def ThreePrismGraph():
r"""
Return 3-prism graph.
EXAMPLES::
sage: from flexrilog import GraphGenerator, FlexRiGraph
sage: FlexRiGraph([(0, 3), (0, 4), (0, 5), (1, 2), (1, 4), (1, 5), (2, 3), (2, 5), (3, 4)]) == GraphGenerator.ThreePrismGraph()
True
.. PLOT::
:scale: 70
from flexrilog import GraphGenerator
G = GraphGenerator.ThreePrismGraph()
sphinx_plot(G)
"""
G = FlexRiGraph(Integer(7916),
name='3-prism',
pos={
4: [0.6, 0.4],
5: [0, 1.4],
2: [1, 1.4],
3: [1, 0],
0: [0, 0],
1: [0.6, 1]
})
G._swap_xy()
return G
def Deltoid(cls, par_type='rational'):
r"""
Return a deltoid motion.
"""
if par_type == 'rational':
FF = FunctionField(QQ, 't')
t = FF.gen()
C = {
_sage_const_0 : vector((_sage_const_0 , _sage_const_0 )),
_sage_const_1 : vector((_sage_const_1 , _sage_const_0 )),
_sage_const_2 : vector((_sage_const_4 *(t**_sage_const_2 - _sage_const_2
)/(t**_sage_const_2 + _sage_const_4 ), _sage_const_12 *t/(t**_sage_const_2 + _sage_const_4 ))),
_sage_const_3 : vector(((t**_sage_const_4 - _sage_const_13 *t**_sage_const_2 + _sage_const_4
)/(t**_sage_const_4 + _sage_const_5 *t**_sage_const_2 + _sage_const_4 ),
_sage_const_6 *(t**_sage_const_3 - _sage_const_2 *t)/(t**_sage_const_4 + _sage_const_5 *t**_sage_const_2 + _sage_const_4 )))
}
G = FlexRiGraph([[0, 1], [1, 2], [2, 3], [0, 3]])
return GraphMotion.ParametricMotion(G, C, 'rational', sampling_type='tan', check=False)
elif par_type == 'symbolic':
t = var('t')
C = {
_sage_const_0 : vector((_sage_const_0 , _sage_const_0 )),
_sage_const_1 : vector((_sage_const_1 , _sage_const_0 )),
_sage_const_2 : vector((_sage_const_4 *(t**_sage_const_2 - _sage_const_2
)/(t**_sage_const_2 + _sage_const_4 ), _sage_const_12 *t/(t**_sage_const_2 + _sage_const_4 ))),
_sage_const_3 : vector(((t**_sage_const_4 - _sage_const_13 *t**_sage_const_2 + _sage_const_4
)/(t**_sage_const_4 + _sage_const_5 *t**_sage_const_2 + _sage_const_4 ),
_sage_const_6 *(t**_sage_const_3 - _sage_const_2 *t)/(t**_sage_const_4 + _sage_const_5 *t**_sage_const_2 + _sage_const_4 )))
}
G = FlexRiGraph([[0, 1], [1, 2], [2, 3], [0, 3]])
return ParametricGraphMotion.ParametricMotion(G, C, 'symbolic', sampling_type='tan', check=False)
else:
raise exceptions.ValueError('Deltoid with par_type ' + str(par_type) + ' is not supported.')
def NoNACGraph():
r"""
Return a graph without NAC-coloring.
EXAMPLE::
sage: from flexrilog import GraphGenerator
sage: GraphGenerator.NoNACGraph()
NoNAC: FlexRiGraph with 7 vertices and 12 edges
.. PLOT::
:scale: 70
from flexrilog import GraphGenerator
G = GraphGenerator.NoNACGraph()
sphinx_plot(G)
"""
return FlexRiGraph(Integer(448412),
pos={
0: (-0.5, -0.75),
1: (0.5, 0.5),
2: (1.5, 0.5),
3: (2.5, -0.75),
4: (0.5, 1.5),
5: (1.5, 1.5),
6: (1, -0.25)
},
name='NoNAC')
def S4Graph():
r"""
Return the graph $S_4$.
