def set_edge_map(self,edge_map):
"""
Sets the edge map of ``self``.
``edge_map`` is anything that is accepted by
``Wordmorphism(edge_map)``, the image of the inverse letters
will be calculated: they need not be explicit in ``edge_map``,
only one of the images of each pair [letter,inverse(letter)]
need to be given by ``edge_map``. Images of ``edge_map`` need
not be reduced.
"""
A=self.domain().alphabet()
tmp_map=WordMorphism(edge_map)
m={}
for a in tmp_map._domain.alphabet():
m[a]=self._codomain.reduce_path(tmp_map.image(a))
m[A.inverse_letter(a)]=self._codomain.reverse_path(m[a])
self._edge_map=WordMorphism(m)
self._vertex_map=None
def set_edge_map(self, edge_map):
"""
Sets the edge map of ``self``.
``edge_map`` is anything that is accepted by
``Wordmorphism(edge_map)``, the image of the inverse letters
will be calculated: they need not be explicit in ``edge_map``,
only one of the images of each pair [letter,inverse(letter)]
need to be given by ``edge_map``. Images of ``edge_map`` need
not be reduced.
"""
A = self.domain().alphabet()
tmp_map = WordMorphism(edge_map)
m = {}
for a in tmp_map._domain.alphabet():
m[a] = self._codomain.reduce_path(tmp_map.image(a))
m[A.inverse_letter(a)] = self._codomain.reverse_path(m[a])
self._edge_map = WordMorphism(m)
self._vertex_map = None
def set_edge_map(self, edge_map):
"""
Sets the edge map of ``self``.
``edge_map`` is anything that is accepted by
``WordMorphism(edge_map)``, the image of the inverse letters
will be calculated: they need not be explicit in ``edge_map``,
only one of the images of each pair [letter,inverse(letter)]
need to be given by ``edge_map``. Images of ``edge_map`` need
not be reduced.
INPUT:
- ``edge_map`` -- anything which is accepted by ``WordMorphism(edge_map)``
EXAMPLES::
sage: from train_track import *
sage: from train_track.inverse_graph import GraphWithInverses
sage: from train_track.graph_map import GraphMap
sage: G = GraphWithInverses([[0,0,'a'],[0,1,'b'],[1,1,'c']])
sage: A = AlphabetWithInverses(2)
sage: H = GraphWithInverses.rose_graph(A)
sage: f = GraphMap(G,H,"a->ab,b->b,c->B")
sage: f.set_edge_map('a->b,b->,c->b')
sage: print(f)
Graph map:
a: 0->0, b: 0->1, c: 1->1
a: 0->0, b: 0->0
edge map: a->b, b->, c->b
"""
A = self.domain().alphabet()
tmp_map = WordMorphism(edge_map)
m = {}
for a in tmp_map._domain.alphabet():
m[a] = self._codomain.reduce_path(tmp_map.image(a))
m[A.inverse_letter(a)] = self._codomain.reverse_path(m[a])
self._edge_map = WordMorphism(m)
self._vertex_map = None
def set_edge_map(self, edge_map):
"""
Sets the edge map of ``self``.
``edge_map`` is anything that is accepted by
``WordMorphism(edge_map)``, the image of the inverse letters
will be calculated: they need not be explicit in ``edge_map``,
only one of the images of each pair [letter,inverse(letter)]
need to be given by ``edge_map``. Images of ``edge_map`` need
not be reduced.
INPUT:
- ``edge_map`` -- anything which is accepted by ``WordMorphism(edge_map)``
EXAMPLES::
sage: from train_track import *
sage: from train_track.inverse_graph import GraphWithInverses
sage: from train_track.graph_map import GraphMap
sage: G = GraphWithInverses([[0,0,'a'],[0,1,'b'],[1,1,'c']])
sage: A = AlphabetWithInverses(2)
sage: H = GraphWithInverses.rose_graph(A)
sage: f = GraphMap(G,H,"a->ab,b->b,c->B")
sage: f.set_edge_map('a->b,b->,c->b')
sage: print(f)
Graph map:
a: 0->0, b: 0->1, c: 1->1
a: 0->0, b: 0->0
edge map: a->b, b->, c->b
"""
A = self.domain().alphabet()
tmp_map = WordMorphism(edge_map)
m = {}
for a in tmp_map._domain.alphabet():
m[a] = self._codomain.reduce_path(tmp_map.image(a))
m[A.inverse_letter(a)] = self._codomain.reverse_path(m[a])
self._edge_map = WordMorphism(m)
self._vertex_map = None
class GraphMap():
"""
A GraphMap is a map from a Graph to another. It maps a vertex to
a vertex and an edge to an edge-path. It respects incidence
relation. The inverse of an edge is send to the reverse path.
AUTHORS:
- Thierry Coulbois (2013-05-16): beta.0 version
"""
def __init__(self, *args):
"""
The following forms are accepted:
- ``GraphMap(f)`` where ``f`` is a ``GraphMap``.
- ``GraphMap(domain, codomain, edge_map, vertex_map=None)`` where
``domain`` and ``codomain`` are ``GraphWithInverses`` and
``edge_map`` is anything accepted by
``WordMorphism(edge_map)`` with domain alphabet an
AlphabetWithInverses (note that only one image of the pair
(a,inverse_letter(a)) needs to be defined for each letter).
EXAMPLES::
sage: from train_track import *
sage: from train_track.inverse_graph import GraphWithInverses
sage: from train_track.graph_map import GraphMap
sage: G = GraphWithInverses([[0,0,'a'],[0,1,'b'],[1,1,'c']])
sage: A = AlphabetWithInverses(2)
sage: H = GraphWithInverses.rose_graph(A)
sage: print(GraphMap(G,H,"a->ab,b->b,c->B"))
Graph map:
a: 0->0, b: 0->1, c: 1->1
a: 0->0, b: 0->0
edge map: a->ab, b->b, c->B
"""
if isinstance(args[0], GraphMap):
self._domain = args[0]._domain
self._codomain = args[0]._codomain
self._edge_map = args[0]._edge_map
self._vertex_map = args[0]._vertex_map
else:
self._domain = args[0]
self._codomain = args[1]
self.set_edge_map(args[2])
if len(args) > 3:
self._vertex_map = args[3]
def __call__(self, argument):
"""
Applies ``self`` to ``argument`` which is either a vertex
of ``self`` or an edge path.
INPUT:
-``argument`` a vertex of ``self`` or a edge path
OUTPUT:
A vertex of ``self`` or an edge path
Applies ``self`` to ``argument`` which is either .
.. SEEALSO::
To compute the image of a letter of the alphabet use
:meth:`train_track.graph_map.GraphMap.image(a)``.
EXAMPLES::
sage: from train_track import *
sage: from train_track.inverse_graph import GraphWithInverses
sage: from train_track.graph_map import GraphMap
sage: G = GraphWithInverses([[0,0,'a'],[0,1,'b'],[1,1,'c']])
sage: A = AlphabetWithInverses(2)
sage: H = GraphWithInverses.rose_graph(A)
sage: f = GraphMap(G,H,"a->ab,b->b,c->B")
sage: f('abc')
word: ab
"""
if self._domain.has_vertex(argument):
if self._vertex_map is None:
self.update_vertex_map()
return self._vertex_map[argument]
else:
return self._codomain.reduce_path(self._edge_map(argument))
def __mul__(self, other):
"""
Compose ``self`` with ``other``.
INPUT:
- ``other`` -- other GraphMap to compute multiplication
OUTPUT:
GraphMap obtains with other domain as first
input and ``self`` codomain for second input and vertex_map is
construct applying ``self`` to ``other`` edge_map image
EXAMPLES::
sage: from train_track import *
sage: from train_track.inverse_graph import GraphWithInverses
sage: from train_track.graph_map import GraphMap
sage: G = GraphWithInverses([[0,0,'a'],[0,1,'b'],[1,1,'c']])
sage: A = AlphabetWithInverses(2)
sage: H = GraphWithInverses.rose_graph(A)
sage: f = GraphMap(G,H,"a->ab,b->b,c->B")
sage: g = GraphMap(H,H,"a->aba,b->ba")
sage: print(g * f)
Graph map:
a: 0->0, b: 0->1, c: 1->1
a: 0->0, b: 0->0
edge map: a->ababa, b->ba, c->AB
"""
A = other._domain.alphabet()
result_map = {}
for a in A.positive_letters():
result_map[a] = self(other._edge_map.image(a))
return GraphMap(other._domain, self._codomain, result_map)
def __repr__(self):
"""
String represetation of ``self``.
OUTPUT:
a string representation of ``self``
EXAMPLES::
sage: from train_track import *
sage: from train_track.inverse_graph import GraphWithInverses
sage: from train_track.graph_map import GraphMap
sage: G = GraphWithInverses([[0,0,'a'],[0,1,'b'],[1,1,'c']])
sage: A = AlphabetWithInverses(2)
sage: H = GraphWithInverses.rose_graph(A)
sage: GraphMap(G,H,"a->ab,b->b,c->B").__repr__()
'Graph map:\na: 0->0, b: 0->1, c: 1->1\na: 0->0, b: 0->0\nedge map: a->ab, b->b, c->B'
"""
result = "Graph map:\n" + self._domain.__str__() + "\n"
result += self._codomain.__str__() + "\n"
result += "edge map: "
for a in self._domain._alphabet.positive_letters():
result += a + "->" + self.image(a).__str__() + ", "
result = result[:-2] # + "\n"
if self._vertex_map is not None:
result = result + "\n"
result = result + "vertex map: " +\
self._vertex_map.__str__() # + "\n"
return result
def domain(self):
"""
Domain of ``self``: this is a graph.
OUTPUT:
Domain of ``self`` which is a GrapMap
EXAMPLES::
sage: from train_track import *
sage: from train_track.inverse_graph import GraphWithInverses
sage: from train_track.graph_map import GraphMap
sage: G = GraphWithInverses([[0,0,'a'],[0,1,'b'],[1,1,'c']])
sage: A = AlphabetWithInverses(2)
sage: H = GraphWithInverses.rose_graph(A)
sage: f = GraphMap(G,H,"a->ab,b->b,c->B")
sage: print(f.domain())
a: 0->0, b: 0->1, c: 1->1
"""
return self._domain
def codomain(self):
"""
Codomain of ``self``: this is a graph.
OUTPUT:
Codomain of ``self`` which is a GrapMap
EXAMPLES::
sage: from train_track import *
sage: from train_track.inverse_graph import GraphWithInverses
sage: from train_track.graph_map import GraphMap
sage: G = GraphWithInverses([[0,0,'a'],[0,1,'b'],[1,1,'c']])
sage: A = AlphabetWithInverses(2)
sage: H = GraphWithInverses.rose_graph(A)
sage: f = GraphMap(G,H,"a->ab,b->b,c->B")
sage: print(f.codomain())
a: 0->0, b: 0->0
"""
return self._codomain
def set_edge_map(self, edge_map):
"""
Sets the edge map of ``self``.
``edge_map`` is anything that is accepted by
``WordMorphism(edge_map)``, the image of the inverse letters
will be calculated: they need not be explicit in ``edge_map``,
only one of the images of each pair [letter,inverse(letter)]
need to be given by ``edge_map``. Images of ``edge_map`` need
not be reduced.
INPUT:
- ``edge_map`` -- anything which is accepted by ``WordMorphism(edge_map)``
EXAMPLES::
sage: from train_track import *
sage: from train_track.inverse_graph import GraphWithInverses
sage: from train_track.graph_map import GraphMap
sage: G = GraphWithInverses([[0,0,'a'],[0,1,'b'],[1,1,'c']])
sage: A = AlphabetWithInverses(2)
sage: H = GraphWithInverses.rose_graph(A)
sage: f = GraphMap(G,H,"a->ab,b->b,c->B")
sage: f.set_edge_map('a->b,b->,c->b')
sage: print(f)
Graph map:
a: 0->0, b: 0->1, c: 1->1
a: 0->0, b: 0->0
edge map: a->b, b->, c->b
"""
A = self.domain().alphabet()
tmp_map = WordMorphism(edge_map)
m = {}
for a in tmp_map._domain.alphabet():
m[a] = self._codomain.reduce_path(tmp_map.image(a))
m[A.inverse_letter(a)] = self._codomain.reverse_path(m[a])
self._edge_map = WordMorphism(m)
self._vertex_map = None
def compose_edge_map(self, edge_morph):
"""Compose ``self`` with the morphism ``edge_morph``.
Update the edge_map of ``self`` with (``edge_morph`` o ``self``).
INPUT:
- ``edge_morph`` -- A ``WordMorphism`` from the alphabet labeling
the codomain of ``self`` to itself.
