+str(v)+"$",(center[0],center[1]+0.15), fontsize=15) \
                    + text("$"+str(k)+"$",(center[0],center[1]-0.2), fontsize=15)
            plot_sequence = [ quiver_sequence[i].plot( circular=True, center=(i*width_factor,0) ) for i in range(len(quiver_sequence)) ]
            arrow_sequence = [ _plot_arrow( sequence[i],i+1,center=((i+0.5)*width_factor,0) ) for i in range(len(sequence)) ]
            sequence = []
            for i in range( len( plot_sequence ) ):
                if i < len( arrow_sequence ):
                    sequence.append( plot_sequence[i] + arrow_sequence[i] )
                else:
                    sequence.append( plot_sequence[i] )
            plot_obj = Graphics()
            for elem in sequence:
                plot_obj += elem
            plot_obj.show(axes=False, figsize=[fig_size*len(quiver_sequence),fig_size])
        return quiver_sequence

    def reorient( self, data ):
        """
        Reorients ``self`` with respect to the given total order, or with respect to an iterator of edges in ``self`` to be reverted.

        WARNING::

            This operation might change the mutation type of ``self``.

        INPUT:

        - ``data`` -- an iterator defining a total order on ``self.vertices()``, or an iterator of edges in ``self`` to be reoriented.

        EXAMPLES::

            sage: Q = ClusterQuiver(['A',(2,3),1])
            sage: Q.mutation_type()
            ['A', [2, 3], 1]

            sage: Q.reorient([(0,1),(1,2),(2,3),(3,4)])
            sage: Q.mutation_type()
            ['D', 5]

            sage: Q.reorient([0,1,2,3,4])
            sage: Q.mutation_type()
            ['A', [1, 4], 1]
        """
        if all( 0 <= i and i < self._n + self._m for i in data ) and len( set( data ) ) == self._n+self._m :
            dg_new = DiGraph()
            for edge in self._digraph.edges():
                if data.index( edge[0] ) < data.index( edge[1] ):
                    dg_new.add_edge( edge[0],edge[1],edge[2] )
                else:
                    dg_new.add_edge( edge[1],edge[0],edge[2] )
            self._digraph = dg_new
            self._M = _edge_list_to_matrix( dg_new.edges(), self._n, self._m )
            self._M.set_immutable()
            self._mutation_type = None
        elif all( type(edge) in [list,tuple] and len(edge)==2 for edge in data ):
            edges = self._digraph.edges(labels=False)
            for edge in data:
                if (edge[1],edge[0]) in edges:
                    label = self._digraph.edge_label(edge[1],edge[0])
                    self._digraph.delete_edge(edge[1],edge[0])
                    self._digraph.add_edge(edge[0],edge[1],label)
            self._M = _edge_list_to_matrix( self._digraph.edges(), self._n, self._m )
            self._M.set_immutable()
            self._mutation_type = None
        else:
            raise ValueError('The order is no total order on the vertices of the quiver or a list of edges to be oriented.')

    def mutation_class_iter( self, depth=infinity, show_depth=False, return_paths=False, data_type="quiver", up_to_equivalence=True, sink_source=False ):
        """
        Returns an iterator for the mutation class of self together with certain constrains.

        INPUT:

        - ``depth`` -- (default: infinity) integer, only quivers with distance at most depth from self are returned.
        - ``show_depth`` -- (default: False) if True, the actual depth of the mutation is shown.
        - ``return_paths`` -- (default: False) if True, a shortest path of mutation sequences from self to the given quiver is returned as well.
        - ``data_type`` -- (default: "quiver") can be one of the following::

            * "quiver"
            * "matrix"
            * "digraph"
            * "dig6"
            * "path"

        - ``up_to_equivalence`` -- (default: True) if True, only one quiver for each graph-isomorphism class is recorded.
        - ``sink_source`` -- (default: False) if True, only mutations at sinks and sources are applied.