EXAMPLE::
sage: from flexrilog import GraphGenerator
sage: GraphGenerator.S4Graph()
S_4: FlexRiGraph with 8 vertices and 14 edges
.. PLOT::
:scale: 70
from flexrilog import GraphGenerator
G = GraphGenerator.S4Graph()
sphinx_plot(G)
"""
return FlexRiGraph(
[(1, 4), (2, 3), (3, 6), (6, 5), (2, 5), (5, 4), (6, 1), (3, 4),
(2, 1), (5, 7), (5, 8), (7, 4), (7, 8), (4, 8)],
pos={
1: (0.5, -0.866025),
2: (1., 0.),
3: (0.5, 0.866025),
4: (-0.5, 0.866025),
5: (-1., 0.),
6: (-0.5, -0.866025),
7: (-0.519, 0.30),
8: (-0.952, 0.550),
},
name='S_4')
def S3Graph():
r"""
Return the graph $S_3$.
EXAMPLE::
sage: from flexrilog import GraphGenerator
sage: GraphGenerator.S3Graph()
S_3: FlexRiGraph with 8 vertices and 14 edges
.. PLOT::
:scale: 70
from flexrilog import GraphGenerator
G = GraphGenerator.S3Graph()
sphinx_plot(G)
"""
return FlexRiGraph(
[(7, 3), (2, 6), (5, 1), (2, 5), (7, 4), (0, 6), (2, 7), (4, 5),
(3, 6), (4, 6), (3, 5), (7, 1), (0, 7), (0, 1)],
pos={
1: (-0.8, -1.4),
7: (1., 0.),
3: (-1., 0.),
6: (-0.5, -0.866025),
2: (0.5, -0.866025),
5: (-0.5, 0.866025),
4: (0.5, 0.866025),
0: (0.8, -1.4)
},
name='S_3')
def S1Graph():
r"""
Return the graph $S_1$.
EXAMPLE::
sage: from flexrilog import GraphGenerator
sage: GraphGenerator.S1Graph()
S_1: FlexRiGraph with 8 vertices and 14 edges
.. PLOT::
:scale: 70
from flexrilog import GraphGenerator
G = GraphGenerator.S1Graph()
sphinx_plot(G)
"""
return FlexRiGraph(
[[6, 1], [2, 3], [5, 4], [4, 3], [6, 3], [6, 5], [1, 2], [8, 3],
[8, 5], [7, 4], [6, 7], [8, 7], [1, 5], [4, 2]],
pos={
1: (-1, 0.8),
2: (-2, -0.2),
3: (0, -1),
4: (-1, 0),
5: (0, 1),
6: (1, 0),
7: (0, -1.75),
8: (1.8, 0)
},
name='S_1')
def Q6Graph():
r"""
Return the graph $Q_6$.
EXAMPLE::
sage: from flexrilog import GraphGenerator
sage: GraphGenerator.Q6Graph()
Q_6: FlexRiGraph with 8 vertices and 13 edges
.. PLOT::
:scale: 70
from flexrilog import GraphGenerator
G = GraphGenerator.Q6Graph()
sphinx_plot(G)
"""
G = FlexRiGraph(
[(0, 1), (0, 2), (0, 5), (1, 4), (1, 7), (2, 3), (2, 6), (3, 6),
(3, 7), (4, 6), (4, 7), (5, 6), (5, 7)],
pos={
0: (0.0, -0.433),
1: (-0.5, -0.866),
2: (0.5, -0.866),
3: (1.0, 0.0),
4: (-1.0, 0.0),
5: (0.0, 0.0),
6: (0.5, 0.866),
7: (-0.5, 0.866)
},
name='Q_6')
return G
def Q5Graph():
r"""
Return the graph $Q_5$.