EXAMPLES::
sage: from train_track import *
sage: from train_track.inverse_graph import GraphWithInverses
sage: from train_track.graph_map import GraphMap
sage: G = GraphWithInverses([[0,0,'a'],[0,1,'b'],[1,1,'c']])
sage: A = AlphabetWithInverses(2)
sage: H = GraphWithInverses.rose_graph(A)
sage: f = GraphMap(G,H,"a->ab,b->b,c->B")
sage: f.compose_edge_map(FreeGroupAutomorphism('a->aba,b->ba'))
sage: print(f)
Graph map:
a: 0->0, b: 0->1, c: 1->1
a: 0->0, b: 0->0
edge map: a->ababa, b->ba, c->AB
"""
edge_map = dict(
(a,
edge_morph.to_word_morphism(use_str=True,
upper_case_as_inverse=True)(
self._edge_map.image(a)))
for a in self._domain._alphabet.positive_letters())
self.set_edge_map(edge_map)
def update_vertex_map(self):
"""
Computes the vertex map of ``self`` from its edge map.
EXAMPLES::
sage: from train_track import *
sage: from train_track.inverse_graph import GraphWithInverses
sage: from train_track.graph_map import GraphMap
sage: G = GraphWithInverses([[0,0,'a'],[0,1,'b'],[1,1,'c']])
sage: A = AlphabetWithInverses(2)
sage: H = GraphWithInverses.rose_graph(A)
sage: f = GraphMap(G,H,"a->ab,b->b,c->B")
sage: f.update_vertex_map()
sage: print(f)
Graph map:
a: 0->0, b: 0->1, c: 1->1
a: 0->0, b: 0->0
edge map: a->ab, b->b, c->B
vertex map: {0: 0, 1: 0}
"""
vertex_map = {}
for e in self._domain._alphabet.positive_letters():
p = self.image(e)
if len(p) > 0:
vertex_map[self._domain.initial_vertex(e)] = \
self._codomain.initial_vertex(p[0])
vertex_map[self._domain.terminal_vertex(e)] = \
self._codomain.terminal_vertex(p[-1])
self._vertex_map = vertex_map
def vertex_map(self):
"""
The vertex map of ``self``: this is a dictionnary.
OUTPUT:
The vertex map of ``self``
EXAMPLES::
sage: from train_track import *
sage: from train_track.inverse_graph import GraphWithInverses
sage: from train_track.graph_map import GraphMap
sage: G = GraphWithInverses([[0,0,'a'],[0,1,'b'],[1,1,'c']])
sage: A = AlphabetWithInverses(2)
sage: H = GraphWithInverses.rose_graph(A)
sage: f = GraphMap(G,H,"a->ab,b->b,c->B")
sage: f.vertex_map()
{0: 0, 1: 0}
"""
if self._vertex_map is None:
self.update_vertex_map()
return self._vertex_map
def edge_map(self):
"""
The edge map of ``self``: this is a word morphism.
OUTPUT:
The edge map of ``self``
EXAMPLES::
sage: from train_track import *
sage: from train_track.inverse_graph import GraphWithInverses
sage: from train_track.graph_map import GraphMap
sage: G = GraphWithInverses([[0,0,'a'],[0,1,'b'],[1,1,'c']])
sage: A = AlphabetWithInverses(2)
sage: H = GraphWithInverses.rose_graph(A)
sage: f = GraphMap(G,H,"a->ab,b->b,c->B")
sage: f.edge_map()
WordMorphism: A->BA, B->B, C->b, a->ab, b->b, c->B
"""
return self._edge_map
def image(self, letter, iter=1):
"""The image of a letter.
if ``iter>1`` then returns ``self^iter(letter)``
INPUT:
- ``iter`` (default: 1) -- a positive integer
- ``letter`` -- a letter of the alphabet of the domain of ``self``.
OUTPUT:
if ``iter`` > 1 then returns ``self``^iter(letter)``
if ``iter`` = 1then returns the image of letter
.. WARNING::
``iter`` may be greater than 1 only if the domain and codomain
of ``self`` are equal (that is to say, ``self`` is a
GraphSelfMap)
EXAMPLES::
sage: from train_track import *
sage: from train_track.inverse_graph import GraphWithInverses
sage: from train_track.graph_map import GraphMap
sage: G = GraphWithInverses([[0,0,'a'],[0,1,'b'],[1,1,'c']])
sage: A = AlphabetWithInverses(2)
sage: H = GraphWithInverses.rose_graph(A)
sage: f = GraphMap(G,H,"a->ab,b->b,c->B")
sage: f.image('a')
word: ab
"""
if iter == 1:
return self._edge_map.image(letter)
else:
return self._codomain.reduce_path(
self._edge_map(self._edge_map(letter), iter - 1))
def inverse(self):
"""A homotopy inverse of ``self``.
For ``t1=self.domain().spanning_tree()`` and
``t2=self.codomain().spanning_tree()``. The alphabet ``A`` is
made of edges not in ``t1`` identified (by their order in the
alphabets of the domain and codomain) to letters not in
``t2``. The automorphism ``phi`` of ``FreeGroup(A)`` is
defined using ``t1`` and ``t2``. The inverse map is given by
``phi.inverse()`` using ``t1`` and edges from ``t2`` are
mapped to a single point (the root of ``t1``).
In particular the inverse maps all vertices to the root of ``t1``.
OUTPUT:
A homotopy inverse of ``self``.
.. WARNING::
``self`` is assumed to be a homotopy equivalence.
EXAMPLES::
sage: from train_track import *
sage: from train_track.inverse_graph import GraphWithInverses
sage: from train_track.graph_map import GraphMap
sage: G = GraphWithInverses([[0,0,'a'],[0,1,'b'],[1,1,'c']])
sage: A = AlphabetWithInverses(2)
sage: H = GraphWithInverses.rose_graph(A)
sage: f = GraphMap(G,H,"a->ab,b->b,c->B")
sage: print(f.inverse())
Graph map:
a: 0->0, b: 0->0
a: 0->0, b: 0->1, c: 1->1
edge map: a->abcB, b->bCB
"""
from .free_group import FreeGroup
from .free_group_automorphism import FreeGroupAutomorphism
G1 = self.domain()
A1 = G1.alphabet()
T1 = G1.spanning_tree()
G2 = self.codomain()
A2 = G2.alphabet()
T2 = G2.spanning_tree()
A = AlphabetWithInverses(len(A1) - len(G1.vertices()) + 1)
F = FreeGroup(A.positive_letters())
map = dict()
translate = dict()
i = 0
for a in A1.positive_letters():
l = len(T1[G1.initial_vertex(a)]) - len(T1[G1.terminal_vertex(a)])
if (l != 1 or T1[G1.initial_vertex(a)][-1] != A1.inverse_letter(a)
) and (l != -1 or T1[G1.terminal_vertex(a)][-1] != a):
# a is not in the spanning tree
map[A[i]] = self(T1[G1.initial_vertex(a)] * Word([a]) *
G1.reverse_path(T1[G1.terminal_vertex(a)]))
translate[A[i]] = a
translate[A.inverse_letter(A[i])] = A1.inverse_letter(a)
i += 1
rename = dict()
edge_map = dict()
i = 0
for a in A2.positive_letters():
l = len(T2[G2.initial_vertex(a)]) - len(T2[G2.terminal_vertex(a)])
if (l != 1 or T2[G2.initial_vertex(a)][-1] != A2.inverse_letter(a)
) and (l != -1 or T2[G2.terminal_vertex(a)][-1] != a):
# a is not in the spanning tree
rename[a] = A[i]
rename[A2.inverse_letter(a)] = A.inverse_letter(A[i])
i += 1
else:
edge_map[a] = Word()
for a in map:
map[a] = F([rename[b] for b in map[a] if b in rename])
phi = FreeGroupAutomorphism(map, F)
psi = phi.inverse()
i = 0
for a in A2.positive_letters():
if a not in edge_map:
result = Word()
for b in psi(A[i]).to_word():
c = translate[b]
result = result * T1[G1.initial_vertex(c)] * Word(
[c]) * G1.reverse_path(T1[G1.terminal_vertex(c)])
edge_map[a] = G1.reduce_path(result)
i += 1
return GraphMap(G2, G1, edge_map)
def tighten(self, verbose=False):
"""
Tighten ``self`` such that there are at least three gates at each
vertex of the domain.
A vertex has at least three gates if there exists outgoing
reduced paths ``u``, ``v``, ``w`` with the reduced paths ``f(u)``,
``f(v)`` and ``f(w)`` starting with different edges.
A map is tight if the tuple ``(f(e1),f(e2),...)`` is minimal
first for the sum of the length, then in lexicographic order.
``self`` and ``self.tighten()`` are homotopic.
OUTPUT:
Edge map of tighten ``self`` such that there are at least two
gates at each vertex of the domain.
.. WARNING::
It is assumed that ``self`` is injective in homotopy.
The result may send edges to trivial edge-paths.
EXAMPLES::
sage: from train_track import *
sage: from train_track.inverse_graph import GraphWithInverses
sage: from train_track.graph_map import GraphMap
sage: A = AlphabetWithInverses(2)
sage: G = GraphWithInverses.rose_graph(A)
sage: H = GraphWithInverses.rose_graph(A)
sage: f = GraphMap(G,H,"a->baabAB,b->babAB")
sage: f.tighten()
WordMorphism: A->AB, B->B, a->ba, b->b
sage: print(f)
Graph map:
a: 0->0, b: 0->0
a: 0->0, b: 0->0
edge map: a->ba, b->b
"""
G1 = self.domain()
A1 = G1.alphabet()
G2 = self.codomain()
edge_map = dict((a, self.image(a)) for a in A1)
adjacent_vertex = dict((v, v) for v in G1.vertices()) # a
# class of vertices linked by a tree which is contracted by
# self to a point
done = False
while not done:
done = True
gates = dict((v, []) for v in G1.vertices()) # the gates
# outgoing from a contracted tree
minimal_edge = dict() # The minimal edge (in lexicographic
# order) outgoing from a contracted forest.
for a in A1:
u = edge_map[a]
v = G1.initial_vertex(a)
vv = adjacent_vertex[v]
if len(u) > 0:
gates[vv].append(u)
if vv not in minimal_edge or a < minimal_edge[vv]:
minimal_edge[vv] = a
else: # we need to increase the adjacent_vertex
w = G1.terminal_vertex(a)
ww = adjacent_vertex[w]
if vv != ww: # we need to fusion the two contracted forests
gates[vv] = gates[vv] + gates[ww]
if ww in minimal_edge:
if vv not in minimal_edge:
minimal_edge[vv] = minimal_edge[ww]
elif minimal_edge[ww] < minimal_edge[vv]:
minimal_edge[vv] = minimal_edge[ww]
for t in G1.vertices():
if adjacent_vertex[t] == ww:
adjacent_vertex[t] = vv
for v in G1.vertices():
if v == adjacent_vertex[v]:
gates[v].sort()
i = 0
while i < len(gates[v]):
prefix = gates[v][i]
j = i + 1
while j < len(
gates[v]) and gates[v][j][0] == prefix[0]:
prefix = prefix[:G2.common_prefix_length(
gates[v][j], prefix)]
j += 1
if 2 * (j - i) > len(gates[v]) or (
2 * (j - i) == len(gates[v])
and edge_map[minimal_edge[v]][0] == prefix[0]):
if verbose:
print("Moving vertex", v, "towards", prefix)
for a in A1:
if v == adjacent_vertex[G1.initial_vertex(a)]:
aa = A1.inverse_letter(a)
edge_map[a] = G2.reduce_path(
G2.reverse_path(prefix) * edge_map[a])
edge_map[aa] = G2.reduce_path(
edge_map[aa] * prefix)
if verbose:
out = ""
for a in A1.positive_letters():
out += a + "->" + str(edge_map[a]) + ","
print(out[:-1])
done = False
break
i = j
if not done:
break
self.set_edge_map(edge_map)
return self._edge_map
def subdivide_domain(self, e):
"""
Subdivide an edge in the domain graph.
The edge ``e`` is subdivided into ``l`` edges where ``l`` is
the length of the image of ``e`` by ``self``.
Update the edge map of ``self``.
Intended to be used by Stalling's folding algorithm to get an
immersion.
.. SEEALSO::
:meth:`train_track.inverse_graph.GraphWithInverses.subdivide_edge()`
INPUT:
- ``e`` -- and edge of the domain of ``self``.
OUTPUT:
A dictionnary that maps old edges of the domain to new edges of the domain.