        EXAMPLES::

            sage: Q = ClusterQuiver(['A',3])
            sage: it = Q.mutation_class_iter()
            sage: for T in it: print(T)
            Quiver on 3 vertices of type ['A', 3]
            Quiver on 3 vertices of type ['A', 3]
            Quiver on 3 vertices of type ['A', 3]
            Quiver on 3 vertices of type ['A', 3]

            sage: it = Q.mutation_class_iter(depth=1)
            sage: for T in it: print(T)
            Quiver on 3 vertices of type ['A', 3]
            Quiver on 3 vertices of type ['A', 3]
            Quiver on 3 vertices of type ['A', 3]

            sage: it = Q.mutation_class_iter(show_depth=True)
            sage: for T in it: pass
            Depth: 0     found: 1          Time: ... s
            Depth: 1     found: 3          Time: ... s
            Depth: 2     found: 4          Time: ... s

            sage: it = Q.mutation_class_iter(return_paths=True)
            sage: for T in it: print(T)
            (Quiver on 3 vertices of type ['A', 3], [])
            (Quiver on 3 vertices of type ['A', 3], [1])
            (Quiver on 3 vertices of type ['A', 3], [0])
            (Quiver on 3 vertices of type ['A', 3], [0, 1])

            sage: it = Q.mutation_class_iter(up_to_equivalence=False)
            sage: for T in it: print(T)
            Quiver on 3 vertices of type ['A', 3]
            Quiver on 3 vertices of type ['A', 3]
            Quiver on 3 vertices of type ['A', 3]
            Quiver on 3 vertices of type ['A', 3]
            Quiver on 3 vertices of type ['A', 3]
            Quiver on 3 vertices of type ['A', 3]
            Quiver on 3 vertices of type ['A', 3]
            Quiver on 3 vertices of type ['A', 3]
            Quiver on 3 vertices of type ['A', 3]
            Quiver on 3 vertices of type ['A', 3]
            Quiver on 3 vertices of type ['A', 3]
            Quiver on 3 vertices of type ['A', 3]
            Quiver on 3 vertices of type ['A', 3]
            Quiver on 3 vertices of type ['A', 3]

            sage: it = Q.mutation_class_iter(return_paths=True,up_to_equivalence=False)
            sage: for T in it: print(T)
            (Quiver on 3 vertices of type ['A', 3], [])
            (Quiver on 3 vertices of type ['A', 3], [2])
            (Quiver on 3 vertices of type ['A', 3], [1])
            (Quiver on 3 vertices of type ['A', 3], [0])
            (Quiver on 3 vertices of type ['A', 3], [2, 1])
            (Quiver on 3 vertices of type ['A', 3], [0, 1])
            (Quiver on 3 vertices of type ['A', 3], [0, 1, 2])
            (Quiver on 3 vertices of type ['A', 3], [0, 1, 0])
            (Quiver on 3 vertices of type ['A', 3], [2, 1, 2])
            (Quiver on 3 vertices of type ['A', 3], [2, 1, 0])
            (Quiver on 3 vertices of type ['A', 3], [2, 1, 0, 2])
            (Quiver on 3 vertices of type ['A', 3], [2, 1, 0, 1])
            (Quiver on 3 vertices of type ['A', 3], [2, 1, 2, 1])
            (Quiver on 3 vertices of type ['A', 3], [2, 1, 2, 0])

            sage: Q = ClusterQuiver(['A',3])
            sage: it = Q.mutation_class_iter(data_type='path')
            sage: for T in it: print(T)
            []
            [1]
            [0]
            [0, 1]

            sage: Q = ClusterQuiver(['A',3])
            sage: it = Q.mutation_class_iter(return_paths=True,data_type='matrix')
            sage: next(it)
            (
            [ 0  0  1]
            [ 0  0  1]
            [-1 -1  0], []
            )

        """
        if data_type == 'path':
            return_paths = False
        if data_type == "dig6":
            return_dig6 = True
        else:
            return_dig6 = False
        dg = DiGraph( self._digraph )
        MC_iter = _mutation_class_iter( dg, self._n, self._m, depth=depth, return_dig6=return_dig6, show_depth=show_depth, up_to_equivalence=up_to_equivalence, sink_source=sink_source )
        for data in MC_iter:
            if data_type == "quiver":
                next_element = ClusterQuiver( data[0], frozen=self._m )
                next_element._mutation_type = self._mutation_type
            elif data_type == "matrix":
                next_element = ClusterQuiver( data[0], frozen=self._m )._M
            elif data_type == "digraph":
                next_element = data[0]
            elif data_type == "dig6":
                next_element = data[0]
            elif data_type == "path":
                next_element = data[1]
            else:
                raise ValueError("The parameter for data_type was "
                                 "not recognized.")
            if return_paths:
                yield (next_element, data[1])
            else:
                yield next_element

    def mutation_class( self, depth=infinity, show_depth=False, return_paths=False, data_type="quiver", up_to_equivalence=True, sink_source=False ):
        """
        Returns the mutation class of self together with certain constrains.