EXAMPLE::
sage: from flexrilog import GraphGenerator
sage: GraphGenerator.Q5Graph()
Q_5: FlexRiGraph with 8 vertices and 13 edges
.. PLOT::
:scale: 70
from flexrilog import GraphGenerator
G = GraphGenerator.Q5Graph()
sphinx_plot(G)
"""
G = FlexRiGraph(
[(0, 1), (0, 3), (0, 6), (1, 2), (1, 6), (2, 4), (2, 7), (3, 5),
(3, 7), (4, 6), (4, 7), (5, 6), (5, 7)],
pos={
0: (-0.587, -0.809),
1: (0.587, -0.809),
2: (0.951, 0.309),
3: (-0.951, 0.309),
4: (0.235, 0.323),
5: (-0.235, 0.323),
6: (0.0, -0.4),
7: (0.0, 1.0)
},
name='Q_5')
return G
def SmallestFlexibleLamanGraph():
r"""
Return the smallest Laman graph that has a flexible labeling.
EXAMPLES::
sage: from flexrilog import GraphGenerator, FlexRiGraph
sage: FlexRiGraph([[0,1],[1,2],[0,2],[0,3],[1,3],[2,4],[3,4]]) == GraphGenerator.SmallestFlexibleLamanGraph()
True
.. PLOT::
:scale: 70
from flexrilog import GraphGenerator
G = GraphGenerator.SmallestFlexibleLamanGraph()
sphinx_plot(G)
"""
G = FlexRiGraph(
[[0, 1], [1, 2], [0, 2], [0, 3], [1, 3], [2, 4], [3, 4]],
name='SmallestFlexibleLamanGraph',
pos={
0: [0, 0],
1: [2, 0],
2: [1, 1],
3: [1, -1],
4: [3, 0]
})
return G
def S5Graph(old_labeling=False):
r"""
Return the graph $S_5$.
EXAMPLE::
sage: from flexrilog import GraphGenerator
sage: GraphGenerator.S5Graph()
S_5: FlexRiGraph with 8 vertices and 13 edges
.. PLOT::
:scale: 70
from flexrilog import GraphGenerator
G = GraphGenerator.S5Graph()
sphinx_plot(G)
"""
if not old_labeling:
return FlexRiGraph(
[(1, 2), (1, 3), (1, 4), (1, 5), (2, 6), (2, 3), (3, 7),
(4, 5), (4, 6), (4, 7), (5, 8), (6, 8), (7, 8)],
pos={
2: (-0.588, -0.809),
3: (0.588, -0.809),
7: (0.588, 0.309),
6: (-0.588, 0.309),
4: (0.235, 0),
5: (-0.235, 0),
1: (0.000, -0.400),
8: (0.000, 0.7)
},
name='S_5')
return FlexRiGraph(
[(0, 3), (0, 4), (0, 5), (1, 2), (1, 4), (1, 6), (2, 3), (2, 6),
(3, 7), (4, 7), (5, 6), (5, 7), (6, 7)],
pos={
1: (-0.588, -0.809),
2: (0.588, -0.809),
3: (0.588, 0.309),
4: (-0.588, 0.309),
7: (0.235, 0),
5: (-0.235, 0),
6: (0.000, -0.400),
0: (0.000, 0.7)
},
name='S_5')
def K23Graph():
r"""
Return the graph $K_{2,3}$.
EXAMPLES::
sage: from flexrilog import GraphGenerator, FlexRiGraph
sage: FlexRiGraph(graphs.CompleteBipartiteGraph(2,3)).is_isomorphic(GraphGenerator.K23Graph())
True
.. PLOT::
:scale: 70
from flexrilog import GraphGenerator
G = GraphGenerator.K23Graph()
sphinx_plot(G)
"""
K23 = FlexRiGraph([(1, 2), (1, 4), (3, 2), (3, 4), (5, 2), (5, 4)],
name='K23',
pos={
4: (0, 1),
5: (1, 0),
3: (0.00, 0.000),
2: (0, -1),
1: (-1, 0)
})
for delta in K23.NAC_colorings():
for edges in [delta.red_edges(), delta.blue_edges()]:
if len(edges) == 2:
delta.set_name('alpha' +
str(edges[0].intersection(edges[1])[0]))
elif len(edges) == 3:
if Set(edges[0]).intersection(Set(edges[1])).intersection(
Set(edges[2])):
delta.set_name('gamma')
else:
for e in edges:
if not e.intersection(
Set(
flatten([
list(e2) for e2 in edges if e2 != e
]))):
delta.set_name('beta' + str(e[0]) + str(e[1]))
return K23
def K33Graph():
r"""
Return the graph $K_{3,3}$.