EXAMPLES::
sage: from train_track import *
sage: from train_track.inverse_graph import GraphWithInverses
sage: from train_track.graph_map import GraphMap
sage: A = AlphabetWithInverses(2)
sage: G = GraphWithInverses.rose_graph(A)
sage: H = GraphWithInverses.rose_graph(A)
sage: f = GraphMap(G,H,"a->aba,b->ab")
sage: f.subdivide_domain('a')
{'A': word: DCA, 'B': word: B, 'a': word: acd, 'b': word: b}
sage: print(f)
Graph map:
a: 0->1, b: 0->0, c: 1->2, d: 2->0
a: 0->0, b: 0->0
edge map: a->a, b->ab, c->b, d->a
"""
G = self.domain()
A = G.alphabet()
result_map = dict((a, Word([a])) for a in A)
w = self.image(e)
d = dict((a, self.image(a)) for a in A.positive_letters())
new_edges = A.add_new_letters(len(w) - 1)
new_vertices = G.new_vertices(len(w) - 1)
for i, a in enumerate(new_edges):
v = new_vertices[i]
if i == 0:
vi = G.initial_vertex(e)
vt = G.terminal_vertex(e)
f = new_edges[i][0]
ee = A.inverse_letter(e)
ff = new_edges[i][1]
G.set_terminal_vertex(e, v)
G.add_edge(v, vt, [f, ff])
result_map[e] = result_map[e] * Word([f])
result_map[ee] = Word([ff]) * result_map[ee]
d[a[0]] = w[i + 1]
else:
vi = self._domain.initial_vertex(new_edges[i - 1][0])
vt = self._domain.terminal_vertex(new_edges[i - 1][0])
f = new_edges[i][0]
# ee=A.inverse_letter(e) #already done
ff = new_edges[i][1]
self._domain.set_terminal_vertex(new_edges[i - 1][0], v)
self._domain.add_edge(v, vt, [f, ff])
result_map[e] = result_map[e] * Word([f])
result_map[ee] = Word([ff]) * result_map[ee]
d[a[0]] = w[i + 1]
# updating self.edge_map after subdivision
if A.is_positive_letter(e):
d[e] = Word([self.image(e)[0]])
else:
d[ee] = Word([self.image(ee)[-1]])
self.set_edge_map(d)
return result_map
def illegal_turns(self, turns=None):
"""
List of illegal turns in the domain graph.
A turn is illegal if it is mapped by ``self`` to a degenerate
turn.
INPUT:
- ``turns`` -- a list of turns of the domain graph. Default is None
meaning all the turns of the graph. I not ``None`` return the
sublist of ``turns`` consisting of illegal turns.
EXAMPLES::
sage: from train_track import *
sage: from train_track.inverse_graph import GraphWithInverses
sage: from train_track.graph_map import GraphMap
sage: A = AlphabetWithInverses(2)
sage: G = GraphWithInverses.rose_graph(A)
sage: H = GraphWithInverses.rose_graph(A)
sage: f = GraphMap(G,H,"a->aba,b->ab")
sage: f.illegal_turns()
[('a', 'b')]
"""
result = []
if turns is None:
turns = self.domain().turns()
for turn in turns:
u = self.image(turn[0])
v = self.image(turn[1])
if len(u) == 0 or len(v) == 0 or u[0] == v[0]:
result.append(turn)
return result
def stallings_folding(self):
"""
Implement Stallings' folding to get an immersion from ``self``.
The domain of ``self`` is fold until we get an
immersion. Intended to be used to compute the pullback of two
graph maps and the intersection of subgroupes of a free group.
ALGORITHM:
We first subdivide edges of the domain according to length of
their image.
Then fold one gate at one vertex and update the edge map and
illegal turns list.
Repeat the process till no illegal turns remain.
EXAMPLES::
sage: from train_track import *
sage: from train_track.inverse_graph import GraphWithInverses
sage: from train_track.graph_map import GraphMap
sage: A = AlphabetWithInverses(2)
sage: G = GraphWithInverses.rose_graph(A)
sage: H = GraphWithInverses.rose_graph(A)
sage: f = GraphMap(G,H,"a->aba,b->ab")
sage: f.stallings_folding()
sage: print(f)
Graph map:
a: 1->1, c: 1->1
a: 0->0, b: 0->0
edge map: a->a, c->b
REFERENCES:
.. [Stallings] J. Stallings, Topology of Finite Graphs,
AUTHOR:
- Radhika GUPTA
"""
A = self.domain().alphabet()
for a in A:
if len(self.image(a)) > 1:
self.subdivide_domain(a)
Turns = self._domain.turns()
# list of all turns in domain after subdivision
Il_turns = self.illegal_turns(Turns)
# list of illegal turns in domain after subdivision
counter = 0
while len(Il_turns) > 0:
counter = counter + 1
# find edge_list (list of edges in the gate correspoding to e1)
# to fold at exactly one vertex
e1 = Il_turns[0][0]
edge_list = [e1]
for a in A:
if self._domain.initial_vertex(a) == \
self._domain.initial_vertex(e1) and e1 != a:
if (e1, a) in Il_turns or (a, e1) in Il_turns:
edge_list.append(a)
edge_list = list(set(edge_list)) # remove duplicates
# fold at initial_vertex of e1 ( this function updates the
# domain and edge_map)
self._domain.fold(edge_list, [])
# update edge_map again
d = {}
d[edge_list[0]] = self.image(edge_list[0])
for a in A:
if a not in edge_list:
d[a] = self.image(a)
wm = WordMorphism(d)
self.set_edge_map(wm)
Turns = self._domain.turns() # update list of all turns in domain
Il_turns = self.illegal_turns(Turns)
# update list of illegal turns in domain
def pullback(self, other):
r"""
Pullback of the graph maps ``self`` and ``other``.
The codomain of ``self`` and ``other`` must be the same graph.
The pullback is a graph map `f : G_3 \to G` that makes the diagram commute::
G3 -----> G1
| \ |
| \ | self
| \f |
| \ |
V _\| V
G2 -----> G
....other
The pullback method can be used to find intersection of two subgroups
of a Free Group.
INPUT:
- ``other`` -- a graph map `G_2 \to G` with ``self`` (a
graph map `G_1 \to G`)
OUTPUT:
A ``GraphMap``.
EXAMPLES::
sage: from train_track import *
sage: from train_track.inverse_graph import GraphWithInverses
sage: from train_track.graph_map import GraphMap
sage: G1 = GraphWithInverses.rose_graph(AlphabetWithInverses(2,type='x0'))
sage: G2 = GraphWithInverses.rose_graph(AlphabetWithInverses(2,type='a0'))
sage: G = GraphWithInverses.rose_graph(AlphabetWithInverses(2))
sage: n1 = WordMorphism({'x0':['a','a'],'x1':['b','a']})
sage: n2 = WordMorphism({'a0':['b','a'],'a1':['b','b','b','a','B','a']})
sage: f1 = GraphMap(G1,G,n1)
sage: f2 = GraphMap(G2,G,n2)
sage: print(f1.pullback(f2))
Graph map:
a0: (0, 0)->(1, 2), a1: (0, 2)->(1, 0), a2: (0, 2)->(1, 3), a3: (0, 3)->(1, 4), a4: (0, 4)->(1, 3), a5: (1, 2)->(0, 0), a6: (1, 4)->(0, 3)
a: 0->0, b: 0->0
edge map: a0->b, a1->a, a2->b, a3->b, a4->a, a5->a, a6->a
AUTHORS:
- Radhika GUPTA
"""
import itertools
# First convert self and f2 into immersions
self.stallings_folding()
other.stallings_folding()
G3 = GraphWithInverses()
A = AlphabetWithInverses(0, type='a0')
d = {}
# get set of vertices
V = []
for i in itertools.product(self.domain().vertices(),
other.domain().vertices()):
V.append(i)
# add edges
for v in V:
for w in V:
for e1 in self.domain().alphabet().positive_letters():
if self.domain().initial_vertex(e1) == v[0] \
and self.domain().terminal_vertex(e1) == w[0]:
for e2 in other.domain().alphabet().positive_letters():
if other.domain().initial_vertex(e2) == v[1] \
and other.domain().terminal_vertex(e2) == w[1]:
if self.image(e1) == other.image(e2):
e = A.add_new_letter()
G3.add_edge(v, w, e)
# update dictionary to define map on G3
d[e[0]] = self.image(e1)
G3._alphabet = A
n3 = WordMorphism(d)
G = self.codomain() # same as other.codomain()
return GraphMap(G3, G, n3)
@staticmethod
def rose_map(automorphism):
"""
The graph map of the rose representing the automorphism.
The rose is built on a copy of the alphabet of the domain of
``automorphism``.
OUTPUT:
The graph map of the rose representing the automorphism.
EXAMPLES::
sage: from train_track import *
sage: from train_track.inverse_graph import GraphWithInverses
sage: from train_track.graph_map import GraphMap
sage: phi=FreeGroupAutomorphism('a->ab,b->ac,c->a')
sage: print(GraphMap.rose_map(phi))
Graph map:
a: 0->0, b: 0->0, c: 0->0
a: 0->0, b: 0->0, c: 0->0
edge map: a->ab, b->ac, c->a
"""
A = AlphabetWithInverses(automorphism.domain().variable_names())
graph = GraphWithInverses.rose_graph(A)
return GraphMap(
graph, graph,
automorphism.to_word_morphism(use_str=True,
upper_case_as_inverse=True))
class GraphMap():
"""
A GraphMap is a map from a Graph to another . It maps a vertex to
a vertex and an edge to an edge-path. It respects incidence
relation. The inverse of an edge is send to the reverse path.
AUTHORS:
- Thierry Coulbois (2013-05-16): beta.0 version
"""
def __init__(self,domain,codomain,edge_map,vertex_map=None):
self._domain=domain
self._codomain=codomain
self.set_edge_map(edge_map)
self._vertex_map=vertex_map
def __call__(self,argument):
"""
Applies ``self`` to ``argument`` which is either a vertex of ``self`` or
an edge path.
SEE ALSO:
To compute the image of a letter of the alphabet use
``self.image(a)``.
"""
if self._domain.has_vertex(argument):
if self._vertex_map==None:
self.update_vertex_map()
return self._vertex_map[argument]
else:
return self._codomain.reduce_path(self._edge_map(argument))
def __mul__(self,other):
"""
Compose ``self`` with ``other``.
"""
A=other._domain.alphabet()
result_map={}
for a in A.positive_letters():
result_map[a]=self(other._edge_map.image(a))
return GraphMap(other._domain,self._codomain,result_map)
def __str__(self):
"""
String represetation of ``self``.
"""
result="Graph map:\n"+self._domain.__str__()+"\n"
result+=self._codomain.__str__()+"\n"
result=result+"edge map: "
for a in self._domain._alphabet.positive_letters(): result+=a+"->"+self.image(a).__str__()+", "
result=result[:-2]+"\n"
if self._vertex_map!=None:
result=result+"vertex map: "+self._vertex_map.__str__()+"\n"
return result
def domain(self):
"""
Domain of ``self``: this is a graph.
"""
return self._domain
def codomain(self):
"""
Codomain of ``self``: this is a graph.
"""
return self._codomain
def set_edge_map(self,edge_map):
"""
Sets the edge map of ``self``.
``edge_map`` is anything that is accepted by
``Wordmorphism(edge_map)``, the image of the inverse letters
will be calculated: they need not be explicit in ``edge_map``,
only one of the images of each pair [letter,inverse(letter)]
need to be given by ``edge_map``. Images of ``edge_map`` need
not be reduced.
"""
A=self.domain().alphabet()
tmp_map=WordMorphism(edge_map)
m={}
for a in tmp_map._domain.alphabet():
m[a]=self._codomain.reduce_path(tmp_map.image(a))
m[A.inverse_letter(a)]=self._codomain.reverse_path(m[a])
self._edge_map=WordMorphism(m)
self._vertex_map=None
def compose_edge_map(self,edge_morph):
"""
Compose ``self`` with the morphism ``edge_morph``.
Update the edge_map of ``self`` with (``edge_morph`` o ``self``).
"""
edge_map=dict((a,edge_morph(self._edge_map.image(a))) for a in self._domain._alphabet.positive_letters())
self.set_edge_map(edge_map)
def update_vertex_map(self):
"""
Computes the vertex map of ``self`` from its edge map.
"""
vertex_map={}
for e in self._domain._alphabet.positive_letters():
p=self.image(e)
if len(p)>0:
vertex_map[self._domain.initial_vertex(e)]=self._codomain.initial_vertex(p[0])
vertex_map[self._domain.terminal_vertex(e)]=self._codomain.terminal_vertex(p[-1])
self._vertex_map=vertex_map
def edge_map(self):
"""
The edge map of ``self``: this is a word morphism.
