/
minrank.py
1002 lines (852 loc) · 40.2 KB
/
minrank.py
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# -*- coding: utf-8 -*-
"""
A collection of bounds for minimum rank
This module implements a collection of bounds for minimum rank, plus a
driver function to apply each of these in turn to a graph.
This code was initially published in http://arxiv.org/abs/0812.1616 by
Laura DeLoss, Jason Grout, Tracy McKay, Jason Smith, and Geoff Tims.
"""
#######################################################################
#
# Copyright (C) 2008, 2009 Laura DeLoss, Jason Grout, Tracy McKay,
# Jason Smith, and Geoff Tims.
#
#
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 2 of the License, or
# (at your option) any later version.
#
# This program is distributed in the hope that it will be useful, but
# WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
# General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program. If not, see http://www.gnu.org/licenses/.
#######################################################################
##########################################################
# Imports all the graphs of order 7 or less and stores #
# them in a list called atlas_graphs so that #
# atlas_graphs[i] is the ith graph in the atlas of #
# graphs #
##########################################################
from sage.graphs.graph import Graph
import networkx.generators.atlas
atlas_graphs = [Graph(i) for i in \
networkx.generators.atlas.graph_atlas_g()]
##########################################################
# A list "database" of all the minimum ranks of graphs #
# of order 7 or less. #
# #
# The minimum ranks are stored as ordered pairs: the #
# first coordinate is the graph number, the second #
# coordinate is the minimum rank. The first tuple in #
# the list min_ranks is just a position holder. #
##########################################################
min_ranks = [(0,None), (1,0), (2,0), (3,1), (4,0), (5,1), (6,2),
(7,1), (8,0), (9,1), (10,2), (11,2), (12,1), (13,2), (14,3), (15,2),
(16,2), (17,2), (18,1), (19,0), (20,1), (21,2), (22,2), (23,1),
(24,2), (25,3), (26,3), (27,2), (28,2), (29,2), (30,3), (31,4),
(32,2), (33,2), (34,3), (35,3), (36,3), (37,3), (38,3), (39,1),
(40,3), (41,3), (42,2), (43,3), (44,2), (45,2), (46,2), (47,3),
(48,2), (49,2), (50,2), (51,2), (52,1), (53,0), (54,1), (55,2),
(56,2), (57,1), (58,2), (59,3), (60,3), (61,3), (62,2), (63,2),
(64,2), (65,3), (66,4), (67,2), (68,3), (69,4), (70,4), (71,2),
(72,3), (73,3), (74,3), (75,3), (76,3), (77,2), (78,3), (79,4),
(80,4), (81,4), (82,3), (83,5), (84,3), (85,3), (86,1), (87,3),
(88,3), (89,2), (90,3), (91,2), (92,3), (93,4), (94,4), (95,4),
(96,4), (97,4), (98,4), (99,4), (100,3), (101,3), (102,4), (103,4),
(104,4), (105,4), (106,2), (107,2), (108,2), (109,3), (110,2),
(111,4), (112,4), (113,4), (114,3), (115,4), (116,2), (117,3),
(118,4), (119,3), (120,4), (121,3), (122,4), (123,4), (124,4),
(125,3), (126,3), (127,4), (128,4), (129,3), (130,3), (131,2),
(132,2), (133,3), (134,3), (135,3), (136,4), (137,4), (138,3),
(139,4), (140,3), (141,3), (142,3), (143,3), (144,3), (145,3),
(146,2), (147,4), (148,4), (149,3), (150,3), (151,3), (152,4),
(153,3), (154,3), (155,2), (156,3), (157,3), (158,3), (159,3),
(160,3), (161,2), (162,3), (163,3), (164,4), (165,2), (166,3),
(167,4), (168,3), (169,3), (170,3), (171,3), (172,3), (173,3),
(174,3), (175,2), (176,1), (177,3), (178,3), (179,3), (180,3),
(181,3), (182,3), (183,3), (184,3), (185,3), (186,3), (187,3),
(188,3), (189,2), (190,2), (191,2), (192,3), (193,3), (194,2),
(195,2), (196,3), (197,2), (198,3), (199,2), (200,2), (201,2),
(202,3), (203,2), (204,2), (205,2), (206,2), (207,2), (208,1),
(209,0), (210,1), (211,2), (212,2), (213,1), (214,2), (215,3),
(216,3), (217,3), (218,2), (219,2), (220,2), (221,3), (222,4),
(223,2), (224,3), (225,4), (226,4), (227,4), (228,2), (229,3),
(230,3), (231,3), (232,3), (233,3), (234,2), (235,3), (236,4),
(237,4), (238,4), (239,3), (240,5), (241,3), (242,3), (243,3),
(244,4), (245,4), (246,5), (247,5), (248,3), (249,1), (250,3),
(251,3), (252,2), (253,3), (254,2), (255,3), (256,4), (257,4),
(258,4), (259,4), (260,4), (261,4), (262,3), (263,4), (264,3),
(265,4), (266,4), (267,4), (268,4), (269,2), (270,2), (271,3),