        INPUT:

        - ``depth`` -- (default: infinity) integer, only seeds with distance at most depth from self are returned.
        - ``show_depth`` -- (default: False) if True, the actual depth of the mutation is shown.
        - ``return_paths`` -- (default: False) if True, a shortest path of mutation sequences from self to the given quiver is returned as well.
        - ``data_type`` -- (default: "quiver") can be one of the following::

            * "quiver" -- the quiver is returned
            * "dig6" -- the dig6-data is returned
            * "path" -- shortest paths of mutation sequences from self are returned

        - ``sink_source`` -- (default: False) if True, only mutations at sinks and sources are applied.

        EXAMPLES::

            sage: Q = ClusterQuiver(['A',3])
            sage: Ts = Q.mutation_class()
            sage: for T in Ts: print(T)
            Quiver on 3 vertices of type ['A', 3]
            Quiver on 3 vertices of type ['A', 3]
            Quiver on 3 vertices of type ['A', 3]
            Quiver on 3 vertices of type ['A', 3]

            sage: Ts = Q.mutation_class(depth=1)
            sage: for T in Ts: print(T)
            Quiver on 3 vertices of type ['A', 3]
            Quiver on 3 vertices of type ['A', 3]
            Quiver on 3 vertices of type ['A', 3]

            sage: Ts = Q.mutation_class(show_depth=True)
            Depth: 0     found: 1          Time: ... s
            Depth: 1     found: 3          Time: ... s
            Depth: 2     found: 4          Time: ... s

            sage: Ts = Q.mutation_class(return_paths=True)
            sage: for T in Ts: print(T)
            (Quiver on 3 vertices of type ['A', 3], [])
            (Quiver on 3 vertices of type ['A', 3], [1])
            (Quiver on 3 vertices of type ['A', 3], [0])
            (Quiver on 3 vertices of type ['A', 3], [0, 1])

            sage: Ts = Q.mutation_class(up_to_equivalence=False)
            sage: for T in Ts: print(T)
            Quiver on 3 vertices of type ['A', 3]
            Quiver on 3 vertices of type ['A', 3]
            Quiver on 3 vertices of type ['A', 3]
            Quiver on 3 vertices of type ['A', 3]
            Quiver on 3 vertices of type ['A', 3]
            Quiver on 3 vertices of type ['A', 3]
            Quiver on 3 vertices of type ['A', 3]
            Quiver on 3 vertices of type ['A', 3]
            Quiver on 3 vertices of type ['A', 3]
            Quiver on 3 vertices of type ['A', 3]
            Quiver on 3 vertices of type ['A', 3]
            Quiver on 3 vertices of type ['A', 3]
            Quiver on 3 vertices of type ['A', 3]
            Quiver on 3 vertices of type ['A', 3]

            sage: Ts = Q.mutation_class(return_paths=True,up_to_equivalence=False)
            sage: for T in Ts: print(T)
            (Quiver on 3 vertices of type ['A', 3], [])
            (Quiver on 3 vertices of type ['A', 3], [2])
            (Quiver on 3 vertices of type ['A', 3], [1])
            (Quiver on 3 vertices of type ['A', 3], [0])
            (Quiver on 3 vertices of type ['A', 3], [2, 1])
            (Quiver on 3 vertices of type ['A', 3], [0, 1])
            (Quiver on 3 vertices of type ['A', 3], [0, 1, 2])
            (Quiver on 3 vertices of type ['A', 3], [0, 1, 0])
            (Quiver on 3 vertices of type ['A', 3], [2, 1, 2])
            (Quiver on 3 vertices of type ['A', 3], [2, 1, 0])
            (Quiver on 3 vertices of type ['A', 3], [2, 1, 0, 2])
            (Quiver on 3 vertices of type ['A', 3], [2, 1, 0, 1])
            (Quiver on 3 vertices of type ['A', 3], [2, 1, 2, 1])
            (Quiver on 3 vertices of type ['A', 3], [2, 1, 2, 0])

            sage: Ts = Q.mutation_class(show_depth=True)
            Depth: 0     found: 1          Time: ... s
            Depth: 1     found: 3          Time: ... s
            Depth: 2     found: 4          Time: ... s

            sage: Ts = Q.mutation_class(show_depth=True, up_to_equivalence=False)
            Depth: 0     found: 1          Time: ... s
            Depth: 1     found: 4          Time: ... s
            Depth: 2     found: 6          Time: ... s
            Depth: 3     found: 10        Time: ... s
            Depth: 4     found: 14        Time: ... s