"""
return self._edge_map
def image(self,letter,iter=1):
"""
The image of a letter.
if ``iter>1`` then returns ``self^iter(letter)``
"""
if iter==1:
return self._edge_map.image(letter)
else:
return self._codomain.reduce_path(self._edge_map(self._edge_map(letter),iter-1))
def inverse(self):
"""
A homotopy inverse of ``self``.
WARNING:
``self`` is assumed to be a homotopy equivalence.
"""
G1=self.domain()
A1=G1.alphabet()
t1=G1.spanning_tree()
G2=self.codomain()
A2=G2.alphabet()
t2=G2.spanning_tree()
A=AlphabetWithInverses(len(A1)-len(G1.vertices())+1)
F=FreeGroup(A)
map=dict()
translate=dict()
i=0
for a in A1.positive_letters():
l=len(t1[G1.initial_vertex(a)])-len(t1[G1.terminal_vertex(a)])
if (l!=1 or t1[G1.initial_vertex(a)][-1]!=A1.inverse_letter(a)) and\
(l!=-1 or t1[G1.terminal_vertex(a)][-1]!=a): # a is not in the spanning tree
map[A[i]]=self(t1[G1.initial_vertex(a)]*Word([a])*G1.reverse_path(t1[G1.terminal_vertex(a)]))
translate[A[i]]=a
translate[A.inverse_letter(A[i])]=A1.inverse_letter(a)
i+=1
rename=dict()
edge_map=dict()
i=0
for a in A2.positive_letters():
l=len(t2[G2.initial_vertex(a)])-len(t2[G2.terminal_vertex(a)])
if (l!=1 or t2[G2.initial_vertex(a)][-1]!=A2.inverse_letter(a)) and\
(l!=-1 or t2[G2.terminal_vertex(a)][-1]!=a): # a is not in the spanning tree
rename[a]=A[i]
rename[A2.inverse_letter(a)]=A.inverse_letter(A[i])
i+=1
else:
edge_map[a]=Word()
for a in map:
map[a]=F([rename[b] for b in map[a] if b in rename])
phi=FreeGroupAutomorphism(map,F)
psi=phi.inverse()
i=0
for a in A2.positive_letters():
if a not in edge_map:
result=Word()
for b in psi.image(A[i]):
c=translate[b]
result=result*t1[G1.initial_vertex(c)]*Word([c])*G1.reverse_path(t1[G1.terminal_vertex(c)])
edge_map[a]=G1.reduce_path(result)
i+=1
return GraphMap(G2,G1,edge_map)
def tighten(self):
"""
Tighten ``self`` such that there are at least two gates at
each vertex of the domain.
A map is tight if for each vertex ``v`` of the domain, there
exist reduced edge paths ``u`` and ``v`` in the domain with
``self(u)`` and ``self(v)`` non-trivial reduced paths starting
with different edges.
``self`` and ``self.tighten()`` are homotopic.
"""
G1=self.domain()
A1=G1.alphabet()
G2=self.domain()
edge_map=dict((a,self.image(a)) for a in A1)
done=False
while not done:
done=True
prefix=dict()
for a in A1:
u=edge_map[a]
v=G1.initial_vertex(a)
if len(u)>0:
if v in prefix:
if len(prefix[v])>0:
p=G2.common_prefix_length(u,prefix[v])
prefix[v]=prefix[v][:p]
else:
prefix[v]=u
for a in A1:
v=G1.initial_vertex(a)
if v in prefix and len(prefix[v])>0:
done=False
aa=A1.inverse_letter(a)
if len(edge_map[a])>0:
edge_map[a]=edge_map[a][len(prefix[v]):]
edge_map[aa]=edge_map[aa][:-len(prefix[v])]
else:
edge_map[a]=G2.reverse_path(prefix[v])
edge_map[aa]=prefix[v]
self.set_edge_map(edge_map)
return self
@staticmethod
def rose_map(automorphism):
"""
The graph map of the rose representing the automorphism.
The rose is built on a copy of the alphabet of the domain of
``automorphism``.
"""
graph=GraphWithInverses.rose_graph(automorphism.domain().alphabet().copy())
return GraphMap(graph,graph,automorphism)
class GraphMap():
"""
A GraphMap is a map from a Graph to another. It maps a vertex to
a vertex and an edge to an edge-path. It respects incidence
relation. The inverse of an edge is send to the reverse path.
AUTHORS:
- Thierry Coulbois (2013-05-16): beta.0 version
"""
def __init__(self,*args):
"""
The following forms are accepted:
- ``GraphMap(f)`` where ``f`` is a ``GraphMap``.
- ``GraphMap(domain,codomain,edge_map,vertex_map=None)`` where
``domain`` and ``codomain`` are ``GraphWithInverses`` and
``edge_map`` is anything accepted by
``WordMorphism(edge_map)`` with domain alphabet an
AlphabetWithInverses (note that only one image of the pair
(a,inverse_letter(a)) needs to be defined for each letter).
"""
if isinstance(args[0],GraphMap):
self._domain=args[0]._domain
self._codomain=args[0]._codomain
self._edge_map=args[0]._edge_map
self._vertex_map=args[0]._vertex_map
else:
self._domain=args[0]
self._codomain=args[1]
self.set_edge_map(args[2])
if len(args)>3:
self._vertex_map=args[3]
def __call__(self,argument):
"""
Applies ``self`` to ``argument`` which is either a vertex of ``self`` or
an edge path.
SEE ALSO:
To compute the image of a letter of the alphabet use
``self.image(a)``.
"""
if self._domain.has_vertex(argument):
if self._vertex_map==None:
self.update_vertex_map()
return self._vertex_map[argument]
else:
return self._codomain.reduce_path(self._edge_map(argument))
def __mul__(self,other):
"""
Compose ``self`` with ``other``.
"""
A=other._domain.alphabet()
result_map={}
for a in A.positive_letters():
result_map[a]=self(other._edge_map.image(a))
return GraphMap(other._domain,self._codomain,result_map)
def __str__(self):
"""
String represetation of ``self``.
"""
result="Graph map:\n"+self._domain.__str__()+"\n"
result+=self._codomain.__str__()+"\n"
result=result+"edge map: "
for a in self._domain._alphabet.positive_letters(): result+=a+"->"+self.image(a).__str__()+", "
result=result[:-2]+"\n"
if self._vertex_map!=None:
result=result+"vertex map: "+self._vertex_map.__str__()+"\n"
return result
def domain(self):
"""
Domain of ``self``: this is a graph.
"""
return self._domain
def codomain(self):
"""
Codomain of ``self``: this is a graph.
"""
return self._codomain
def set_edge_map(self,edge_map):
"""
Sets the edge map of ``self``.
``edge_map`` is anything that is accepted by
``Wordmorphism(edge_map)``, the image of the inverse letters
will be calculated: they need not be explicit in ``edge_map``,
only one of the images of each pair [letter,inverse(letter)]
need to be given by ``edge_map``. Images of ``edge_map`` need
not be reduced.
"""
A=self.domain().alphabet()
tmp_map=WordMorphism(edge_map)
m={}
for a in tmp_map._domain.alphabet():
m[a]=self._codomain.reduce_path(tmp_map.image(a))
m[A.inverse_letter(a)]=self._codomain.reverse_path(m[a])
self._edge_map=WordMorphism(m)
self._vertex_map=None
def compose_edge_map(self,edge_morph):
"""
Compose ``self`` with the morphism ``edge_morph``.
Update the edge_map of ``self`` with (``edge_morph`` o ``self``).
"""
edge_map=dict((a,edge_morph(self._edge_map.image(a))) for a in self._domain._alphabet.positive_letters())
self.set_edge_map(edge_map)
def update_vertex_map(self):
"""
Computes the vertex map of ``self`` from its edge map.
"""
vertex_map={}
for e in self._domain._alphabet.positive_letters():
p=self.image(e)
if len(p)>0:
vertex_map[self._domain.initial_vertex(e)]=self._codomain.initial_vertex(p[0])
vertex_map[self._domain.terminal_vertex(e)]=self._codomain.terminal_vertex(p[-1])
self._vertex_map=vertex_map
def edge_map(self):
"""
The edge map of ``self``: this is a word morphism.
"""
return self._edge_map
def image(self,letter,iter=1):
"""
The image of a letter.
if ``iter>1`` then returns ``self^iter(letter)``
"""
if iter==1:
return self._edge_map.image(letter)
else:
return self._codomain.reduce_path(self._edge_map(self._edge_map(letter),iter-1))
def inverse(self):
"""A homotopy inverse of ``self``.
For ``t1=self.domain().spanning_tree()`` and
``t2=self.codomain().spanning_tree()``. The alphabet ``A`` is
made of edges not in ``t1`` identified (by their order in the
alphabets of the domain and codomain) to letters not in
``t2``. The automorphism ``phi`` of ``FreeGroup(A)`` is
defined using ``t1`` and ``t2``. The inverse map is given by
``phi.inverse()`` using ``t1`` and edges from ``t2`` are
mapped to a single point (the root of ``t1``).
In particular the inverse maps all vertices to the root of ``t1``.
WARNING:
``self`` is assumed to be a homotopy equivalence.
"""
from free_group import FreeGroup
from free_group_automorphism import FreeGroupAutomorphism
G1=self.domain()
A1=G1.alphabet()
t1=G1.spanning_tree()
G2=self.codomain()
A2=G2.alphabet()
t2=G2.spanning_tree()
A=AlphabetWithInverses(len(A1)-len(G1.vertices())+1)
F=FreeGroup(A)
map=dict()
translate=dict()
i=0
for a in A1.positive_letters():
l=len(t1[G1.initial_vertex(a)])-len(t1[G1.terminal_vertex(a)])
if (l!=1 or t1[G1.initial_vertex(a)][-1]!=A1.inverse_letter(a)) and\
(l!=-1 or t1[G1.terminal_vertex(a)][-1]!=a): # a is not in the spanning tree
map[A[i]]=self(t1[G1.initial_vertex(a)]*Word([a])*G1.reverse_path(t1[G1.terminal_vertex(a)]))
translate[A[i]]=a
translate[A.inverse_letter(A[i])]=A1.inverse_letter(a)
i+=1
rename=dict()
edge_map=dict()
i=0
for a in A2.positive_letters():
l=len(t2[G2.initial_vertex(a)])-len(t2[G2.terminal_vertex(a)])
if (l!=1 or t2[G2.initial_vertex(a)][-1]!=A2.inverse_letter(a)) and\
(l!=-1 or t2[G2.terminal_vertex(a)][-1]!=a): # a is not in the spanning tree
rename[a]=A[i]
rename[A2.inverse_letter(a)]=A.inverse_letter(A[i])
i+=1
else:
edge_map[a]=Word()
for a in map:
map[a]=F([rename[b] for b in map[a] if b in rename])
phi=FreeGroupAutomorphism(map,F)
psi=phi.inverse()
i=0
for a in A2.positive_letters():
if a not in edge_map:
result=Word()
for b in psi.image(A[i]):
c=translate[b]
result=result*t1[G1.initial_vertex(c)]*Word([c])*G1.reverse_path(t1[G1.terminal_vertex(c)])
edge_map[a]=G1.reduce_path(result)
i+=1
return GraphMap(G2,G1,edge_map)
def tighten(self):
"""
Tighten ``self`` such that there are at least two gates at
each vertex of the domain.
A map is tight if for each vertex ``v`` of the domain, there
exist reduced edge paths ``u`` and ``v`` in the domain with
``self(u)`` and ``self(v)`` non-trivial reduced paths starting
with different edges.
``self`` and ``self.tighten()`` are homotopic.
WARNING:
It is assumed that ``self`` is a homotopy equivalence
The result may send edges to trivial edge-paths.