(272,4), (273,4), (274,4), (275,4), (276,5), (277,4), (278,4),
(279,5), (280,5), (281,4), (282,4), (283,4), (284,5), (285,3),
(286,6), (287,4), (288,4), (289,4), (290,2), (291,2), (292,3),
(293,2), (294,4), (295,4), (296,4), (297,3), (298,4), (299,2),
(300,3), (301,4), (302,3), (303,4), (304,3), (305,4), (306,4),
(307,4), (308,3), (309,3), (310,4), (311,3), (312,4), (313,3),
(314,3), (315,4), (316,4), (317,5), (318,4), (319,4), (320,5),
(321,5), (322,5), (323,4), (324,4), (325,4), (326,3), (327,5),
(328,5), (329,4), (330,4), (331,5), (332,5), (333,5), (334,5),
(335,3), (336,5), (337,5), (338,5), (339,4), (340,5), (341,5),
(342,5), (343,4), (344,4), (345,4), (346,4), (347,3), (348,5),
(349,5), (350,5), (351,5), (352,3), (353,5), (354,3), (355,2),
(356,2), (357,3), (358,3), (359,3), (360,4), (361,4), (362,3),
(363,4), (364,3), (365,3), (366,3), (367,3), (368,3), (369,3),
(370,2), (371,4), (372,4), (373,3), (374,3), (375,3), (376,3),
(377,4), (378,3), (379,4), (380,4), (381,5), (382,3), (383,5),
(384,4), (385,5), (386,4), (387,3), (388,4), (389,4), (390,5),
(391,5), (392,4), (393,5), (394,5), (395,4), (396,4), (397,3),
(398,5), (399,5), (400,5), (401,5), (402,5), (403,4), (404,4),
(405,4), (406,4), (407,4), (408,4), (409,4), (410,4), (411,4),
(412,5), (413,5), (414,5), (415,4), (416,4), (417,3), (418,3),
(419,4), (420,4), (421,5), (422,5), (423,5), (424,4), (425,4),
(426,4), (427,5), (428,4), (429,4), (430,4), (431,4), (432,5),
(433,5), (434,5), (435,5), (436,4), (437,5), (438,5), (439,5),
(440,4), (441,4), (442,4), (443,4), (444,4), (445,5), (446,5),
(447,4), (448,4), (449,4), (450,4), (451,3), (452,2), (453,3),
(454,3), (455,3), (456,3), (457,3), (458,2), (459,3), (460,3),
(461,4), (462,2), (463,3), (464,4), (465,3), (466,3), (467,3),
(468,3), (469,3), (470,3), (471,3), (472,2), (473,3), (474,4),
(475,4), (476,4), (477,4), (478,5), (479,5), (480,4), (481,4),
(482,5), (483,4), (484,5), (485,4), (486,4), (487,4), (488,5),
(489,5), (490,4), (491,4), (492,4), (493,4), (494,4), (495,4),
(496,3), (497,5), (498,4), (499,4), (500,4), (501,3), (502,3),
(503,4), (504,4), (505,4), (506,4), (507,3), (508,5), (509,5),
(510,4), (511,4), (512,5), (513,4), (514,4), (515,5), (516,5),
(517,5), (518,5), (519,4), (520,4), (521,4), (522,4), (523,4),
(524,4), (525,3), (526,5), (527,5), (528,5), (529,5), (530,5),
(531,4), (532,4), (533,5), (534,4), (535,4), (536,4), (537,4),
(538,4), (539,4), (540,4), (541,4), (542,4), (543,4), (544,4),
(545,4), (546,4), (547,4), (548,5), (549,4), (550,4), (551,3),
(552,4), (553,4), (554,3), (555,4), (556,4), (557,3), (558,3),
(559,5), (560,4), (561,5), (562,4), (563,5), (564,4), (565,4),
(566,5), (567,4), (568,4), (569,4), (570,3), (571,4), (572,4),
(573,4), (574,5), (575,5), (576,4), (577,4), (578,4), (579,4),
(580,4), (581,4), (582,2), (583,1), (584,3), (585,3), (586,3),
(587,3), (588,3), (589,3), (590,3), (591,3), (592,3), (593,3),
(594,3), (595,3), (596,2), (597,2), (598,4), (599,4), (600,4),
(601,4), (602,4), (603,4), (604,4), (605,4), (606,4), (607,4),
(608,4), (609,4), (610,3), (611,3), (612,3), (613,4), (614,4),
(615,3), (616,4), (617,4), (618,5), (619,3), (620,4), (621,4),
(622,5), (623,5), (624,4), (625,4), (626,4), (627,4), (628,4),
(629,4), (630,3), (631,4), (632,5), (633,4), (634,4), (635,4),
(636,4), (637,4), (638,4), (639,4), (640,5), (641,4), (642,4),
(643,4), (644,3), (645,4), (646,5), (647,4), (648,4), (649,4),
(650,4), (651,4), (652,4), (653,4), (654,4), (655,4), (656,4),
(657,4), (658,4), (659,4), (660,4), (661,4), (662,4), (663,4),
(664,4), (665,4), (666,4), (667,3), (668,3), (669,3), (670,2),
(671,4), (672,4), (673,4), (674,4), (675,4), (676,4), (677,4),
(678,3), (679,4), (680,4), (681,3), (682,5), (683,5), (684,4),
(685,4), (686,3), (687,3), (688,4), (689,3), (690,5), (691,4),
(692,4), (693,4), (694,5), (695,4), (696,4), (697,4), (698,4),
(699,4), (700,4), (701,4), (702,4), (703,4), (704,4), (705,4),
(706,4), (707,4), (708,4), (709,4), (710,5), (711,4), (712,4),
(713,4), (714,4), (715,4), (716,4), (717,4), (718,4), (719,4),
(720,4), (721,3), (722,4), (723,4), (724,4), (725,3), (726,3),
(727,4), (728,4), (729,4), (730,3), (731,2), (732,3), (733,3),
(734,2), (735,2), (736,3), (737,2), (738,3), (739,2), (740,4),