        TESTS::

            sage: all( len(ClusterQuiver(['A',n]).mutation_class()) == ClusterQuiver(['A',n]).mutation_type().class_size() for n in [2..6])
            True

            sage: all( len(ClusterQuiver(['B',n]).mutation_class()) == ClusterQuiver(['B',n]).mutation_type().class_size() for n in [2..6])
            True
        """
        if depth is infinity and not self.is_mutation_finite():
            raise ValueError('The mutation class can - for infinite mutation types - only be computed up to a given depth')
        return [ Q for Q in self.mutation_class_iter( depth=depth, show_depth=show_depth, return_paths=return_paths, data_type=data_type, up_to_equivalence=up_to_equivalence, sink_source=sink_source ) ]

    def is_finite( self ):
        """
        Returns True if self is of finite type.

        EXAMPLES::

            sage: Q = ClusterQuiver(['A',3])
            sage: Q.is_finite()
            True
            sage: Q = ClusterQuiver(['A',[2,2],1])
            sage: Q.is_finite()
            False
            sage: Q = ClusterQuiver([['A',3],['B',3]])
            sage: Q.is_finite()
            True
            sage: Q = ClusterQuiver(['T',[4,4,4]])
            sage: Q.is_finite()
            False
            sage: Q = ClusterQuiver([['A',3],['T',[4,4,4]]])
            sage: Q.is_finite()
            False
            sage: Q = ClusterQuiver([['A',3],['T',[2,2,3]]])
            sage: Q.is_finite()
            True
            sage: Q = ClusterQuiver([['A',3],['D',5]])
            sage: Q.is_finite()
            True
            sage: Q = ClusterQuiver([['A',3],['D',5,1]])
            sage: Q.is_finite()
            False

            sage: Q = ClusterQuiver([[0,1,2],[1,2,2],[2,0,2]])
            sage: Q.is_finite()
            False

            sage: Q = ClusterQuiver([[0,1,2],[1,2,2],[2,0,2],[3,4,1],[4,5,1]])
            sage: Q.is_finite()
            False
        """
        mt = self.mutation_type()
        if type( mt ) in [QuiverMutationType_Irreducible, QuiverMutationType_Reducible] and mt.is_finite():
            return True
        else:
            return False

    def is_mutation_finite( self, nr_of_checks=None, return_path=False ):
        """
        Uses a non-deterministic method by random mutations in various directions. Can result in a wrong answer.

        INPUT:

        - ``nr_of_checks`` -- (default: None) number of mutations applied. Standard is 500*(number of vertices of self).
        - ``return_path`` -- (default: False) if True, in case of self not being mutation finite, a path from self to a quiver with an edge label (a,-b) and a*b > 4 is returned.

        ALGORITHM:

        A quiver is mutation infinite if and only if every edge label (a,-b) satisfy a*b > 4.
        Thus, we apply random mutations in random directions

        EXAMPLES::

            sage: Q = ClusterQuiver(['A',10])
            sage: Q._mutation_type = None
            sage: Q.is_mutation_finite()
            True

            sage: Q = ClusterQuiver([(0,1),(1,2),(2,3),(3,4),(4,5),(5,6),(6,7),(7,8),(2,9)])
            sage: Q.is_mutation_finite()
            False
        """
        if self._n <= 2:
            is_finite = True
            path = None
        elif not return_path and self._mutation_type == 'undetermined infinite mutation type':
            is_finite = False
        elif type( self._mutation_type ) in [QuiverMutationType_Irreducible, QuiverMutationType_Reducible] and self._mutation_type.is_mutation_finite():
            is_finite = True
            path = None
        elif not return_path and type( self._mutation_type ) in [QuiverMutationType_Irreducible, QuiverMutationType_Reducible] and not self._mutation_type.is_mutation_finite():
            is_finite = False
        else:
            # turning dg_component into a canonical form
            dig6 = _digraph_to_dig6(self.digraph())
            # and getting the corresponding matrix
            M = _dig6_to_matrix(dig6)

            is_finite, path = is_mutation_finite(M,nr_of_checks=nr_of_checks)
        if return_path:
            return is_finite, path
        else:
            return is_finite

    def number_of_edges(self):
        r"""
        Return the total number of edges on the quiver