EXAMPLES::
sage: from flexrilog import GraphGenerator, FlexRiGraph
sage: FlexRiGraph(graphs.CompleteBipartiteGraph(3,3)).is_isomorphic(GraphGenerator.K33Graph())
True
.. PLOT::
:scale: 70
from flexrilog import GraphGenerator
G = GraphGenerator.K33Graph()
sphinx_plot(G)
"""
K33 = FlexRiGraph(
[(1, 2), (1, 4), (1, 6), (3, 2), (3, 4), (3, 6), (5, 2), (5, 4),
(5, 6)],
name='K33',
pos={
4: (0.500, 0.866),
5: (-0.500, 0.866),
6: (-1.00, 0.000),
3: (1.00, 0.000),
2: (0.500, -0.866),
1: (-0.500, -0.866)
})
for delta in K33.NAC_colorings():
for edges in [delta.red_edges(), delta.blue_edges()]:
if len(edges) == 3:
delta.set_name('omega' + str(edges[0].intersection(
edges[1]).intersection(edges[2])[0]))
elif len(edges) == 5:
for e in edges:
if not e.intersection(
Set(
flatten(
[list(e2)
for e2 in edges if e2 != e]))):
delta.set_name('epsilon' + str(e[0]) + str(e[1]))
return K33
def LamanGraphs(n):
r"""
Return the Laman graphs with ``n`` vertices.
See [CGGKLS2018b]_.
INPUT:
- ``n`` an integer from 3 to 8
EXAMPLE::
sage: from flexrilog import GraphGenerator
sage: [len(GraphGenerator.LamanGraphs(n)) for n in range(3,8)]
[1, 1, 3, 13, 70]
sage: GraphGenerator.ThreePrismGraph() in GraphGenerator.LamanGraphs(6)
True
"""
if n == 3:
return [FlexRiGraph(Integer(i)) for i in [7]]
elif n == 4:
return [FlexRiGraph(Integer(i)) for i in [31]]
elif n == 5:
return [FlexRiGraph(Integer(i)) for i in [254, 239, 223]]
elif n == 6:
return [
FlexRiGraph(Integer(i)) for i in [
3326, 4011, 7672, 7916, 3934, 10479, 6891, 5791, 3447,
12511, 3451, 3311, 3295
]
]
elif n == 7:
return [
FlexRiGraph(Integer(i)) for i in [
120478, 127198, 190686, 104371, 183548, 412894, 102238,
167646, 101630, 103932, 103805, 560509, 104055, 112469,
112525, 111070, 127575, 190103, 104365, 174558, 189853,
186013, 192733, 174823, 111335, 102253, 127567, 167773,
113483, 560927, 312735, 102262, 298486, 481867, 1269995,
1270351, 190875, 414941, 1256267, 186617, 298919, 401101,
313719, 567671, 104126, 123255, 400857, 312759, 102206,
120414, 202335, 200059, 331599, 330991, 222443, 567647,
182879, 169631, 659039, 410847, 167711, 174303, 101751,
396511, 298223, 104171, 173279, 101743, 101615, 101599
]
]
elif n == 8:
return [
FlexRiGraph(Integer(i)) for i in [
7510520, 19111408, 6739377, 6740393, 8000953, 6462968,
6475132, 6411644, 69373436, 6393718, 18995565, 20125169,
20140275, 19617907, 11357278, 6411934, 12324062, 6418654,
6418797, 12127482, 69311341, 10964186, 69325181, 20126170,
19111826, 170989214, 170990174, 170957470, 204540766,
104368318, 104400062, 20584859, 93882590, 210028766,
20093843, 20126547, 210799326, 19127059, 160959710,
6419230, 20126603, 35765022, 19118979, 36778783, 7877407,
7038046, 7935582, 7947390, 7750878, 12094716, 11636460,