(741,4), (742,4), (743,3), (744,4), (745,2), (746,4), (747,4),
(748,4), (749,4), (750,4), (751,4), (752,4), (753,4), (754,4),
(755,4), (756,4), (757,4), (758,4), (759,4), (760,4), (761,4),
(762,4), (763,4), (764,4), (765,4), (766,4), (767,4), (768,4),
(769,4), (770,4), (771,4), (772,4), (773,4), (774,4), (775,4),
(776,4), (777,4), (778,4), (779,4), (780,3), (781,3), (782,4),
(783,4), (784,4), (785,4), (786,3), (787,3), (788,4), (789,3),
(790,2), (791,3), (792,4), (793,4), (794,4), (795,4), (796,3),
(797,4), (798,4), (799,4), (800,4), (801,3), (802,4), (803,4),
(804,4), (805,4), (806,4), (807,4), (808,4), (809,3), (810,4),
(811,4), (812,3), (813,5), (814,3), (815,3), (816,4), (817,4),
(818,4), (819,4), (820,4), (821,5), (822,4), (823,4), (824,4),
(825,4), (826,4), (827,4), (828,4), (829,3), (830,4), (831,3),
(832,3), (833,4), (834,4), (835,4), (836,4), (837,4), (838,4),
(839,4), (840,5), (841,4), (842,4), (843,4), (844,4), (845,4),
(846,3), (847,4), (848,4), (849,4), (850,4), (851,3), (852,4),
(853,4), (854,4), (855,4), (856,3), (857,4), (858,4), (859,4),
(860,4), (861,4), (862,4), (863,3), (864,4), (865,3), (866,4),
(867,4), (868,4), (869,3), (870,4), (871,4), (872,3), (873,3),
(874,3), (875,4), (876,3), (877,4), (878,3), (879,2), (880,2),
(881,3), (882,2), (883,2), (884,3), (885,3), (886,4), (887,4),
(888,4), (889,4), (890,4), (891,4), (892,4), (893,3), (894,3),
(895,3), (896,3), (897,4), (898,3), (899,4), (900,4), (901,3),
(902,3), (903,4), (904,4), (905,4), (906,3), (907,4), (908,4),
(909,3), (910,4), (911,3), (912,3), (913,3), (914,4), (915,4),
(916,4), (917,4), (918,3), (919,4), (920,4), (921,4), (922,4),
(923,4), (924,3), (925,3), (926,4), (927,4), (928,4), (929,4),
(930,4), (931,3), (932,3), (933,4), (934,4), (935,4), (936,4),
(937,4), (938,4), (939,4), (940,4), (941,4), (942,4), (943,4),
(944,3), (945,3), (946,4), (947,3), (948,3), (949,3), (950,4),
(951,4), (952,4), (953,3), (954,4), (955,4), (956,3), (957,3),
(958,3), (959,4), (960,4), (961,4), (962,4), (963,4), (964,4),
(965,4), (966,4), (967,4), (968,4), (969,4), (970,3), (971,4),
(972,4), (973,3), (974,4), (975,3), (976,4), (977,3), (978,3),
(979,4), (980,4), (981,4), (982,4), (983,3), (984,3), (985,4),
(986,4), (987,3), (988,3), (989,4), (990,3), (991,3), (992,4),
(993,4), (994,3), (995,3), (996,3), (997,4), (998,4), (999,4),
(1000,3), (1001,3), (1002,3), (1003,3), (1004,3), (1005,3), (1006,4),
(1007,2), (1008,4), (1009,2), (1010,2), (1011,2), (1012,3), (1013,3),
(1014,3), (1015,4), (1016,4), (1017,3), (1018,3), (1019,3), (1020,3),
(1021,4), (1022,3), (1023,3), (1024,3), (1025,4), (1026,4), (1027,4),
(1028,3), (1029,4), (1030,4), (1031,4), (1032,2), (1033,4), (1034,3),
(1035,3), (1036,3), (1037,3), (1038,3), (1039,4), (1040,3), (1041,4),
(1042,3), (1043,4), (1044,3), (1045,3), (1046,4), (1047,4), (1048,4),
(1049,3), (1050,4), (1051,4), (1052,4), (1053,4), (1054,4), (1055,4),
(1056,3), (1057,3), (1058,4), (1059,4), (1060,3), (1061,4), (1062,3),
(1063,3), (1064,3), (1065,4), (1066,3), (1067,3), (1068,3), (1069,4),
(1070,3), (1071,4), (1072,3), (1073,3), (1074,3), (1075,3), (1076,3),
(1077,3), (1078,4), (1079,3), (1080,4), (1081,3), (1082,4), (1083,4),
(1084,3), (1085,3), (1086,3), (1087,3), (1088,2), (1089,4), (1090,3),
(1091,4), (1092,3), (1093,4), (1094,3), (1095,3), (1096,3), (1097,4),
(1098,3), (1099,3), (1100,3), (1101,4), (1102,3), (1103,3), (1104,3),
(1105,3), (1106,3), (1107,2), (1108,3), (1109,3), (1110,3), (1111,3),
(1112,3), (1113,3), (1114,3), (1115,3), (1116,3), (1117,4), (1118,4),
(1119,3), (1120,3), (1121,4), (1122,3), (1123,3), (1124,3), (1125,3),
(1126,3), (1127,4), (1128,3), (1129,3), (1130,3), (1131,3), (1132,3),
(1133,3), (1134,3), (1135,3), (1136,3), (1137,3), (1138,3), (1139,3),
(1140,2), (1141,4), (1142,4), (1143,3), (1144,3), (1145,4), (1146,3),
(1147,3), (1148,3), (1149,3), (1150,4), (1151,3), (1152,3), (1153,3),
(1154,4), (1155,3), (1156,3), (1157,3), (1158,3), (1159,3), (1160,4),
(1161,3), (1162,3), (1163,3), (1164,2), (1165,3), (1166,3), (1167,3),
(1168,3), (1169,3), (1170,3), (1171,2), (1172,1), (1173,3), (1174,3),
(1175,3), (1176,3), (1177,3), (1178,3), (1179,3), (1180,3), (1181,3),
(1182,3), (1183,3), (1184,2), (1185,3), (1186,3), (1187,4), (1188,2),