        Note: This only works with non-valued quivers. If used on a
        non-valued quiver then the positive value is taken to be the number of edges added

        OUTPUT:
        
        Returns an integer of the number of edges

        EXAMPLES::
        
            sage: S = ClusterQuiver(['A',4]); S.number_of_edges()
            3
            
            sage: S = ClusterQuiver(['B',4]); S.number_of_edges()
            3

        """

        digraph_edges = self.digraph().edges()

        total_edges = 0
        for edge in digraph_edges:
            total_edges += edge[2][0]

        return total_edges

    def relabel(self, relabelling, inplace=True):
        r"""
        Returns the quiver after doing a relabelling
        
        Will relabel the vertices of the quiver
        
        INPUT:
        
        - ``relabelling`` -- Dictionary of labels to move around
        - ``inplace`` -- (default:True) if True, will return a duplicate of the quiver
        
        EXAMPLES::
        
            sage: S = ClusterQuiver(['A',4]).relabel({1:'5',2:'go'})
        
        """
        if inplace:
            quiver = self
        else:
            quiver = ClusterQuiver(self)
        quiver._digraph.relabel(relabelling)
        quiver._vertex_dictionary = relabelling
        return quiver
        
    def d_vector_fan(self):
        r"""
        Return the d-vector fan associated with the quiver.

        It is the fan whose maximal cones are generated by the
        d-matrices of the clusters.

        This is a complete simplicial fan (and even smooth when the
        initial quiver is acyclic). It only makes sense for quivers of
        finite type.

        EXAMPLES::

            sage: Fd = ClusterQuiver([[1,2]]).d_vector_fan(); Fd
            Rational polyhedral fan in 2-d lattice N
            sage: Fd.ngenerating_cones()
            5

            sage: Fd = ClusterQuiver([[1,2],[2,3]]).d_vector_fan(); Fd
            Rational polyhedral fan in 3-d lattice N
            sage: Fd.ngenerating_cones()
            14
            sage: Fd.is_smooth()
            True

            sage: Fd = ClusterQuiver([[1,2],[2,3],[3,1]]).d_vector_fan(); Fd
            Rational polyhedral fan in 3-d lattice N
            sage: Fd.ngenerating_cones()
            14
            sage: Fd.is_smooth()
            False

        TESTS::

            sage: ClusterQuiver(['A',[2,2],1]).d_vector_fan()
            Traceback (most recent call last):
            ...
            ValueError: only makes sense for quivers of finite type
        """
        from .cluster_seed import ClusterSeed
        from sage.geometry.fan import Fan
        from sage.geometry.cone import Cone

        if not(self.is_finite()):
            raise ValueError('only makes sense for quivers of finite type')
        seed = ClusterSeed(self)
        return Fan([Cone(s.d_matrix().columns())
                    for s in seed.mutation_class()])

    def g_vector_fan(self):
        r"""
        Return the g-vector fan associated with the quiver.

        It is the fan whose maximal cones are generated by the
        g-matrices of the clusters.

        This is a complete simplicial fan. It is only supported for
        quivers of finite type.

        EXAMPLES::

            sage: Fg = ClusterQuiver([[1,2]]).g_vector_fan(); Fg
            Rational polyhedral fan in 2-d lattice N
            sage: Fg.ngenerating_cones()
            5

            sage: Fg = ClusterQuiver([[1,2],[2,3]]).g_vector_fan(); Fg
            Rational polyhedral fan in 3-d lattice N
            sage: Fg.ngenerating_cones()
            14
            sage: Fg.is_smooth()
            True

            sage: Fg = ClusterQuiver([[1,2],[2,3],[3,1]]).g_vector_fan(); Fg
            Rational polyhedral fan in 3-d lattice N
            sage: Fg.ngenerating_cones()
            14
            sage: Fg.is_smooth()
            True

        TESTS::

            sage: ClusterQuiver(['A',[2,2],1]).g_vector_fan()
            Traceback (most recent call last):
            ...
            ValueError: only supported for quivers of finite type
        """
        from .cluster_seed import ClusterSeed
        from sage.geometry.fan import Fan
        from sage.geometry.cone import Cone

        if not(self.is_finite()):
            raise ValueError('only supported for quivers of finite type')
        seed = ClusterSeed(self).principal_extension()
        return Fan([Cone(s.g_matrix().columns())
                    for s in seed.mutation_class()])