"""
G1=self.domain()
A1=G1.alphabet()
G2=self.domain()
edge_map=dict((a,self.image(a)) for a in A1)
done=False
while not done:
done=True
prefix=dict() # the common prefix of all edges outgoing from the class of a vertex
adjacent_vertex=dict() # a class of vertices linked by a tree which is contracted by self to a point
for a in A1:
u=edge_map[a]
v=G1.initial_vertex(a)
if len(u)>0:
if v not in adjacent_vertex:
adjacent_vertex[v]=set([v])
for w in adjacent_vertex[v]:
if w in prefix:
if len(prefix[w])>0:
p=G2.common_prefix_length(u,prefix[w])
prefix[w]=prefix[w][:p]
else:
prefix[w]=u
else: #we need to increase the adjacent_vertex
vv=G1.terminal_vertex(a) #note that v!=vv because else the loop a is contracted contrary to homotopy equivalence
if v in adjacent_vertex and vv in adjacent_vertex:
adjacent_vertex[v].update(adjacent_vertex[vv])
elif v in adjacent_vertex:
adjacent_vertex[v].add(vv)
elif vv in adjacent_vertex:
adjacent_vertex[v]=adjacent_vertex[vv]
adjacent_vertex[v].add(v)
else:
adjacent_vertex[v]=set([v,vv])
if v in prefix:
prefixv=prefix[v]
else:
prefixv=False
for w in adjacent_vertex[v]:
if v!=w:
adjacent_vertex[w]=adjacent_vertex[v]
if w in prefix:
if prefixv:
p=G2.common_prefix_length(prefixv,prefix[w])
prefixv=prefixv[:p]
else:
prefixv=prefix[w]
if prefixv:
for w in adjacent_vertex[v]:
prefix[w]=prefixv
for a in A1:
v=G1.initial_vertex(a)
if v in prefix and len(prefix[v])>0:
done=False
aa=A1.inverse_letter(a)
if len(edge_map[a])>0:
edge_map[a]=edge_map[a][len(prefix[v]):]
edge_map[aa]=edge_map[aa][:-len(prefix[v])]
# else:
# edge_map[a]=G2.reverse_path(prefix[v])
# edge_map[aa]=prefix[v]
break
self.set_edge_map(edge_map)
return self
def subdivide_domain(self, e):
"""
Subdivide an edge in the domain graph.
The edge ``e`` is subdivided into ``l`` edges where ``l`` is
the length of the image of ``e`` by ``self``.
Update the edge map of ``self``.
Intended to be used by Stalling's folding algorithm to get an
immersion.
SEE ALSO::
GraphWithInverses.subdivide_edge()
"""
G=self.domain()
A = G.alphabet()
result_map=dict((a,Word([a])) for a in A)
w = self.image(e)
new_edges=A.add_new_letters(len(w)-1)
new_vertices=G.new_vertices(len(w)-1)
d={(a,self.image(a)) for a in A.positive_letters()}
for i,a in enumerate(new_edges):
v=new_vertices[i]
if i==0:
vi=G.initial_vertex(e)
vt=G.terminal_vertex(e)
f=new_edges[i][0]
ee=A.inverse_letter(e)
ff=new_edges[i][1]
G.set_terminal_vertex(e,v)
G.add_edge(v,vt,[f,ff])
result_map[e]=result_map[e]*Word([f])
result_map[ee]=Word([ff])*result_map[ee]
d[a[0]] = w[i+1]
else:
vi=self._domain.initial_vertex(new_edges[i-1][0])
vt=self._domain.terminal_vertex(new_edges[i-1][0])
f=new_edges[i][0]
#ee=A.inverse_letter(e) #already done
ff=new_edges[i][1]
self._domain.set_terminal_vertex(new_edges[i-1][0],v)
self._domain.add_edge(v,vt,[f,ff])
result_map[e]=result_map[e]*Word([f])
result_map[ee]=Word([ff])*result_map[ee]
d[a[0]] = w[i+1]
# updating self.edge_map after subdivision
if A.is_positive_letter(e):
d[e] = Word([self.image(e)[0]])
else:
d[ee]=Word([self.image(ee)[-1]])
self.set_edge_map(d)
return result_map
def illegal_turns(self,turns=None):
"""
List of illegal turns in the domain graph.
A turn is illegal if it is mapped by ``self`` to a degenerate
turn.
INPUT:
``turns`` a list of turns of the domain graph. Default is None
meaning all the turns of the graph. I not ``None`` return the
sublist of ``turns`` consisting of illegal turns.
"""
result = []
if turns is None:
turns=self.domain().turns()
for turn in turns:
u=self.image(turn[0])
v=self.image(turn[1])
if len(u)==0 or len(v)==0 or u[0]==v[0]:
result.append(turn)
return result
def stallings_folding(self):
"""
Implement Stallings' folding to get an immersion from ``self``.
The domain of ``self`` is fold until we get an immersion.
ALGORITHM:
We first subdivide edges of the domain according to length of
their image.
Then fold one gate at one vertex and update the edge map and
illegal turns list.
Repeat the process till no illegal turns remain.
REFERENCES:
[Stallings] J. Stallings, Topology of Finite Graphs,
"""
A = self.domain().alphabet()
for a in A:
if len(self.image(a)) >1:
self.subdivide_domain(a)
Turns = self._domain.turns() #list of all turns in domain after subdivision
Il_turns = self.illegal_turns(Turns) # list of illegal turns in domain after subdivision
counter = 0
while len(Il_turns)>0:
counter = counter +1
# find edge_list (list of edges in the gate correspoding to e1) to fold at exactly one vertex
e1=Il_turns[0][0]
edge_list = [e1]
for a in A:
if self._domain.initial_vertex(a) == self._domain.initial_vertex(e1) and e1!=a:
if (e1,a) in Il_turns or (a,e1) in Il_turns:
edge_list.append(a)
edge_list = list(set(edge_list)) # remove duplicates
# fold at initial_vertex of e1 ( this function updates the domain and edge_map)
self._domain.fold(edge_list,[])
#update edge_map again
d={}
d[edge_list[0]] = self.image(edge_list[0])
for a in A:
if a not in edge_list:
d[a] = self.image(a)
wm = WordMorphism(d)
self.set_edge_map(wm)
Turns = self._domain.turns() #update list of all turns in domain
Il_turns = self.illegal_turns(Turns) # update list of illegal turns in domain
return self
def pullback(self,f2,G3,A):
"""
INPUT : Two Graph maps f1:G1->G, f2:G2->G, an empty GraphWithInverses G3 and an empty AlphabetWithInverses A
OPERATION : Find the pullback G3 and a graph map f:G3->G
OUTPUT : Graphmap f
The pullback method can be used to find intersection of two subgroups of a Free Group.
Example :
G1 = GraphWithInverses.rose_graph(AlphabetWithInverses(2,type='x0'))
G2 = GraphWithInverses.rose_graph(AlphabetWithInverses(2,type='a0'))
G = GraphWithInverses.rose_graph(AlphabetWithInverses(2))
n1 = WordMorphism({'x0':['a','a'],'x1':['b','a']})
n2 = WordMorphism({'a0':['b','a'],'a1':['b','b','b','a','B','a']})
f1 = GraphMap(G1,G,n1)
f2 = GraphMap(G2,G,n2)
G3 = GraphWithInverses()
A = AlphabetWithInverses(0,type='a0')
f1.pullback(f2,G3,A)
"""
import itertools
#First convert self and f2 into immersions
self.stallings_folding()
f2.stallings_folding()
# G3 = GraphWithInverses()
#A = AlphabetWithInverses(0,type='a0')
d={}
#get set of vertices
V = []
for i in itertools.product(self.domain().vertices(),f2.domain().vertices()):
V.append(i)
#add edges
for v in V:
for w in V:
for e1 in self.domain().alphabet().positive_letters():
if self.domain().initial_vertex(e1) == v[0] and self.domain().terminal_vertex(e1) == w[0]:
for e2 in f2.domain().alphabet().positive_letters():
if f2.domain().initial_vertex(e2) == v[1] and f2.domain().terminal_vertex(e2) == w[1]:
if self.image(e1)==f2.image(e2):
e = A.add_new_letter()
G3.add_edge(v,w,e )
#update dictionary to define map on G3
d[e[0]] = self.image(e1)
G3._alphabet = A
n3 = WordMorphism(d)
G = self.codomain() #same as f2.codomain()
return GraphMap(G3,G,n3)
@staticmethod
def rose_map(automorphism):
"""
The graph map of the rose representing the automorphism.
The rose is built on a copy of the alphabet of the domain of
``automorphism``.
"""
graph=GraphWithInverses.rose_graph(automorphism.domain().alphabet().copy())
return GraphMap(graph,graph,automorphism)
class GraphMap:
"""
A GraphMap is a map from a Graph to another. It maps a vertex to
a vertex and an edge to an edge-path. It respects incidence
relation. The inverse of an edge is send to the reverse path.
AUTHORS:
- Thierry Coulbois (2013-05-16): beta.0 version
"""
def __init__(self, *args):
"""
The following forms are accepted:
- ``GraphMap(f)`` where ``f`` is a ``GraphMap``.
- ``GraphMap(domain, codomain, edge_map, vertex_map=None)`` where
``domain`` and ``codomain`` are ``GraphWithInverses`` and
``edge_map`` is anything accepted by
``WordMorphism(edge_map)`` with domain alphabet an
AlphabetWithInverses (note that only one image of the pair
(a,inverse_letter(a)) needs to be defined for each letter).
EXAMPLES::
sage: G = GraphWithInverses([[0,0,'a'],[0,1,'b'],[1,1,'c']])
sage: A = AlphabetWithInverses(2)
sage: H = GraphWithInverses.rose_graph(A)
sage: print GraphMap(G,H,"a->ab,b->b,c->B")
Graph map:
Graph with inverses: a: 0->0, b: 0->1, c: 1->1
Graph with inverses: a: 0->0, b: 0->0
edge map: a->ab, b->b, c->B
"""
if isinstance(args[0], GraphMap):
self._domain = args[0]._domain
self._codomain = args[0]._codomain
self._edge_map = args[0]._edge_map
self._vertex_map = args[0]._vertex_map
else:
self._domain = args[0]
self._codomain = args[1]
self.set_edge_map(args[2])
if len(args) > 3:
self._vertex_map = args[3]
def __call__(self, argument):
"""
Applies ``self`` to ``argument`` which is either a vertex
of ``self`` or an edge path.
INPUT:
-``argument`` a vertex of ``self`` or a edge path
OUTPUT:
A vertex of ``self`` or an edge path
Applies ``self`` to ``argument`` which is either .
SEE ALSO:
To compute the image of a letter of the alphabet use
``self.image(a)``.
EXAMPLES::
sage: G = GraphWithInverses([[0,0,'a'],[0,1,'b'],[1,1,'c']])
sage: A = AlphabetWithInverses(2)
sage: H = GraphWithInverses.rose_graph(A)
sage: f = GraphMap(G,H,"a->ab,b->b,c->B")
sage: f('abc')
word: ab
"""
if self._domain.has_vertex(argument):
if self._vertex_map is None:
self.update_vertex_map()
return self._vertex_map[argument]
else:
return self._codomain.reduce_path(self._edge_map(argument))
def __mul__(self, other):
"""
Compose ``self`` with ``other``.
INPUT:
- ``other`` other GraphMap to compute multiplication
OUTPUT:
GraphMap obtains with other domain as first
input and ``self`` codomain for second input and vertex_map is
construct applying ``self`` to ``other`` edge_map image
EXAMPLES::
sage: G = GraphWithInverses([[0,0,'a'],[0,1,'b'],[1,1,'c']])
sage: A = AlphabetWithInverses(2)
sage: H = GraphWithInverses.rose_graph(A)
sage: f = GraphMap(G,H,"a->ab,b->b,c->B")
sage: g = GraphMap(H,H,"a->aba,b->ba")
sage: print g * f
Graph map:
Graph with inverses: a: 0->0, b: 0->1, c: 1->1
Graph with inverses: a: 0->0, b: 0->0
edge map: a->ababa, b->ba, c->AB
"""
A = other._domain.alphabet()
result_map = {}
for a in A.positive_letters():
result_map[a] = self(other._edge_map.image(a))
return GraphMap(other._domain, self._codomain, result_map)
def __str__(self):
"""
String represetation of ``self``.
OUTPUT:
a string representation of ``self``
EXAMPLES::
sage: G = GraphWithInverses([[0,0,'a'],[0,1,'b'],[1,1,'c']])
sage: A = AlphabetWithInverses(2)
sage: H = GraphWithInverses.rose_graph(A)
sage: GraphMap(G,H,"a->ab,b->b,c->B").__str__()
'Graph map:\nGraph with inverses: a: 0->0, b: 0->1, c: 1->1\nGraph with inverses: a: 0->0, b: 0->0\nedge map: a->ab, b->b, c->B'
"""
result = "Graph map:\n" + self._domain.__str__() + "\n"
result += self._codomain.__str__() + "\n"
result += "edge map: "
for a in self._domain._alphabet.positive_letters():
result += a + "->" + self.image(a).__str__() + ", "
result = result[:-2] # + "\n"
if self._vertex_map is not None:
result = result + "\n"
result = result + "vertex map: " +\
self._vertex_map.__str__() # + "\n"
return result
def domain(self):
"""
Domain of ``self``: this is a graph.
OUTPUT:
Domain of ``self`` which is a GrapMap
EXAMPLES::
sage: G = GraphWithInverses([[0,0,'a'],[0,1,'b'],[1,1,'c']])
sage: A = AlphabetWithInverses(2)
sage: H = GraphWithInverses.rose_graph(A)
sage: f = GraphMap(G,H,"a->ab,b->b,c->B")
sage: print f.domain()
Graph with inverses: a: 0->0, b: 0->1, c: 1->1
"""
return self._domain
def codomain(self):
"""
Codomain of ``self``: this is a graph.