(1189,3), (1190,4), (1191,3), (1192,3), (1193,3), (1194,3), (1195,3),
(1196,3), (1197,3), (1198,3), (1199,3), (1200,3), (1201,3), (1202,3),
(1203,3), (1204,3), (1205,3), (1206,2), (1207,3), (1208,2), (1209,3),
(1210,3), (1211,2), (1212,3), (1213,2), (1214,3), (1215,3), (1216,2),
(1217,3), (1218,3), (1219,3), (1220,3), (1221,3), (1222,3), (1223,3),
(1224,3), (1225,3), (1226,2), (1227,3), (1228,3), (1229,2), (1230,2),
(1231,3), (1232,3), (1233,2), (1234,2), (1235,3), (1236,3), (1237,2),
(1238,2), (1239,3), (1240,2), (1241,3), (1242,2), (1243,2), (1244,2),
(1245,2), (1246,3), (1247,2), (1248,2), (1249,2), (1250,2), (1251,2),
(1252,1)]
###########################################################
# The atlas of graphs is ordered by number of vertices, then number of
# edges. To make our search for minimum ranks more efficient, we
# store the first and last indices of the graphs having a certain
# number of vertices and edges.
#
# graph_help[n][m] is a tuple (i,j), meaning that the graphs with n
# vertices and m edges start at index i and end on index j in the
# atlas of graphs.
###########################################################
graph_help = {
1: {0: (1,1)},
2: {0: (2,2), 1: (3,3)},
3: {0: (4,4), 1: (5,5), 2: (6,6), 3: (7,7)},
4: {0: (8,8), 1: (9,9), 2: (10,11), 3: (12,14), 4: (15,16),
5: (17,17), 6: (18,18)},
5: {0: (19,19), 1: (20,20), 2: (21,22), 3: (23,26), 4: (27,32), 5:
(33,38), 6: (39,44), 7: (45,48), 8: (49,50), 9: (51,51), 10:
(52,52)},
6: {0: (53,53), 1:(54,54), 2:(55,56), 3: (57,61), 4: (62,70),
5: (71,85), 6: (86,106), 7: (107,130), 8: (131,154), 9: (155,175),
10: (176,190), 11: (191,199), 12: (200,204), 13: (205,206),
14: (207,207), 15: (208,208)},
7: {0: (209,209), 1: (210,210), 2: (211,212), 3: (213,217),
4: (218,227), 5: (228,248), 6: (249,289), 7: (290,354), 8: (355,451),
9: (452,582), 10: (583,730), 11: (731,878), 12: (879,1009),
13: (1010,1106), 14: (1107,1171), 15: (1172,1212), 16: (1213,1233),
17: (1234,1243), 18: (1244,1248), 19: (1249,1250), 20: (1251,1251),
21: (1252,1252)}}
def get_mr_from_list(graph):
"""
If the graph's minimum rank is stored in the table, returns the
minimum rank; otherwise returns False.
:param graph: graph to look up
:returns: the minimum rank if the graph is in the list or False if
it is not
EXAMPLES::
sage: from sage.graphs.minrank import get_mr_from_list
sage: get_mr_from_list(Graph({0:[2,3],1:[2],2:[3,4],3:[4]}))
3
sage: get_mr_from_list(graphs.PathGraph(8))
False
"""
#check to make sure graph can be found in list
if graph.order()>7:
return False
order = graph.order() #number of vertices
size = graph.size() #number of edges
starting_index, ending_index = graph_help[order][size]
#look for graph and return minimum rank
for i in range(starting_index,ending_index+1):
if graph.is_isomorphic(atlas_graphs[i]):
return min_ranks[i][1]
raise ValueError("This should never happen!")
def zerosgame(graph, initial_set=[]):
"""
Apply the color-change rule to a given graph given an optional
initial set.
:param graph: the graph on which to apply the rule
:param initial_set: the set of "zero" (black) vertices in the graph
:return: the list of zero (black) vertices in the resulting derived
coloring
EXAMPLES::
sage: from sage.graphs.minrank import zerosgame
sage: zerosgame(graphs.PathGraph(5))
[]
sage: zerosgame(graphs.PathGraph(5),[0])
[0, 1, 2, 3, 4]
"""
new_zero_set=set(initial_set)
zero_set=set([])
zero_neighbors={}
active_zero_set = set([])
inactive_zero_set = set([])
another_run=True
while another_run:
another_run=False
# Add the new zero vertices
zero_set.update(new_zero_set)
active_zero_set.update(new_zero_set)
active_zero_set.difference_update(inactive_zero_set)
zero_neighbors.update([[i,
set(graph.neighbors(i)).difference(zero_set)]
for i in new_zero_set])
# Find the next set of zero vertices
new_zero_set.clear()
inactive_zero_set.clear()
for v in active_zero_set:
zero_neighbors[v].difference_update(zero_set)
if len(zero_neighbors[v])==1:
new_zero_set.add(zero_neighbors[v].pop())
inactive_zero_set.add(v)
another_run=True
return list(zero_set)
def zero_forcing_set_bruteforce(graph, bound=None, all_sets=False):
"""
Return a zero forcing set of minimum order that also has order
less than the given bound.