11636216, 12703097, 6396410, 13944030, 26774750, 10610910,
6396654, 37850494, 12707181, 14191866, 13733610, 69311198,
21091693, 26316494, 22058477, 50437485, 6462174, 10833118,
6886654, 10816766, 25283806, 6410494, 25267454, 6411870,
10609886, 6393694, 69310174, 6395757, 10800350, 10587870,
10587390, 6393086, 6465020, 19174901, 7459635, 6478445,
35829997, 10672733, 19060973, 35838077, 7459258, 20026868,
6470121, 10672377, 69376505, 6482285, 6484205, 10676461,
69376749, 69380589, 36739453, 10678493, 69390589, 6470554,
7444402, 6927794, 7903674, 7935418, 20093404, 19111764,
50568561, 19111820, 7909811, 6470492, 6482718, 7942515,
7688563, 8132019, 7950651, 6484765, 13669881, 12899677,
13153629, 13161597, 12760537, 21143017, 25271785, 25337305,
6461945, 21157213, 25285981, 6395827, 12887533, 21272045,
25464301, 12768861, 21157101, 21149133, 25285869, 25277901,
25351389, 25337549, 26252765, 25730285, 50503005, 6460921,
6461165, 35755389, 35820797, 10652925, 6672166, 20518261,
51481974, 6470539, 10803611, 6465459, 10790807, 6609679,
19056061, 69507983, 19187127, 10821356, 12885740, 6704926,
6692588, 10833694, 6706846, 10836574, 10835614, 69381007,
35969311, 35860269, 6470513, 6482739, 6958893, 37924653,
35860254, 6958878, 6404012, 35765043, 6467444, 6405034,
35764021, 6958766, 8130343, 6484597, 35860142, 37924526,
37938743, 13166135, 13154895, 25281790, 12895486, 25478391,
12887539, 21145075, 21140985, 25269753, 21158479, 12883449,
18993406, 10799358, 25466333, 21551215, 21543247, 19190007,
50444623, 19063031, 6465423, 35753341, 11585183, 6465143,
10587767, 6692344, 20581749, 20137333, 20583837, 20092821,
20093781, 19570453, 50699494, 12897660, 20533367, 6704508,
20042519, 11364431, 11881039, 6403925, 6403981, 6478619,
10672795, 69376923, 69388731, 36739515, 12339295, 6482703,
6419031, 12142715, 6419215, 11880683, 11352267, 11868875,
69325599, 69389087, 12327131, 69391007, 11882715, 35838495,
36739871, 7122342, 7638950, 20042142, 51484077, 19504918,
50699606, 6954620, 13147772, 35757884, 6966846, 12896179,
35868222, 51681511, 21153715, 51491047, 50436019, 13154539,
12892619, 35756475, 6887031, 12904783, 21153211, 12895675,
25478559, 26821727, 10610327, 6396787, 12696395, 21158123,
21158235, 21150155, 21092715, 6493939, 37852603, 6410683,
37850927, 12887951, 18999639, 18998679, 26813647, 12756891,
26559695, 6855287, 19467639, 35982711, 6410871, 50433399,
21164239, 6480315, 21162319, 35778719, 27004111, 10799547,
6624999, 6402791, 21098831, 25898187, 71605071, 22059855,
19190175, 84056783, 10805663, 10610263, 6393709, 6461339,
6473147, 35771517, 35763549, 18976599, 35778655, 10602159,
35753759, 10667679, 10798495, 19434911, 27976853, 28624277,
28887213, 28625293, 6463416, 7653750, 7653806, 8129910,
7645670, 20518302, 20027286, 6478522, 20028246, 20093654,