OUTPUT:
Codomain of ``self`` which is a GrapMap
EXAMPLES::
sage: G = GraphWithInverses([[0,0,'a'],[0,1,'b'],[1,1,'c']])
sage: A = AlphabetWithInverses(2)
sage: H = GraphWithInverses.rose_graph(A)
sage: f = GraphMap(G,H,"a->ab,b->b,c->B")
sage: print f.codomain()
Graph with inverses: a: 0->0, b: 0->0
"""
return self._codomain
def set_edge_map(self, edge_map):
"""
Sets the edge map of ``self``.
``edge_map`` is anything that is accepted by
``WordMorphism(edge_map)``, the image of the inverse letters
will be calculated: they need not be explicit in ``edge_map``,
only one of the images of each pair [letter,inverse(letter)]
need to be given by ``edge_map``. Images of ``edge_map`` need
not be reduced.
INPUT:
- ``edge_map``: anything which is accepted by ``WordMorphism(edge_map)``
EXAMPLES::
sage: G = GraphWithInverses([[0,0,'a'],[0,1,'b'],[1,1,'c']])
sage: A = AlphabetWithInverses(2)
sage: H = GraphWithInverses.rose_graph(A)
sage: f = GraphMap(G,H,"a->ab,b->b,c->B")
sage: f.set_edge_map('a->b,b->,c->b')
sage: print f
Graph map:
Graph with inverses: a: 0->0, b: 0->1, c: 1->1
Graph with inverses: a: 0->0, b: 0->0
edge map: a->b, b->, c->b
"""
A = self.domain().alphabet()
tmp_map = WordMorphism(edge_map)
m = {}
for a in tmp_map._domain.alphabet():
m[a] = self._codomain.reduce_path(tmp_map.image(a))
m[A.inverse_letter(a)] = self._codomain.reverse_path(m[a])
self._edge_map = WordMorphism(m)
self._vertex_map = None
def compose_edge_map(self, edge_morph):
"""Compose ``self`` with the morphism ``edge_morph``.
Update the edge_map of ``self`` with (``edge_morph`` o ``self``).
INPUT:
- ``edge_morph``: A ``WordMorphism`` from the alphabet labeling
the codomain of ``self`` to itself.
EXAMPLES::
sage: G = GraphWithInverses([[0,0,'a'],[0,1,'b'],[1,1,'c']])
sage: A = AlphabetWithInverses(2)
sage: H = GraphWithInverses.rose_graph(A)
sage: f = GraphMap(G,H,"a->ab,b->b,c->B")
sage: f.compose_edge_map(FreeGroupAutomorphism('a->aba,b->ba'))
sage: print f
Graph map:
Graph with inverses: a: 0->0, b: 0->1, c: 1->1
Graph with inverses: a: 0->0, b: 0->0
edge map: a->ababa, b->ba, c->AB
"""
edge_map = dict((a, edge_morph(self._edge_map.image(a))) for a in
self._domain._alphabet.positive_letters())
self.set_edge_map(edge_map)
def update_vertex_map(self):
"""
Computes the vertex map of ``self`` from its edge map.
EXAMPLES::
sage: G = GraphWithInverses([[0,0,'a'],[0,1,'b'],[1,1,'c']])
sage: A = AlphabetWithInverses(2)
sage: H = GraphWithInverses.rose_graph(A)
sage: f = GraphMap(G,H,"a->ab,b->b,c->B")
sage: f.update_vertex_map()
sage: print f
Graph map:
Graph with inverses: a: 0->0, b: 0->1, c: 1->1
Graph with inverses: a: 0->0, b: 0->0
edge map: a->ab, b->b, c->B
vertex map: {0: 0, 1: 0}
"""
vertex_map = {}
for e in self._domain._alphabet.positive_letters():
p = self.image(e)
if len(p) > 0:
vertex_map[self._domain.initial_vertex(e)] = \
self._codomain.initial_vertex(p[0])
vertex_map[self._domain.terminal_vertex(e)] = \
self._codomain.terminal_vertex(p[-1])
self._vertex_map = vertex_map
def edge_map(self):
"""
The edge map of ``self``: this is a word morphism.
OUTPUT:
The edge map of ``self``
EXAMPLES::
sage: G = GraphWithInverses([[0,0,'a'],[0,1,'b'],[1,1,'c']])
sage: A = AlphabetWithInverses(2)
sage: H = GraphWithInverses.rose_graph(A)
sage: f = GraphMap(G,H,"a->ab,b->b,c->B")
sage: f.edge_map()
WordMorphism: A->BA, B->B, C->b, a->ab, b->b, c->B
"""
return self._edge_map
def image(self, letter, iter=1):
"""The image of a letter.
if ``iter>1`` then returns ``self^iter(letter)``
INPUT:
- ``iter``: -- (default 1) a positive integer
- ``letter``: a letter of the alphabet of the domain of ``self``.
OUTPUT:
if ``iter`` > 1 then returns ``self``^iter(letter)``
if ``iter`` = 1then returns the image of letter
WARNING:
``iter`` may be greater than 1 only if the domain and codomain
of ``self`` are equal (that is to say, ``self`` is a
GraphSelfMap)
EXAMPLES::
sage: G = GraphWithInverses([[0,0,'a'],[0,1,'b'],[1,1,'c']])
sage: A = AlphabetWithInverses(2)
sage: H = GraphWithInverses.rose_graph(A)
sage: f = GraphMap(G,H,"a->ab,b->b,c->B")
sage: f.image('a')
word: ab
"""
if iter == 1:
return self._edge_map.image(letter)
else:
return self._codomain.reduce_path(
self._edge_map(self._edge_map(letter), iter - 1))
def inverse(self):
"""A homotopy inverse of ``self``.
For ``t1=self.domain().spanning_tree()`` and
``t2=self.codomain().spanning_tree()``. The alphabet ``A`` is
made of edges not in ``t1`` identified (by their order in the
alphabets of the domain and codomain) to letters not in
``t2``. The automorphism ``phi`` of ``FreeGroup(A)`` is
defined using ``t1`` and ``t2``. The inverse map is given by
``phi.inverse()`` using ``t1`` and edges from ``t2`` are
mapped to a single point (the root of ``t1``).
In particular the inverse maps all vertices to the root of ``t1``.
OUTPUT:
A homotopy inverse of ``self``.
WARNING:
``self`` is assumed to be a homotopy equivalence.
EXAMPLES::
sage: G = GraphWithInverses([[0,0,'a'],[0,1,'b'],[1,1,'c']])
sage: A = AlphabetWithInverses(2)
sage: H = GraphWithInverses.rose_graph(A)
sage: f = GraphMap(G,H,"a->ab,b->b,c->B")
sage: print f.inverse()
Graph map:
Graph with inverses: a: 0->0, b: 0->0
Graph with inverses: a: 0->0, b: 0->1, c: 1->1
edge map: a->abcB, b->bCB
"""
from free_group import FreeGroup
from free_group_automorphism import FreeGroupAutomorphism
g1 = self.domain()
a1 = g1.alphabet()
t1 = g1.spanning_tree()
g2 = self.codomain()
a2 = g2.alphabet()
t2 = g2.spanning_tree()
A = AlphabetWithInverses(len(a1) - len(g1.vertices()) + 1)
f = FreeGroup(A)
map = dict()
translate = dict()
i = 0
for a in a1.positive_letters():
l = len(t1[g1.initial_vertex(a)]) - len(t1[g1.terminal_vertex(a)])
if (l != 1 or t1[g1.initial_vertex(a)][-1] != a1.inverse_letter(
a)) and (l != -1 or t1[g1.terminal_vertex(a)][-1] != a):
# a is not in the spanning tree
map[A[i]] = self(
t1[g1.initial_vertex(a)] * Word([a]) * g1.reverse_path(
t1[g1.terminal_vertex(a)]))
translate[A[i]] = a
translate[A.inverse_letter(A[i])] = a1.inverse_letter(a)
i += 1
rename = dict()
edge_map = dict()
i = 0
for a in a2.positive_letters():
l = len(t2[g2.initial_vertex(a)]) - len(t2[g2.terminal_vertex(a)])
if (l != 1 or t2[g2.initial_vertex(a)][-1] != a2.inverse_letter(
a)) and (l != -1 or t2[g2.terminal_vertex(a)][-1] != a):
# a is not in the spanning tree
rename[a] = A[i]
rename[a2.inverse_letter(a)] = A.inverse_letter(A[i])
i += 1
else:
edge_map[a] = Word()
for a in map:
map[a] = f([rename[b] for b in map[a] if b in rename])
phi = FreeGroupAutomorphism(map, f)
psi = phi.inverse()
i = 0
for a in a2.positive_letters():
if a not in edge_map:
result = Word()
for b in psi.image(A[i]):
c = translate[b]
result = result * t1[g1.initial_vertex(c)] * Word(
[c]) * g1.reverse_path(t1[g1.terminal_vertex(c)])
edge_map[a] = g1.reduce_path(result)
i += 1
return GraphMap(g2, g1, edge_map)
def tighten(self):
"""
Tighten ``self`` such that there are at least two gates at
each vertex of the domain.
A map is tight if for each vertex ``v`` of the domain, there
exist reduced edge paths ``u`` and ``v`` in the domain with
``self(u)`` and ``self(v)`` non-trivial reduced paths starting
with different edges.
``self`` and ``self.tighten()`` are homotopic.
OUTPUT:
Tighten ``self`` such that there are at least two gates at
each vertex of the domain.
WARNING:
It is assumed that ``self`` is a homotopy equivalence
The result may send edges to trivial edge-paths.
EXAMPLES::
sage: A = AlphabetWithInverses(2)
sage: G = GraphWithInverses.rose_graph(A)
sage: H = GraphWithInverses.rose_graph(A)
sage: f = GraphMap(G,H,"a->baabAB,b->babAB")
sage: f.tighten()
WordMorphism: A->BA, B->B, a->ab, b->b
sage: print f
Graph map:
Graph with inverses: a: 0->0, b: 0->0
Graph with inverses: a: 0->0, b: 0->0
edge map: a->ab, b->b
"""
G1 = self.domain()
A1 = G1.alphabet()
G2 = self.codomain()
edge_map = dict((a, self.image(a)) for a in A1)
done = False
while not done:
done = True
prefix = dict() # the common prefix of all edges outgoing from
# the class of a vertex
adjacent_vertex = dict() # a class of vertices linked by a tree
# which is contracted by self to a point
for a in A1:
u = edge_map[a]
v = G1.initial_vertex(a)
if len(u) > 0:
if v not in adjacent_vertex:
adjacent_vertex[v] = set([v])
for w in adjacent_vertex[v]:
if w in prefix:
if len(prefix[w]) > 0:
p = G2.common_prefix_length(u, prefix[w])
prefix[w] = prefix[w][:p]
else:
prefix[w] = u
else: # we need to increase the adjacent_vertex
vv = G1.terminal_vertex(a)
# note that v!=vv because else the loop a is contracted
# contrary to homotopy equivalence
if v in adjacent_vertex and vv in adjacent_vertex:
adjacent_vertex[v].update(adjacent_vertex[vv])
elif v in adjacent_vertex:
adjacent_vertex[v].add(vv)
elif vv in adjacent_vertex:
adjacent_vertex[v] = adjacent_vertex[vv]
adjacent_vertex[v].add(v)
else:
adjacent_vertex[v] = set([v, vv])
if v in prefix:
prefixv = prefix[v]
else:
prefixv = False
for w in adjacent_vertex[v]:
if v != w:
adjacent_vertex[w] = adjacent_vertex[v]
if w in prefix:
if prefixv:
p = G2.common_prefix_length(prefixv,
prefix[w])
prefixv = prefixv[:p]
else:
prefixv = prefix[w]
if prefixv:
for w in adjacent_vertex[v]:
prefix[w] = prefixv
for a in A1:
v = G1.initial_vertex(a)
if v in prefix and len(prefix[v]) > 0:
done = False
aa = A1.inverse_letter(a)
if len(edge_map[a]) > 0:
edge_map[a] = edge_map[a][len(prefix[v]):]
edge_map[aa] = edge_map[aa][:-len(prefix[v])]
self.set_edge_map(edge_map)
return self._edge_map
def subdivide_domain(self, e):
"""
Subdivide an edge in the domain graph.
The edge ``e`` is subdivided into ``l`` edges where ``l`` is
the length of the image of ``e`` by ``self``.
Update the edge map of ``self``.
Intended to be used by Stalling's folding algorithm to get an
immersion.
...SEE ALSO::
:meth:`sage.combinat.words.GraphWithInverses.subdivide_edge()`
INPUT:
- ``e``: and edge of the domain of ``self``.
OUTPUT:
A dictionnary that maps old edges of the domain to new edges of the domain.
EXAMPLES::
sage: A = AlphabetWithInverses(2)
sage: G = GraphWithInverses.rose_graph(A)
sage: H = GraphWithInverses.rose_graph(A)
sage: f = GraphMap(G,H,"a->aba,b->ab")
sage: f.subdivide_domain('a')
{'A': word: DCA, 'B': word: B, 'a': word: acd, 'b': word: b}
sage: print f
Graph map:
Graph with inverses: a: 0->1, b: 0->0, c: 1->2, d: 2->0
Graph with inverses: a: 0->0, b: 0->0
edge map: a->a, b->ab, c->b, d->a
"""
G = self.domain()
A = G.alphabet()
result_map = dict((a, Word([a])) for a in A)
w = self.image(e)
d = dict((a, self.image(a)) for a in A.positive_letters())
new_edges = A.add_new_letters(len(w) - 1)
new_vertices = G.new_vertices(len(w) - 1)
for i, a in enumerate(new_edges):
v = new_vertices[i]
if i == 0:
vi = G.initial_vertex(e)
vt = G.terminal_vertex(e)
f = new_edges[i][0]
ee = A.inverse_letter(e)
ff = new_edges[i][1]
G.set_terminal_vertex(e, v)
G.add_edge(v, vt, [f, ff])
result_map[e] = result_map[e] * Word([f])
result_map[ee] = Word([ff]) * result_map[ee]
d[a[0]] = w[i + 1]
else:
vi = self._domain.initial_vertex(new_edges[i - 1][0])
vt = self._domain.terminal_vertex(new_edges[i - 1][0])
f = new_edges[i][0]
# ee=A.inverse_letter(e) #already done
ff = new_edges[i][1]
self._domain.set_terminal_vertex(new_edges[i - 1][0], v)
self._domain.add_edge(v, vt, [f, ff])
result_map[e] = result_map[e] * Word([f])
result_map[ee] = Word([ff]) * result_map[ee]
d[a[0]] = w[i + 1]
# updating self.edge_map after subdivision
if A.is_positive_letter(e):
d[e] = Word([self.image(e)[0]])
else:
d[ee] = Word([self.image(ee)[-1]])
self.set_edge_map(d)
return result_map
def illegal_turns(self, turns=None):
"""
List of illegal turns in the domain graph.
A turn is illegal if it is mapped by ``self`` to a degenerate
turn.
INPUT:
- ``turns`` a list of turns of the domain graph. Default is None
meaning all the turns of the graph. I not ``None`` return the
sublist of ``turns`` consisting of illegal turns.
EXAMPLES::
sage: A = AlphabetWithInverses(2)
sage: G = GraphWithInverses.rose_graph(A)
sage: H = GraphWithInverses.rose_graph(A)
sage: f = GraphMap(G,H,"a->aba,b->ab")
sage: f.illegal_turns()
[('a', 'b')]
"""
result = []
if turns is None:
turns = self.domain().turns()
for turn in turns:
u = self.image(turn[0])
v = self.image(turn[1])
if len(u) == 0 or len(v) == 0 or u[0] == v[0]:
result.append(turn)
return result
def stallings_folding(self):
"""
Implement Stallings' folding to get an immersion from ``self``.
The domain of ``self`` is fold until we get an
immersion. Intended to be used to compute the pullback of two
graph maps and the intersection of subgroupes of a free group.
ALGORITHM:
We first subdivide edges of the domain according to length of
their image.
Then fold one gate at one vertex and update the edge map and
illegal turns list.
Repeat the process till no illegal turns remain.
EXAMPLES::
sage: A = AlphabetWithInverses(2)
sage: G = GraphWithInverses.rose_graph(A)
sage: H = GraphWithInverses.rose_graph(A)
sage: f = GraphMap(G,H,"a->aba,b->ab")
sage: f.stallings_folding()
sage: print f
Graph map:
Graph with inverses: a: 1->1, c: 1->1
Graph with inverses: a: 0->0, b: 0->0
edge map: a->a, c->b
REFERENCES:
[Stallings] J. Stallings, Topology of Finite Graphs,
AUTHOR:
- Radhika GUPTA
"""
A = self.domain().alphabet()
for a in A:
if len(self.image(a)) > 1:
self.subdivide_domain(a)
Turns = self._domain.turns()
# list of all turns in domain after subdivision
Il_turns = self.illegal_turns(Turns)
# list of illegal turns in domain after subdivision
counter = 0
while len(Il_turns) > 0:
counter = counter + 1
# find edge_list (list of edges in the gate correspoding to e1)
# to fold at exactly one vertex
e1 = Il_turns[0][0]
edge_list = [e1]
for a in A:
if self._domain.initial_vertex(a) == \
self._domain.initial_vertex(e1) and e1 != a:
if (e1, a) in Il_turns or (a, e1) in Il_turns:
edge_list.append(a)
edge_list = list(set(edge_list)) # remove duplicates
# fold at initial_vertex of e1 ( this function updates the
# domain and edge_map)
self._domain.fold(edge_list, [])
# update edge_map again
d = {}
d[edge_list[0]] = self.image(edge_list[0])
for a in A:
if a not in edge_list:
d[a] = self.image(a)
wm = WordMorphism(d)
self.set_edge_map(wm)
Turns = self._domain.turns() # update list of all turns in domain
Il_turns = self.illegal_turns(Turns)
# update list of illegal turns in domain
def pullback(self, other):
r"""
Pullback of the graph maps ``self`` and ``other``.
The codomain of ``self`` and ``other`` must be the same graph.
The pullback is a graph map f:G3 -> G that makes the diagram commute:
G3 -----> G1
| \ |
| \ | self
| \f |
| \ |
V _\| V
G2 -----> G
....other
The pullback method can be used to find intersection of two subgroups
of a Free Group.
INPUT:
- ``other``: a graph map G2->G with ``self``: a graph map G1->G,
OUTPUT:
A ``GraphMap`` f
EXAMPLES::
sage: G1 = GraphWithInverses.rose_graph(AlphabetWithInverses(2,type='x0'))
sage: G2 = GraphWithInverses.rose_graph(AlphabetWithInverses(2,type='a0'))
sage: G = GraphWithInverses.rose_graph(AlphabetWithInverses(2))
sage: n1 = WordMorphism({'x0':['a','a'],'x1':['b','a']})
sage: n2 = WordMorphism({'a0':['b','a'],'a1':['b','b','b','a','B','a']})
sage: f1 = GraphMap(G1,G,n1)
sage: f2 = GraphMap(G2,G,n2)
sage: print f1.pullback(f2)
Graph map:
Graph with inverses: a0: (0, 0)->(1, 2), a1: (0, 2)->(1, 0), a2: (0, 2)->(1, 3), a3: (0, 3)->(1, 4), a4: (0, 4)->(1, 3), a5: (1, 2)->(0, 0), a6: (1, 4)->(0, 3)
Graph with inverses: a: 0->0, b: 0->0
edge map: a0->b, a1->a, a2->b, a3->b, a4->a, a5->a, a6->a
AUTHORS:
- Radhika GUPTA
"""
import itertools
# First convert self and f2 into immersions
self.stallings_folding()
other.stallings_folding()
G3 = GraphWithInverses()
A = AlphabetWithInverses(0, type='a0')
d = {}
# get set of vertices
V = []
for i in itertools.product(self.domain().vertices(),
other.domain().vertices()):
V.append(i)
# add edges
for v in V:
for w in V:
for e1 in self.domain().alphabet().positive_letters():
if self.domain().initial_vertex(e1) == v[0] \
and self.domain().terminal_vertex(e1) == w[0]:
for e2 in other.domain().alphabet().positive_letters():
if other.domain().initial_vertex(e2) == v[1] \
and other.domain().terminal_vertex(e2) == w[
1]:
if self.image(e1) == other.image(e2):
e = A.add_new_letter()
G3.add_edge(v, w, e)
# update dictionary to define map on G3
d[e[0]] = self.image(e1)
G3._alphabet = A
n3 = WordMorphism(d)
G = self.codomain() # same as other.codomain()
return GraphMap(G3, G, n3)
@staticmethod
def rose_map(automorphism):
"""
The graph map of the rose representing the automorphism.
The rose is built on a copy of the alphabet of the domain of
``automorphism``.
OUTPUT:
The graph map of the rose representing the automorphism.
EXAMPLES::
sage: phi=FreeGroupAutomorphism('a->ab,b->ac,c->a')
sage: print GraphMap.rose_map(phi)
Graph map:
Graph with inverses: a: 0->0, b: 0->0, c: 0->0
Graph with inverses: a: 0->0, b: 0->0, c: 0->0
edge map: a->ab, b->ac, c->a
"""
graph = GraphWithInverses.rose_graph(
automorphism.domain().alphabet().copy())
return GraphMap(graph, graph, automorphism)
class GraphMap():
"""
A GraphMap is a map from a Graph to another. It maps a vertex to
a vertex and an edge to an edge-path. It respects incidence
relation. The inverse of an edge is send to the reverse path.
AUTHORS:
- Thierry Coulbois (2013-05-16): beta.0 version
"""
def __init__(self, *args):
"""
The following forms are accepted:
- ``GraphMap(f)`` where ``f`` is a ``GraphMap``.
- ``GraphMap(domain,codomain,edge_map,vertex_map=None)`` where
``domain`` and ``codomain`` are ``GraphWithInverses`` and
``edge_map`` is anything accepted by
``WordMorphism(edge_map)`` with domain alphabet an
AlphabetWithInverses (note that only one image of the pair
(a,inverse_letter(a)) needs to be defined for each letter).
"""
if isinstance(args[0], GraphMap):
self._domain = args[0]._domain
self._codomain = args[0]._codomain
self._edge_map = args[0]._edge_map
self._vertex_map = args[0]._vertex_map
else:
self._domain = args[0]
self._codomain = args[1]
self.set_edge_map(args[2])
if len(args) > 3:
self._vertex_map = args[3]
def __call__(self, argument):
"""
Applies ``self`` to ``argument`` which is either a vertex of ``self`` or
an edge path.
SEE ALSO:
To compute the image of a letter of the alphabet use
``self.image(a)``.
"""
if self._domain.has_vertex(argument):
if self._vertex_map == None:
self.update_vertex_map()
return self._vertex_map[argument]
else:
return self._codomain.reduce_path(self._edge_map(argument))
def __mul__(self, other):
"""
Compose ``self`` with ``other``.
"""
A = other._domain.alphabet()
result_map = {}
for a in A.positive_letters():
result_map[a] = self(other._edge_map.image(a))
return GraphMap(other._domain, self._codomain, result_map)
def __str__(self):
"""
String represetation of ``self``.
"""
result = "Graph map:\n" + self._domain.__str__() + "\n"
result += self._codomain.__str__() + "\n"
result = result + "edge map: "
for a in self._domain._alphabet.positive_letters():
result += a + "->" + self.image(a).__str__() + ", "
result = result[:-2] + "\n"
if self._vertex_map != None:
result = result + "vertex map: " + self._vertex_map.__str__(
) + "\n"
return result
def domain(self):
"""
Domain of ``self``: this is a graph.
"""
return self._domain
def codomain(self):
"""
Codomain of ``self``: this is a graph.
"""
return self._codomain
def set_edge_map(self, edge_map):
"""
Sets the edge map of ``self``.
``edge_map`` is anything that is accepted by
``Wordmorphism(edge_map)``, the image of the inverse letters
will be calculated: they need not be explicit in ``edge_map``,
only one of the images of each pair [letter,inverse(letter)]
need to be given by ``edge_map``. Images of ``edge_map`` need
not be reduced.
"""
A = self.domain().alphabet()
tmp_map = WordMorphism(edge_map)
m = {}
for a in tmp_map._domain.alphabet():
m[a] = self._codomain.reduce_path(tmp_map.image(a))
m[A.inverse_letter(a)] = self._codomain.reverse_path(m[a])
self._edge_map = WordMorphism(m)
self._vertex_map = None
def compose_edge_map(self, edge_morph):
"""
Compose ``self`` with the morphism ``edge_morph``.
Update the edge_map of ``self`` with (``edge_morph`` o ``self``).
"""
edge_map = dict((a, edge_morph(self._edge_map.image(a)))
for a in self._domain._alphabet.positive_letters())
self.set_edge_map(edge_map)
def update_vertex_map(self):
"""
Computes the vertex map of ``self`` from its edge map.
"""
vertex_map = {}
for e in self._domain._alphabet.positive_letters():
p = self.image(e)
if len(p) > 0:
vertex_map[self._domain.initial_vertex(
e)] = self._codomain.initial_vertex(p[0])
vertex_map[self._domain.terminal_vertex(
e)] = self._codomain.terminal_vertex(p[-1])
self._vertex_map = vertex_map
def edge_map(self):
"""
The edge map of ``self``: this is a word morphism.
"""
return self._edge_map
def image(self, letter, iter=1):
"""
The image of a letter.
if ``iter>1`` then returns ``self^iter(letter)``
"""
if iter == 1:
return self._edge_map.image(letter)
else:
return self._codomain.reduce_path(
self._edge_map(self._edge_map(letter), iter - 1))
def inverse(self):
"""A homotopy inverse of ``self``.
For ``t1=self.domain().spanning_tree()`` and
``t2=self.codomain().spanning_tree()``. The alphabet ``A`` is
made of edges not in ``t1`` identified (by their order in the
alphabets of the domain and codomain) to letters not in
``t2``. The automorphism ``phi`` of ``FreeGroup(A)`` is
defined using ``t1`` and ``t2``. The inverse map is given by
``phi.inverse()`` using ``t1`` and edges from ``t2`` are
mapped to a single point (the root of ``t1``).
In particular the inverse maps all vertices to the root of ``t1``.
WARNING:
``self`` is assumed to be a homotopy equivalence.
"""
from free_group import FreeGroup
from free_group_automorphism import FreeGroupAutomorphism
G1 = self.domain()
A1 = G1.alphabet()
t1 = G1.spanning_tree()
G2 = self.codomain()
A2 = G2.alphabet()
t2 = G2.spanning_tree()
A = AlphabetWithInverses(len(A1) - len(G1.vertices()) + 1)
F = FreeGroup(A)
map = dict()
translate = dict()
i = 0
for a in A1.positive_letters():
l = len(t1[G1.initial_vertex(a)]) - len(t1[G1.terminal_vertex(a)])
if (l!=1 or t1[G1.initial_vertex(a)][-1]!=A1.inverse_letter(a)) and\
(l!=-1 or t1[G1.terminal_vertex(a)][-1]!=a): # a is not in the spanning tree
map[A[i]] = self(t1[G1.initial_vertex(a)] * Word([a]) *
G1.reverse_path(t1[G1.terminal_vertex(a)]))
translate[A[i]] = a
translate[A.inverse_letter(A[i])] = A1.inverse_letter(a)
i += 1
rename = dict()
edge_map = dict()
i = 0
for a in A2.positive_letters():
l = len(t2[G2.initial_vertex(a)]) - len(t2[G2.terminal_vertex(a)])
if (l!=1 or t2[G2.initial_vertex(a)][-1]!=A2.inverse_letter(a)) and\
(l!=-1 or t2[G2.terminal_vertex(a)][-1]!=a): # a is not in the spanning tree
rename[a] = A[i]
rename[A2.inverse_letter(a)] = A.inverse_letter(A[i])
i += 1
else:
edge_map[a] = Word()
for a in map:
map[a] = F([rename[b] for b in map[a] if b in rename])
phi = FreeGroupAutomorphism(map, F)
psi = phi.inverse()
i = 0
for a in A2.positive_letters():
if a not in edge_map:
result = Word()
for b in psi.image(A[i]):
c = translate[b]
result = result * t1[G1.initial_vertex(c)] * Word(
[c]) * G1.reverse_path(t1[G1.terminal_vertex(c)])
edge_map[a] = G1.reduce_path(result)
i += 1
return GraphMap(G2, G1, edge_map)
def tighten(self):
"""
Tighten ``self`` such that there are at least two gates at
each vertex of the domain.
A map is tight if for each vertex ``v`` of the domain, there
exist reduced edge paths ``u`` and ``v`` in the domain with
``self(u)`` and ``self(v)`` non-trivial reduced paths starting
with different edges.
``self`` and ``self.tighten()`` are homotopic.
WARNING:
It is assumed that ``self`` is a homotopy equivalence
The result may send edges to trivial edge-paths.
"""
G1 = self.domain()
A1 = G1.alphabet()
G2 = self.domain()
edge_map = dict((a, self.image(a)) for a in A1)
done = False
while not done:
done = True
prefix = dict(
) # the common prefix of all edges outgoing from the class of a vertex
adjacent_vertex = dict(
) # a class of vertices linked by a tree which is contracted by self to a point
for a in A1:
u = edge_map[a]
v = G1.initial_vertex(a)
if len(u) > 0:
if v not in adjacent_vertex:
adjacent_vertex[v] = set([v])
for w in adjacent_vertex[v]:
if w in prefix:
if len(prefix[w]) > 0:
p = G2.common_prefix_length(u, prefix[w])
prefix[w] = prefix[w][:p]
else:
prefix[w] = u
else: #we need to increase the adjacent_vertex
vv = G1.terminal_vertex(
a
) #note that v!=vv because else the loop a is contracted contrary to homotopy equivalence
if v in adjacent_vertex and vv in adjacent_vertex:
adjacent_vertex[v].update(adjacent_vertex[vv])
elif v in adjacent_vertex:
adjacent_vertex[v].add(vv)
elif vv in adjacent_vertex:
adjacent_vertex[v] = adjacent_vertex[vv]
adjacent_vertex[v].add(v)
else:
adjacent_vertex[v] = set([v, vv])
if v in prefix:
prefixv = prefix[v]
else:
prefixv = False
for w in adjacent_vertex[v]:
if v != w:
adjacent_vertex[w] = adjacent_vertex[v]
if w in prefix:
if prefixv:
p = G2.common_prefix_length(
prefixv, prefix[w])
prefixv = prefixv[:p]
else:
prefixv = prefix[w]
if prefixv:
for w in adjacent_vertex[v]:
prefix[w] = prefixv
for a in A1:
v = G1.initial_vertex(a)
if v in prefix and len(prefix[v]) > 0:
done = False
aa = A1.inverse_letter(a)
if len(edge_map[a]) > 0:
edge_map[a] = edge_map[a][len(prefix[v]):]
edge_map[aa] = edge_map[aa][:-len(prefix[v])]
# else:
# edge_map[a]=G2.reverse_path(prefix[v])
# edge_map[aa]=prefix[v]
break
self.set_edge_map(edge_map)
return self
def subdivide_domain(self, e):
"""
Subdivide an edge in the domain graph.
The edge ``e`` is subdivided into ``l`` edges where ``l`` is
the length of the image of ``e`` by ``self``.
Update the edge map of ``self``.
Intended to be used by Stalling's folding algorithm to get an
immersion.
SEE ALSO::
GraphWithInverses.subdivide_edge()
"""
G = self.domain()
A = G.alphabet()
result_map = dict((a, Word([a])) for a in A)
w = self.image(e)
new_edges = A.add_new_letters(len(w) - 1)
new_vertices = G.new_vertices(len(w) - 1)
d = {(a, self.image(a)) for a in A.positive_letters()}
for i, a in enumerate(new_edges):
v = new_vertices[i]
if i == 0:
vi = G.initial_vertex(e)
vt = G.terminal_vertex(e)
f = new_edges[i][0]
ee = A.inverse_letter(e)
ff = new_edges[i][1]
G.set_terminal_vertex(e, v)
G.add_edge(v, vt, [f, ff])
result_map[e] = result_map[e] * Word([f])
result_map[ee] = Word([ff]) * result_map[ee]
d[a[0]] = w[i + 1]
else:
vi = self._domain.initial_vertex(new_edges[i - 1][0])
vt = self._domain.terminal_vertex(new_edges[i - 1][0])
f = new_edges[i][0]
#ee=A.inverse_letter(e) #already done
ff = new_edges[i][1]
self._domain.set_terminal_vertex(new_edges[i - 1][0], v)
self._domain.add_edge(v, vt, [f, ff])
result_map[e] = result_map[e] * Word([f])
result_map[ee] = Word([ff]) * result_map[ee]
d[a[0]] = w[i + 1]
# updating self.edge_map after subdivision
if A.is_positive_letter(e):
d[e] = Word([self.image(e)[0]])
else:
d[ee] = Word([self.image(ee)[-1]])
self.set_edge_map(d)
return result_map
def illegal_turns(self, turns=None):
"""
List of illegal turns in the domain graph.
A turn is illegal if it is mapped by ``self`` to a degenerate
turn.
INPUT:
``turns`` a list of turns of the domain graph. Default is None
meaning all the turns of the graph. I not ``None`` return the
sublist of ``turns`` consisting of illegal turns.
"""
result = []
if turns is None:
turns = self.domain().turns()
for turn in turns:
u = self.image(turn[0])
v = self.image(turn[1])
if len(u) == 0 or len(v) == 0 or u[0] == v[0]:
result.append(turn)
return result
def stallings_folding(self):
"""
Implement Stallings' folding to get an immersion from ``self``.
The domain of ``self`` is fold until we get an immersion.
ALGORITHM:
We first subdivide edges of the domain according to length of
their image.
Then fold one gate at one vertex and update the edge map and
illegal turns list.
Repeat the process till no illegal turns remain.
REFERENCES:
[Stallings] J. Stallings, Topology of Finite Graphs,
"""
A = self.domain().alphabet()
for a in A:
if len(self.image(a)) > 1:
self.subdivide_domain(a)
Turns = self._domain.turns(
) #list of all turns in domain after subdivision
Il_turns = self.illegal_turns(
Turns) # list of illegal turns in domain after subdivision
counter = 0
while len(Il_turns) > 0:
counter = counter + 1
# find edge_list (list of edges in the gate correspoding to e1) to fold at exactly one vertex
e1 = Il_turns[0][0]
edge_list = [e1]
for a in A:
if self._domain.initial_vertex(
a) == self._domain.initial_vertex(e1) and e1 != a:
if (e1, a) in Il_turns or (a, e1) in Il_turns:
edge_list.append(a)
edge_list = list(set(edge_list)) # remove duplicates
# fold at initial_vertex of e1 ( this function updates the domain and edge_map)
self._domain.fold(edge_list, [])
#update edge_map again
d = {}
d[edge_list[0]] = self.image(edge_list[0])
for a in A:
if a not in edge_list:
d[a] = self.image(a)
wm = WordMorphism(d)
self.set_edge_map(wm)
Turns = self._domain.turns() #update list of all turns in domain
Il_turns = self.illegal_turns(
Turns) # update list of illegal turns in domain
return self
def pullback(self, f2, G3, A):
"""
INPUT : Two Graph maps f1:G1->G, f2:G2->G, an empty GraphWithInverses G3 and an empty AlphabetWithInverses A
OPERATION : Find the pullback G3 and a graph map f:G3->G
OUTPUT : Graphmap f
The pullback method can be used to find intersection of two subgroups of a Free Group.
Example :
G1 = GraphWithInverses.rose_graph(AlphabetWithInverses(2,type='x0'))
G2 = GraphWithInverses.rose_graph(AlphabetWithInverses(2,type='a0'))
G = GraphWithInverses.rose_graph(AlphabetWithInverses(2))
n1 = WordMorphism({'x0':['a','a'],'x1':['b','a']})
n2 = WordMorphism({'a0':['b','a'],'a1':['b','b','b','a','B','a']})
f1 = GraphMap(G1,G,n1)
f2 = GraphMap(G2,G,n2)
G3 = GraphWithInverses()
A = AlphabetWithInverses(0,type='a0')
f1.pullback(f2,G3,A)
"""
import itertools
#First convert self and f2 into immersions
self.stallings_folding()
f2.stallings_folding()
# G3 = GraphWithInverses()
#A = AlphabetWithInverses(0,type='a0')
d = {}
#get set of vertices
V = []
for i in itertools.product(self.domain().vertices(),
f2.domain().vertices()):
V.append(i)
#add edges
for v in V:
for w in V:
for e1 in self.domain().alphabet().positive_letters():
if self.domain().initial_vertex(e1) == v[
0] and self.domain().terminal_vertex(e1) == w[0]:
for e2 in f2.domain().alphabet().positive_letters():
if f2.domain().initial_vertex(
e2) == v[1] and f2.domain(
).terminal_vertex(e2) == w[1]:
if self.image(e1) == f2.image(e2):
e = A.add_new_letter()
G3.add_edge(v, w, e)
#update dictionary to define map on G3
d[e[0]] = self.image(e1)
G3._alphabet = A
n3 = WordMorphism(d)
G = self.codomain() #same as f2.codomain()
return GraphMap(G3, G, n3)
@staticmethod
def rose_map(automorphism):
"""
The graph map of the rose representing the automorphism.
The rose is built on a copy of the alphabet of the domain of
``automorphism``.
"""
graph = GraphWithInverses.rose_graph(
automorphism.domain().alphabet().copy())
return GraphMap(graph, graph, automorphism)