:param graph: the graph on which to find the zero-forcing set
:param int bound: the maximum acceptable order for a zero-forcing set
:param bool all_sets: whether to return all zero forcing sets
or just the first one
:return: a zero-forcing set (or list of all zero-forcing sets if all_sets is True)
of minimum order that also has order less than the bound if one exists;
False if no such zero-forcing set can be found.
EXAMPLES::
sage: from sage.graphs.minrank import zero_forcing_set_bruteforce
sage: zero_forcing_set_bruteforce(graphs.CompleteGraph(5))
{0, 1, 2, 3}
sage: zero_forcing_set_bruteforce(graphs.CompleteGraph(5),2)
False
sage: zero-forcing_set_bruteforce(graphs.CompleteGraph(5), all_sets=True)
[{0, 1, 2, 3}, {0, 1, 2, 4}, {0, 1, 3, 4}, {0, 2, 3, 4}, {1, 2, 3, 4}]
"""
from sage.all import Subsets
order=graph.order()
if bound is None:
bound = order
if bound<0:
bound=1
found_zfs=False
zfs_sets=[]
vertices=graph.vertices()
mindegree=min(graph.degree())
for i in range(mindegree,bound+1):
if found_zfs:
break
for subset in Subsets(vertices,i):
outcome=zerosgame(graph,subset)
if len(outcome)==order:
if all_sets:
found_zfs=True
zfs_sets.append(subset)
else:
return subset
if found_zfs:
return zfs_sets
else:
return False
def find_Z(graph):
"""
Returns the order of a smallest zero-forcing set of a graph
:param graph: the graph on which to find a smallest zero-forcing set
:return: the minimum possible order for a zero-forcing set of the graph
EXAMPLES::
sage: from sage.graphs.minrank import find_Z
sage: find_Z(graphs.CompleteGraph(5))
4
"""
return len(zero_forcing_set_bruteforce(graph))
def has_forbidden_induced_subgraph(graph):
"""
Check for a forbidden induced subgraph (a path on 4 vertices,
fish, dart, or complete tripartite graph).
:param graph: the graph to be checked
:return: True if the graph contains an induced copy of P_4, fish,
dart, or K_{3,3,3}; False if it does not
EXAMPLES::
sage: from sage.graphs.minrank import has_forbidden_induced_subgraph
sage: has_forbidden_induced_subgraph(graphs.CompleteGraph(10))
False
sage: has_forbidden_induced_subgraph(graphs.PathGraph(3))
False
sage: K333 = Graph({0: [3,4,5,6,7,8], 1: [3,4,5,6,7,8], 2: [3,4,5,6,7,8], 3: [6,7,8], 4: [6,7,8], 5: [6,7,8]})
sage: has_forbidden_induced_subgraph(K333)
True
sage: g = Graph({0:[1,2,3,4], 1: [2]}) # fish
sage: has_forbidden_induced_subgraph(g)
True
sage: g.add_edge((0,6))
sage: has_forbidden_induced_subgraph(g)
True
"""
from sage.all import Combinations
order = graph.order()
vertices = graph.vertices()
path = atlas_graphs[14]
fish = atlas_graphs[34]
dart = atlas_graphs[40]
K333 = Graph({0: [3,4,5,6,7,8], 1: [3,4,5,6,7,8], \
2: [3,4,5,6,7,8], 3: [6,7,8], \
4: [6,7,8], 5: [6,7,8]})
if order < 4:
return False
for sub_vertices in Combinations(vertices,4):
# Finds all order 4 induced subgraphs
if graph.subgraph(sub_vertices).is_isomorphic(path):
return True
if order < 5:
return False
for sub_vertices in Combinations(vertices,5):
# Finds all order 5 induced subgraphs
if graph.subgraph(sub_vertices).is_isomorphic(dart):
return True
if graph.subgraph(sub_vertices).is_isomorphic(fish):
return True
if order < 9:
return False
for sub_vertices in Combinations(vertices,9):
# Finds all order 9 induced subgraphs
if graph.subgraph(sub_vertices).is_isomorphic(K333):
return True
return False
# if in library mode, we need to import this.
# if it is just included in a sage notebook, then it is in the global namespace
# so we try importing it if we can. If we can't import it, then just trust that
# everything is in the global namespace.
try:
from zero_forcing_set_wavefront import zero_forcing_set_wavefront
except ImportError:
pass
def min_rank_by_bounds(graph, tests = ['precomputed', 'order', 'zero forcing', 'not path', 'forbidden minrank 2', 'not planar', 'not outer planar', 'clique cover', 'diameter']):
"""
Return dictionaries giving the upper and lower bounds from running
the specified tests. If tests is not set, then all applicable
tests are run.
:param graph: the graph for which to find bounds
:return: a list of 2 dictionaries; the upper and lower bounds,
respectively.