51484083, 19570326, 7652790, 20486558, 19126870, 20125581,
35764910, 19603213, 19118917, 50699683, 7137575, 7654183,
60082263, 60082319, 60081303, 20533535, 6478460, 119047351,
119048367, 117884055, 117885071, 20518679, 20107927,
35859898, 6475580, 6958522, 37924282, 6413114, 35763894,
35779127, 38837051, 36772667, 7871291, 8136103, 37938493,
20486935, 20585047, 19585559, 13165885, 42067261,
110888095, 177489055, 235291807, 20140687, 51678631,
236193183, 19512071, 20130631, 50713767, 35772990,
10817516, 50967719, 36771518, 38835902, 12881900, 7870142,
50705799, 36743991, 25268204, 19618319, 211042527,
19610183, 35778750, 50711847, 36769598, 6462456, 7868222,
6773323, 7547723, 13671147, 13674987, 35854587, 7235663,
7752271, 8133199, 8145007, 12141693, 12127853, 11357293,
10970317, 8020063, 12324077, 13675343, 11422813, 13689071,
11865577, 11226233, 11218153, 25511151, 19077839, 12193017,
25576671, 13460559, 12773723, 71474011, 12708203, 71479771,
71477739, 71414251, 22059499, 71866747, 83991019, 14435423,
14238843, 12773967, 14225003, 13454443, 13067467, 18999695,
22073583, 71872863, 84068703, 20006303, 19485855, 42067103,
14421227, 71480015, 13519963, 71478095, 71491935, 69325471,
71414607, 22073695, 71872751, 84068591, 51419375, 84449519,
36771487, 38835871, 19470575, 21534959, 100767391,
71426415, 22071663, 71870831, 84003183, 36778207, 75620575,
6415725, 6419023, 8143983, 10606173, 19453087, 7234639,
6772299, 35771935, 36738719, 12895471, 36745439, 12805355,
21187947, 6494955, 6887803, 71517563, 38931835, 38933755,
100863355, 37950715, 12782971, 38868347, 6508795, 31004235,
55776843, 14197997, 26330711, 6888047, 21564783, 21370331,
12985819, 21582943, 12888303, 10823919, 25562587, 25496815,
30106699, 155852363, 147594827, 147150411, 14191993,
12963177, 21287151, 37971055, 26318547, 25796179, 38937839,
38937951, 25479407, 84100335, 12902639, 25288943, 19076823,
38868575, 10692255, 21566703, 10660511, 10689119, 25800263,
21160175, 6953567, 6639855, 6893807, 7084271, 38872431,
25407171, 51449199, 51451119, 25481439, 26297567, 76096735,
19091679, 25340127, 50501855, 6462239, 26741983, 6705375,
25562335, 13145339, 12684911, 7902814, 13145695, 12750431,
12687083, 21140955, 21138923, 12750075, 21091919, 37850735,
36000863, 11617887, 42067039, 10587823, 6393647, 21073135,
13944043, 21141199, 36771423, 12756575, 100767327,
21139279, 21153007, 21153119, 50435311, 35760479, 18991455,
50433375, 26264799, 75810015, 21087599, 12686971, 50500831,
19437791, 10653343, 12684655, 25726175, 10587935, 10816735,
6393207, 25267423, 6395259, 35753583, 18975983, 6481131,
19434847, 10798303, 10653279, 10783967, 6393199, 6393071,
6393055
]
]
def Q2Graph(old_labeling=False):
r"""
Return the graph $Q_2$.
EXAMPLE::
sage: from flexrilog import GraphGenerator
sage: GraphGenerator.Q2Graph()
Q_2: FlexRiGraph with 8 vertices and 13 edges
.. PLOT::
:scale: 70
from flexrilog import GraphGenerator
G = GraphGenerator.Q2Graph()
sphinx_plot(G)
"""
if old_labeling:
G = FlexRiGraph(
[(0, 4), (0, 5), (0, 6), (1, 2), (1, 3), (1, 6), (2, 5),
(2, 7), (3, 4), (3, 7), (4, 7), (5, 7), (6, 7)],
pos={
0: (-1, 0),
1: (1, 0),
2: (0.5, -0.866025),
3: (0.5, 0.866025),
4: (-0.5, 0.866025),
5: (-0.5, -0.866025),
6: (0, 0.3),
7: (0, -0.3)
},
name='Q_2')
else:
G = FlexRiGraph(
[(1, 3), (3, 4), (2, 4), (2, 7), (6, 7), (1, 6), (3, 5),
(4, 5), (5, 6), (5, 7), (5, 8), (1, 8), (2, 8)],
pos={
1: (-1, 0),
2: (1, 0),
4: (0.5, -0.866025),
7: (0.5, 0.866025),
6: (-0.5, 0.866025),
3: (-0.5, -0.866025),
8: (0, 0.3),
5: (0, -0.3)
},
name='Q_2')
for col in G.NAC_colorings():
num = col.NAC2int()
if num == 15487:
col.set_name('alpha2')
elif num == 15229:
col.set_name('beta')
elif num == 14589:
col.set_name('gamma2')
elif num == 14466:
col.set_name('gamma1')
elif num == 15106:
col.set_name('delta')
elif num == 15360:
col.set_name('alpha1')
elif num == 14066:
col.set_name('zeta34')
elif num == 13682:
col.set_name('epsilon27')
elif num == 12912:
col.set_name('zeta67')
elif num == 12784:
col.set_name('epsilon24')
elif num == 12687:
col.set_name('epsilon13')
elif num == 13581:
col.set_name('epsilon16')
return G
def Q1Graph(old_labeling=False):
r"""
Return the graph $Q_1$.
EXAMPLE::
sage: from flexrilog import GraphGenerator
sage: GraphGenerator.Q1Graph()
Q_1: FlexRiGraph with 7 vertices and 11 edges
.. PLOT::
:scale: 70
from flexrilog import GraphGenerator
G = GraphGenerator.Q1Graph()
sphinx_plot(G)
"""
if old_labeling:
return FlexRiGraph(
[(0, 1), (0, 2), (0, 6), (1, 2), (1, 4), (1, 5), (2, 3),
(3, 4), (3, 5), (4, 6), (5, 6)],
pos={
5: (0.500, 0.866),
4: (-0.500, 0.866),
6: (-1.00, 0.000),
3: (1.00, 0.000),
2: (0.500, -0.866),
0: (-0.500, -0.866),
1: (0.000, 0.000)
},
name='Q_1')
G = FlexRiGraph(
[[5, 6], [5, 7], [6, 7], [1, 5], [2, 6], [2, 4], [1, 3], [3, 7],
[4, 7], [1, 4], [2, 3]],
pos={
4: (0.500, 0.866),
3: (-0.500, 0.866),
1: (-1.00, 0.000),
2: (1.00, 0.000),
6: (0.500, -0.866),
5: (-0.500, -0.866),
7: (0.000, 0.000)
},
name='Q_1')
for cls in G.NAC_colorings_isomorphism_classes():
if len(cls) == 1:
delta = cls[0]
if len(delta.blue_edges()) in [4, 7]:
delta.set_name('eta')
else:
delta.set_name('zeta')
else:
for delta in cls:
for edges in [delta.red_edges(), delta.blue_edges()]:
if len(cls) == 4 and len(edges) == 7:
u, v = [
comp
for comp in Graph([list(e) for e in edges
]).connected_components()
if len(comp) == 2
][0]
delta.set_name('epsilon' + (
str(u) + str(v) if u < v else str(v) + str(u)))
break
if len(edges) == 3:
vertex = edges[0].intersection(
edges[1]).intersection(edges[2])[0]
name = 'phi' if [
w for w in G.neighbors(vertex)
if G.degree(w) == 4
] else 'psi'
delta.set_name(name + str(vertex))
break
if len(edges) == 5:
u, v = [
comp
for comp in Graph([list(e) for e in edges
]).connected_components()
if len(comp) == 2
][0]
delta.set_name('gamma' + str(min(u, v)))
break
return G
def MaxEmbeddingsLamanGraph(n, labeled_from_one=True):
r"""
Return the Laman graph with ``n`` vertices with the maximum number of complex embeddings.
See [GKT2018]_.
INPUT:
- ``n`` an integer from 6 to 12
EXAMPLES::
sage: from flexrilog import GraphGenerator
sage: GraphGenerator.MaxEmbeddingsLamanGraph(6).is_isomorphic(GraphGenerator.ThreePrismGraph())
True
The graphs:
.. PLOT::
:width: 70%
from flexrilog import GraphGenerator
G = GraphGenerator.MaxEmbeddingsLamanGraph(6)
sphinx_plot(G)
.. PLOT::
:width: 70%
from flexrilog import GraphGenerator
G = GraphGenerator.MaxEmbeddingsLamanGraph(7)
sphinx_plot(G)
.. PLOT::
:width: 70%
from flexrilog import GraphGenerator
G = GraphGenerator.MaxEmbeddingsLamanGraph(8)
sphinx_plot(G)
.. PLOT::
:width: 70%
from flexrilog import GraphGenerator
G = GraphGenerator.MaxEmbeddingsLamanGraph(9)
sphinx_plot(G)
.. PLOT::
:width: 70%
from flexrilog import GraphGenerator
G = GraphGenerator.MaxEmbeddingsLamanGraph(10)
sphinx_plot(G)
.. PLOT::
:width: 70%
from flexrilog import GraphGenerator
G = GraphGenerator.MaxEmbeddingsLamanGraph(11)
sphinx_plot(G)
.. PLOT::
:width: 70%
from flexrilog import GraphGenerator
G = GraphGenerator.MaxEmbeddingsLamanGraph(12)
sphinx_plot(G)
"""
graph_repr = {
6: 7916,
7: 1269995,
8: 170989214,
9: 11177989553,
10: 4778440734593,
11: 18120782205838348,
12: 252590061719913632,
}
positions = {
7: {
5: (-1, -1),
2: (1, -1),
1: (2.5, 2),
4: (-1, 1),
3: (1, 1),
0: (-2.5, 2),
6: (0, 0)
},
8: {
6: (-0.00, -17.00),
7: (2.5, 7.24),
4: (7.50, 0.00),
5: (-0.00, -7.24),
3: (20.00, 0.00),
1: (-20.00, 0.00),
2: (-7.50, 0.00),
0: (-0.00, 19.00)
},
9: {
8: (21.00, 2.00),
7: (-5.10, 20.00),
6: (0.17, -5.70),
3: (7.90, -2.80),
2: (-5.10, -16.00),
5: (0.17, 9.70),
4: (-8.20, 2.00),
0: (7.90, 6.80),
1: (-19.00, 2.00),
},
10: {
9: (-20.00, -16.00),
8: (20.00, -16.00),
6: (-0.00, -2.30),
7: (-0.00, 13.00),
4: (-9.10, -0.20),
5: (9.10, -0.20),
1: (-6.00, -11.00),
0: (-12.00, 21.00),
2: (6.00, -11.00),
3: (12.00, 21.00)
},
11: {
9: (7.20, 6.80),
8: (20.00, 2.00),
10: (-7.20, 6.80),
7: (-20.00, 2.00),
6: (0.00, -6.60),
2: (7.20, -2.80),
3: (0.00, 2.00),
4: (0.00, -16.00),
5: (-7.20, -2.80),
0: (0.00, 10.60),
1: (0.00, 20.00)
},
12: {
11: (20.00, -12.00),
6: (-20.00, -12.00),
10: (7.80, 6.20),
7: (-8.30, -3.90),
9: (8.30, -3.90),
8: (-7.80, 6.20),
0: (15.00, 0.85),
1: (20.00, 14.00),
2: (0.00, -2.10),
3: (-20.00, 14.00),
4: (0.00, -7.80),
5: (-15.00, 0.85)
}
}
if n == 6:
G = GraphGenerator.ThreePrismGraph()
elif n > 6 and n < 13:
G = FlexRiGraph(Integer(graph_repr[n]),
pos=positions[n],
name='MaxEmbeddingsLamanGraph_' + str(n) + 'vert')
else:
raise exceptions.ValueError(
'Only graphs with 6-12 vertices are supported.')
if not labeled_from_one:
return G
else:
return FlexRiGraph([(u + 1, v + 1)
for u, v in G.edges(labels=False)],
pos={v + 1: G._pos[v]
for v in G._pos},
name=G.name())