EXAMPLE::
sage: from sage.graphs.minrank import min_rank_by_bounds
sage: g = Graph({0: [1,2,4,6,7], 1: [3,5,6,7,8], 2: [4,6,8], 3: [4,7,6], 4: [6], 5: [6,8,7]})
sage: min_rank_by_bounds(g)
({'zero forcing': 4},
{'clique cover': 9,
'not outer planar': 6,
'not path': 7,
'not planar': 5,
'order': 8})
sage: min_rank_by_bounds(g, tests=['zero forcing', 'order', 'not path'])
({'zero forcing': 4}, {'not path': 7, 'order': 8})
"""
if isinstance(tests, str):
tests = [tests]
order = graph.order()
lower_bound = {}
upper_bound = {}
if 'precomputed' in tests:
mr = get_mr_from_list(graph)
if mr is not False:
lower_bound['precomputed'] = mr
upper_bound['precomputed'] = mr
if 'order' in tests:
upper_bound['order'] = order - 1
if 'zero forcing' in tests:
lower_bound['zero forcing'] = order - find_Z(graph)
# Check if graph is a tree.
# If yes, then the ZFS will determine minimum rank.
if graph.is_tree():
upper_bound['zero forcing (tree)'] = lower_bound['zero forcing']
if 'zero forcing fast' in tests:
lower_bound['zero forcing fast'] = order - zero_forcing_set_wavefront(graph)[0]
# Check if graph is a tree.
# If yes, then the ZFS will determine minimum rank.
if graph.is_tree():
upper_bound['zero forcing fast (tree)'] = lower_bound['zero forcing fast']
if 'not path' in tests:
if graph.diameter() < order - 1:
upper_bound['not path'] = order - 2
if 'forbidden minrank 2' in tests:
if has_forbidden_induced_subgraph(graph):
lower_bound['forbidden minrank 2'] = 3
else:
upper_bound['forbidden minrank 2'] = 2
if 'diameter' in tests:
lower_bound['diameter'] = graph.diameter()
if 'not planar' in tests:
# Old versions of Sage assume that planar testing does not
# have vertices of degree zero. We can delete vertices of
# degree zero without affecting the planarity.
h = graph.copy()
h.delete_vertices([v for v in h.vertices() if h.degree(v) == 0])
if h.order()>0 and h.is_planar() is False:
upper_bound['not planar'] = order - 4
if 'not outer planar' in tests:
if is_outerplanar(graph) is False:
upper_bound['not outer planar'] = order - 3
if 'clique cover' in tests:
upper_bound['clique cover'] = len(edge_clique_cover_minimum(graph))
return (lower_bound, upper_bound)
def find_rank_spread(vertex, graph):
"""
Returns the exact rank spread for a graph and a vertex (i.e.,
mr(G)-mr(G-v)) if the minimum ranks of both the graph and the
graph without the vertex can be calculated using the minrank_bounds
function.
:param vertex: the vertex
:param graph: the graph
:return: the rank spread and mr(graph-vertex) if both can be
calculated using the minrank_bounds program, or False and
False if either cannot be calculated exactly.
EXAMPLES::
sage: from sage.graphs.minrank import find_rank_spread
sage: find_rank_spread(2, Graph({0:[1,2,3],1:[2,3],2:[3]}))
(0, 1)
sage: g = Graph({0:[1,2,4,6,7],1:[3,5,6,7,8], 2:[4,6,8],3:[4,7,6],4:[6],5:[6,8,7]})
sage: find_rank_spread(2,g)
(False, False)
"""
subgraph=graph.copy()
subgraph.delete_vertex(vertex)
graph_bounds=minrank_bounds(graph)
if graph_bounds[0]==graph_bounds[1]:
# We have an actual min rank for graph
subgraph_bounds=minrank_bounds(subgraph)
if subgraph_bounds[0]==subgraph_bounds[1]:
# We have an actual min rank for the subgraph
return graph_bounds[0]-subgraph_bounds[0], subgraph_bounds[0]
return False,False
def cut_vertex_connected_graph_mr(c_vertex,graph):
"""
Given a cut vertex and a graph, attempt to calculate the minimum
rank of the graph by applying the cut vertex method to the graph
and vertex.
:param c_vertex: the cut vertex
:param graph: the graph in which the cut vertex is contained
:return: a list of length 2 with the minimum rank as all entries,
if the minimum rank can be calculated in this way
False if the minimum rank cannot be calculated in this way
EXAMPLE::
sage: from sage.graphs.minrank import cut_vertex_connected_graph_mr
sage: cut_vertex_connected_graph_mr(0,Graph({0:[1,2,3],2:[3]}))
(2, 2)
sage: cut_vertex_connected_graph_mr(2,Graph({0:[1,2,3],2:[3]}))
Traceback (most recent call last):
...
ValueError: Supplied vertex is not a cut vertex
"""
g=graph.copy()
if g.is_connected() is False:#this should never happen
raise ValueError("Graph is not connected")
if c_vertex not in graph.vertices():#again, should never happen
raise ValueError("Supplied vertex is not in the graph")
g.delete_vertex(c_vertex)
subgraphs=g.connected_components_subgraphs()
if len(subgraphs) <= 1: # c_vertex is not a cut-vertex
raise ValueError("Supplied vertex is not a cut vertex")
index=0
rank_spread=0
subgraph_mr_sum=0
for subgraph in subgraphs:
subgraph_with_v = graph.subgraph(subgraph.vertices()+[c_vertex])
new_rank_spread, subgraph_mr = \
find_rank_spread(c_vertex, subgraph_with_v)
if new_rank_spread is False:
return False
else:
rank_spread += new_rank_spread
subgraph_mr_sum += subgraph_mr
rank_spread = min(rank_spread, 2)
return subgraph_mr_sum+rank_spread, subgraph_mr_sum+rank_spread
def minrank_bounds(graph, all_bounds=False, tests=['precomputed', 'order', 'zero forcing', 'zero forcing fast', 'not path', 'forbidden minrank 2', 'not planar', 'not outer planar', 'clique cover', 'cut vertex', 'disconnected', 'diameter']):
"""
Find lower and upper bounds for the minimum rank of a graph. If
all_bounds is False, then only return the best lower and upper
bounds. If True, return two dictionaries giving all applicable
lower bounds and upper bounds, respectively.
:param graph: the graph whose minimum rank is bounded
:param all_bounds: if False, then only return the best lower and
upper bounds. If True, return dictionaries giving all
applicable lower bounds and upper bounds.
:param tests: a list of tests to get bounds. Possible values are
'precomputed', 'order', 'zero forcing', 'zero forcing
fast', 'not path', 'no forbidden', 'not planar', 'not
outer planar', 'clique cover', 'cut vertex',
'disconnected'
:return: the lower and upper bounds for the minimum rank, in that
order
EXAMPLES::
sage: from sage.graphs.minrank import minrank_bounds
sage: minrank_bounds(graphs.CompleteGraph(3))
(1, 1)
sage: minrank_bounds(graphs.CompleteGraph(3), all_bounds=True)
({'precomputed': 1, 'rank': 0, 'zero forcing': 1},
{'clique cover': 1,
'no forbidden': 2,
'not path': 1,
'order': 2,
'precomputed': 1,
'rank': 3})
sage: minrank_bounds(graphs.PathGraph(4), all_bounds=True)
({'cut vertex (1)': 3, 'precomputed': 3, 'rank': 0, 'zero forcing': 3},
{'clique cover': 3,
'cut vertex (1)': 3,
'order': 3,
'precomputed': 3,
'rank': 4,
'zero forcing (tree)': 3})
sage: minrank_bounds(graphs.PathGraph(4), all_bounds=True, tests=['order', 'zero forcing'])
({'rank': 0, 'zero forcing': 3},
{'order': 3, 'rank': 4, 'zero forcing (tree)': 3})
sage: minrank_bounds(graphs.HeawoodGraph())
(8, 10)
sage: minrank_bounds(graphs.HeawoodGraph(), all_bounds=True)
({'rank': 0, 'zero forcing': 8},
{'clique cover': 21,
'not outer planar': 11,
'not path': 12,
'not planar': 10,
'order': 13,
'rank': 14})
sage: minrank_bounds(graphs.PetersenGraph())
(5, 6)
sage: minrank_bounds(graphs.PetersenGraph(), all_bounds=True)
({'rank': 0, 'zero forcing': 5},
{'clique cover': 15,
'not outer planar': 7,
'not path': 8,
'not planar': 6,
'order': 9,
'rank': 10})
"""
if isinstance(tests, str):
tests = [tests]
possible_tests = set(['precomputed', 'order', 'zero forcing', 'zero forcing fast', 'not path', 'forbidden minrank 2', 'not planar', 'not outer planar', 'clique cover', 'cut vertex', 'disconnected', 'diameter'])
# Check tests
unknown_tests = set(tests).difference(possible_tests)
if len(unknown_tests)>0:
print "Unknown tests specified: ", list(unknown_tests)
g=graph.copy()
lower_bound = {'rank': 0}
upper_bound = {'rank': g.order()}
if g.is_connected():
bounds = min_rank_by_bounds(graph, tests=tests)
lower_bound.update(bounds[0])
upper_bound.update(bounds[1])
# Try finding a cut vertex
if 'cut vertex' in tests:
# work around a bug in the cut vertex routines; see
# http://trac.sagemath.org/sage_trac/ticket/7853
if g.order()>1:
c_vertex=cut_vertex_balanced(g)
else:
c_vertex=False
if c_vertex is not False:
cut_vertex_bounds = cut_vertex_connected_graph_mr(c_vertex,graph)
if cut_vertex_bounds is not False:
lower_bound['cut vertex (%s)'%(c_vertex,)] = cut_vertex_bounds[0]
upper_bound['cut vertex (%s)'%(c_vertex,)] = cut_vertex_bounds[1]
else:
if 'disconnected' in tests:
connected_components = g.connected_components_subgraphs()
lower_bound['disconnected'] = 0
upper_bound['disconnected'] = 0
for component in g.connected_components_subgraphs():
sub_bound = minrank_bounds(component, tests=tests)
lower_bound['disconnected'] += sub_bound[0]
upper_bound['disconnected'] += sub_bound[1]
# Make sure that the lower bound is not greater than the upper bound
if max(lower_bound.values()) > min(upper_bound.values()):
raise StandardError("""
Best lower bound is greater than best upper bound; something is wrong:
lower bounds: %s
upper bounds: %s""" % (lower_bound,upper_bound))
if all_bounds is True:
return lower_bound, upper_bound
else:
# Return the best lower and upper bounds
return max(lower_bound.values()), min(upper_bound.values())
# From the patch for generic_graph.py
def is_outerplanar(self, **kwds):
"""
Check if the graph is outer-planar.
A graph is outer-planar if there is a planar embedding of the
graph such that some face includes every vertex (i.e., the graph
can be drawn so that all vertices are on a circle, all edges are
inside the circle, and no edges intersect).
:param graph: the graph to be checked
:return: True if the graph is outer-planar; False if it is not
Any extra keyword options are passed to the ``is_circular_planar``
function on a copy of self.
EXAMPLES::
sage: Graph({0:[1,2,3,4],1:[2,4],2:[3],3:[4]}).is_outerplanar()
False
sage: graphs.CompleteGraph(3).is_outerplanar()
True
"""
h = self.copy()
# Work around a bug in planarity testing by deleting degree 0 vertices
h.delete_vertices([v for v in h.vertices() if h.degree(v) == 0])
if h.order()==0:
return True
h.set_boundary(h.vertices())
kwds['ordered']=False
return h.is_circular_planar(**kwds)
# From the patch for generic_graph.py
def cut_vertex_balanced(self):
"""
Returns a cut vertex which cuts the graph into pieces with
smallest maximum size.
:return: a cut-vertex (if one exists) that either results in
components with a minimum of the maximum component order.
If no cut vertex exists, returns ``False``.
EXAMPLES::
sage: graphs.PathGraph(3).cut_vertex_balanced()
1
sage: graphs.PathGraph(20).cut_vertex_balanced()
9
sage: [graphs.PathGraph(i).cut_vertex_balanced() for i in [1..20]]
[False, False, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 6, 7, 7, 8, 8, 9, 9]
sage: graphs.CompleteGraph(3).cut_vertex_balanced()
False
"""
_,cut_vertices=self.blocks_and_cut_vertices()
vertices=self.vertices()
graph_cc_num=self.connected_components_number()
graph_order=self.order()
#this will hold the "best" cut-vertex and the order of the largest
#connected component after deletion
best_v=(False,graph_order)
#checks each vertex and determines the best one
for v in cut_vertices:
g=self.copy()
g.delete_vertex(v)
max_order = max(len(c) for c in g.connected_components())
if max_order<best_v[1]:
best_v=(v,max_order)
return best_v[0]
# From the patch for graph.py
def cliques_containing_edge(self, edge):
"""
Return all maximal cliques of the graph that contains the
edge.
:param edge: the edge for which to find the maximal clique
:return: the list of vertices in a maximal clique that
contains the edge. If the edge is not in the self, return
``None``.
EXAMPLES::
sage: graphs.CompleteGraph(5).cliques_containing_edge((1,2))
[0, 1, 2, 3, 4]
sage: graphs.PathGraph(5).cliques_containing_edge((1,3))
"""
vertex1,vertex1=edge
potential_cliques=self.cliques_containing_vertex(vertex1)
# sort the cliques containing vertex1 by order, largest first
potential_cliques.sort(key=len, reverse=True)
for clique in potential_cliques:
if vertex2 in clique:
return clique
return None
# From the patch for graph.py
def edge_clique_cover_minimum(self, bound=None):
"""
Returns an minimum edge clique cover for the graph if the
number of covering cliques is at most ``bound``; otherwise,
returns ``None``.
An edge clique cover is a set of cliques which contain all of
the edges of the graph.
.. note::
This function assumes self is connected.
:param bound: the maximum number of cliques to consider in an
edge clique cover
:return: If a minimum edge clique cover is found that has at
most ``bound`` cliques, the edge clique cover is returned
as a list of lists, each sublist being the vertices of a
clique. If a clique cover from this function requires more
than ``bound`` cliques, ``None`` is returned.
EXAMPLES::
sage: graphs.PathGraph(3).edge_clique_cover_minimum()
[[0, 1], [1, 2]]
sage: graphs.CompleteGraph(5).edge_clique_cover_minimum()
[[0, 1, 2, 3, 4]]
sage: graphs.HouseGraph().edge_clique_cover_minimum()
[[2, 3, 4], [0, 1], [0, 2], [1, 3]]
sage: graphs.PetersenGraph().edge_clique_cover_minimum(bound=4)
"""
from sage.all import ceil, Combinations
# Take care of trivial case
if self.size() == 0:
return []
max_cliques=self.cliques_maximal()
max_cliques.sort(key=len)
largest_clique_vertices = len(max_cliques[-1])
max_cliques = [sorted(clique) for clique in max_cliques]
largest_clique_edges = largest_clique_vertices \
*(largest_clique_vertices-1)/2
edges_of_graph=self.edges(labels=False)
num_edges = self.size()
mandatory_cliques=[]
for v in self.vertices():
# If v is contained in only one clique, then that clique must
# be in the clique cover
cliques_containing_v = [c for c in max_cliques if v in c]
if len(cliques_containing_v)==1 \
and (cliques_containing_v[0] not in mandatory_cliques):
mandatory_cliques.append(cliques_containing_v[0])
for e in self.edges():
# If e is contained in only one clique, then that clique must
# be in the clique cover
cliques_containing_e = [c for c in max_cliques
if e[0] in c and e[1] in c]
if len(cliques_containing_e)==1 \
and (cliques_containing_e[0] not in mandatory_cliques):
mandatory_cliques.append(cliques_containing_e[0])
# Check to see if mandatory_cliques contains a clique cover
edges_in_set_of_cliques = set([])
for clique in mandatory_cliques:
edges_in_clique = [(clique[i], clique[j])
for i in xrange(len(clique))
for j in xrange(i+1,len(clique))]
edges_in_set_of_cliques.update(set(edges_in_clique))
if len(edges_in_set_of_cliques) == num_edges:
if bound is None or len(mandatory_cliques) <= bound:
return mandatory_cliques
else:
# There are too many cliques. Return None to be
# consistent with the documentation, even though we
# actually know the clique cover number (and it is greater
# than bound).
return None
max_cliques = [c for c in max_cliques if c not in mandatory_cliques]
if bound==None:
stopping_point=len(max_cliques)
else:
stopping_point=min(len(max_cliques),bound-len(mandatory_cliques))
starting_point = max(1,ceil(float(num_edges) / largest_clique_edges) \
- len(mandatory_cliques))
for i in range(starting_point,stopping_point+1):
for set_of_cliques in Combinations(max_cliques,i):
edges_in_set_of_cliques = set([])
for clique in set_of_cliques+mandatory_cliques:
edges_in_clique = [(clique[i], clique[j])
for i in xrange(len(clique))
for j in xrange(i+1,len(clique))]
edges_in_set_of_cliques.update(set(edges_in_clique))
if len(edges_in_set_of_cliques) == num_edges: