1.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
1.2 +++ b/Orig/ContactMaps.128.txt Tue Nov 22 08:15:16 2016 +0100
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2.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
2.2 +++ b/Orig/ContactMaps.16.txt Tue Nov 22 08:15:16 2016 +0100
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3.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
3.2 +++ b/Orig/ContactMaps.256.txt Tue Nov 22 08:15:16 2016 +0100
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4.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
4.2 +++ b/Orig/ContactMaps.32.txt Tue Nov 22 08:15:16 2016 +0100
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5.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
5.2 +++ b/Orig/ContactMaps.64.txt Tue Nov 22 08:15:16 2016 +0100
5.3 @@ -0,0 +1,90 @@
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6.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
6.2 +++ b/Orig/ContactMaps.8.txt Tue Nov 22 08:15:16 2016 +0100
6.3 @@ -0,0 +1,90 @@
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7.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
7.2 +++ b/Orig/ContactMaps.txt Tue Nov 22 08:15:16 2016 +0100
7.3 @@ -0,0 +1,90 @@
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8.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
8.2 +++ b/Orig/Molecules.16.txt Tue Nov 22 08:15:16 2016 +0100
8.3 @@ -0,0 +1,83 @@
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9.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
9.2 +++ b/Orig/Molecules.32.txt Tue Nov 22 08:15:16 2016 +0100
9.3 @@ -0,0 +1,67 @@
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10.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
10.2 +++ b/Orig/Molecules.64.txt Tue Nov 22 08:15:16 2016 +0100
10.3 @@ -0,0 +1,35 @@
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11.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
11.2 +++ b/Orig/Molecules.8.txt Tue Nov 22 08:15:16 2016 +0100
11.3 @@ -0,0 +1,90 @@
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12.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
12.2 +++ b/Orig/Molecules.txt Tue Nov 22 08:15:16 2016 +0100
12.3 @@ -0,0 +1,90 @@
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13.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
13.2 +++ b/Orig/Proteins.128.txt Tue Nov 22 08:15:16 2016 +0100
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14.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
14.2 +++ b/Orig/Proteins.16.txt Tue Nov 22 08:15:16 2016 +0100
14.3 @@ -0,0 +1,234 @@
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15.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
15.2 +++ b/Orig/Proteins.256.txt Tue Nov 22 08:15:16 2016 +0100
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16.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
16.2 +++ b/Orig/Proteins.32.txt Tue Nov 22 08:15:16 2016 +0100
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17.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
17.2 +++ b/Orig/Proteins.64.txt Tue Nov 22 08:15:16 2016 +0100
17.3 @@ -0,0 +1,234 @@
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18.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
18.2 +++ b/Orig/Proteins.8.txt Tue Nov 22 08:15:16 2016 +0100
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19.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
19.2 +++ b/Orig/Proteins.txt Tue Nov 22 08:15:16 2016 +0100
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20.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
20.2 +++ b/Orig/contactMapIso.txt Tue Nov 22 08:15:16 2016 +0100
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20.58 +265 1.343
20.59 +271 1.404
20.60 +274 1.538
20.61 +275 1.623
20.62 +276 1.503
20.63 +277 1.471
20.64 +280 1.601
20.65 +286 1.612
20.66 +302 1.696
20.67 +327 1.783
20.68 +330 1.761
20.69 +334 1.964
20.70 +336 1.839
20.71 +359 2.097
20.72 +360 2.163
20.73 +385 2.117
20.74 +387 2.149
20.75 +388 2.147
20.76 +393 2.207
20.77 +395 2.236
20.78 +414 2.352
20.79 +428 2.666
20.80 +457 2.834
20.81 +484 3.249
20.82 +501 3.035
20.83 +556 3.407
20.84 +558 3.247
20.85 +564 3.737
20.86 +585 3.696
20.87 +599 3.857
20.88 +600 3.735
20.89 +611 4.12
20.90 +617 3.811
20.91 +694 5.391
20.92 +741 5.231
20.93 +773 4.201
21.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
21.2 +++ b/Orig/contactMapsIso.txt Tue Nov 22 08:15:16 2016 +0100
21.3 @@ -0,0 +1,90 @@
21.4 +99 0.0349
21.5 +102 0.0387
21.6 +104 0.0437
21.7 +105 0.0424
21.8 +107 0.0397
21.9 +114 0.0371
21.10 +117 0.0419
21.11 +120 0.0477
21.12 +122 0.0503
21.13 +123 0.0481
21.14 +124 0.0421
21.15 +125 0.0557
21.16 +130 0.0559
21.17 +133 0.0572
21.18 +136 0.0588
21.19 +137 0.0589
21.20 +138 0.058
21.21 +140 0.0656
21.22 +141 0.061
21.23 +144 0.0596
21.24 +146 0.0335
21.25 +147 0.0552
21.26 +149 0.0484
21.27 +150 0.0605
21.28 +151 0.0592
21.29 +156 0.0686
21.30 +157 0.0706
21.31 +158 0.0682
21.32 +159 0.0659
21.33 +162 0.0685
21.34 +164 0.0631
21.35 +169 0.072
21.36 +181 0.0707
21.37 +185 0.0855
21.38 +186 0.0838
21.39 +187 0.08
21.40 +190 0.0877
21.41 +191 0.0872
21.42 +192 0.0889
21.43 +193 0.093
21.44 +197 0.0924
21.45 +198 0.1006
21.46 +199 0.0912
21.47 +214 0.0904
21.48 +220 0.1014
21.49 +221 0.1003
21.50 +222 0.1043
21.51 +224 0.1029
21.52 +225 0.1053
21.53 +226 0.1076
21.54 +228 0.1089
21.55 +237 0.1136
21.56 +252 0.1203
21.57 +264 0.1302
21.58 +265 0.124
21.59 +271 0.1305
21.60 +274 0.1437
21.61 +275 0.1494
21.62 +276 0.1395
21.63 +277 0.1375
21.64 +280 0.141
21.65 +286 0.1408
21.66 +302 0.1566
21.67 +327 0.1652
21.68 +330 0.1653
21.69 +334 0.1811
21.70 +336 0.1721
21.71 +359 0.194
21.72 +360 0.2031
21.73 +385 0.1981
21.74 +387 0.2019
21.75 +388 0.1988
21.76 +393 0.2099
21.77 +395 0.2083
21.78 +414 0.2197
21.79 +428 0.2437
21.80 +457 0.2567
21.81 +484 0.2966
21.82 +501 0.2779
21.83 +556 0.3125
21.84 +558 0.3062
21.85 +564 0.3498
21.86 +585 0.3466
21.87 +599 0.3545
21.88 +600 0.3504
21.89 +611 0.3838
21.90 +617 0.3602
21.91 +694 0.5001
21.92 +741 0.4809
21.93 +773 0.4004
22.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
22.2 +++ b/Orig/moleculeIso.txt Tue Nov 22 08:15:16 2016 +0100
22.3 @@ -0,0 +1,91 @@
22.4 +8 0.009
22.5 +10 0.009
22.6 +11 0.014
22.7 +12 0.014
22.8 +13 0.011
22.9 +14 0.015
22.10 +15 0.013
22.11 +16 0.012
22.12 +17 0.016
22.13 +18 0.013
22.14 +19 0.01
22.15 +20 0.015
22.16 +21 0.018
22.17 +22 0.016
22.18 +23 0.018
22.19 +24 0.016
22.20 +25 0.023
22.21 +26 0.024
22.22 +27 0.023
22.23 +28 0.022
22.24 +29 0.023
22.25 +30 0.024
22.26 +31 0.025
22.27 +32 0.026
22.28 +33 0.026
22.29 +34 0.03
22.30 +35 0.03
22.31 +36 0.03
22.32 +37 0.031
22.33 +38 0.032
22.34 +39 0.03
22.35 +40 0.03
22.36 +41 0.034
22.37 +42 0.039
22.38 +43 0.032
22.39 +44 0.037
22.40 +45 0.031
22.41 +46 0.038
22.42 +47 0.035
22.43 +48 0.039
22.44 +49 0.04
22.45 +50 0.042
22.46 +51 0.041
22.47 +52 0.048
22.48 +53 0.04
22.49 +54 0.04
22.50 +55 0.046
22.51 +56 0.037
22.52 +57 0.04
22.53 +58 0.055
22.54 +59 0.047
22.55 +60 0.045
22.56 +61 0.051
22.57 +62 0.053
22.58 +63 0.066
22.59 +64 0.043
22.60 +65 0.066
22.61 +66 0.066
22.62 +67 0.065
22.63 +68 0.074
22.64 +69 0.059
22.65 +70 0.055
22.66 +71 0.083
22.67 +72 0.057
22.68 +73 0.06
22.69 +74 0.054
22.70 +75 0.054
22.71 +76 0.062
22.72 +77 0.064
22.73 +78 0.071
22.74 +79 0.062
22.75 +80 0.075
22.76 +81 0.069
22.77 +82 0.07
22.78 +83 0.07
22.79 +84 0.069
22.80 +85 0.073
22.81 +86 0.086
22.82 +87 0.092
22.83 +88 0.085
22.84 +89 0.082
22.85 +90 0.097
22.86 +91 0.083
22.87 +92 0.111
22.88 +93 0.073
22.89 +94 0.124
22.90 +95 0.087
22.91 +96 0.097
22.92 +97 0.087
22.93 +98 0.088
22.94 +99 0.112
23.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
23.2 +++ b/Orig/moleculesIso.txt Tue Nov 22 08:15:16 2016 +0100
23.3 @@ -0,0 +1,89 @@
23.4 +8 0.00214
23.5 +10 0.00289
23.6 +11 0.00354
23.7 +12 0.00469
23.8 +13 0.00355
23.9 +14 0.00412
23.10 +15 0.00449
23.11 +16 0.00418
23.12 +17 0.00585
23.13 +18 0.00507
23.14 +19 0.00426
23.15 +20 0.00666
23.16 +21 0.00682
23.17 +22 0.00644
23.18 +23 0.00785
23.19 +24 0.00687
23.20 +25 0.01061
23.21 +26 0.00831
23.22 +27 0.00838
23.23 +28 0.01307
23.24 +29 0.0105
23.25 +30 0.01221
23.26 +31 0.01033
23.27 +32 0.01276
23.28 +33 0.01496
23.29 +34 0.01311
23.30 +35 0.01318
23.31 +36 0.01419
23.32 +37 0.01445
23.33 +38 0.01471
23.34 +39 0.01562
23.35 +40 0.01423
23.36 +41 0.01704
23.37 +42 0.01971
23.38 +43 0.01602
23.39 +44 0.02022
23.40 +45 0.01618
23.41 +46 0.02056
23.42 +47 0.01739
23.43 +48 0.01871
23.44 +49 0.01775
23.45 +50 0.01764
23.46 +51 0.01992
23.47 +52 0.01969
23.48 +53 0.01888
23.49 +54 0.01948
23.50 +55 0.02089
23.51 +56 0.01696
23.52 +57 0.01885
23.53 +58 0.02133
23.54 +59 0.02238
23.55 +60 0.0212
23.56 +61 0.02596
23.57 +62 0.02761
23.58 +63 0.02743
23.59 +64 0.02153
23.60 +65 0.02993
23.61 +66 0.03349
23.62 +67 0.0314
23.63 +68 0.0351
23.64 +69 0.0246
23.65 +70 0.02679
23.66 +71 0.03377
23.67 +72 0.0278
23.68 +73 0.02835
23.69 +74 0.02895
23.70 +75 0.02715
23.71 +76 0.03037
23.72 +77 0.03238
23.73 +78 0.03211
23.74 +79 0.03462
23.75 +80 0.03723
23.76 +81 0.03793
23.77 +82 0.03227
23.78 +83 0.03539
23.79 +84 0.03406
23.80 +85 0.03441
23.81 +86 0.04042
23.82 +87 0.03769
23.83 +88 0.03697
23.84 +89 0.04212
23.85 +91 0.03941
23.86 +93 0.04149
23.87 +94 0.04669
23.88 +95 0.04263
23.89 +96 0.04419
23.90 +97 0.04054
23.91 +98 0.04179
23.92 +99 0.04812
24.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
24.2 +++ b/Orig/proteinIso.txt Tue Nov 22 08:15:16 2016 +0100
24.3 @@ -0,0 +1,234 @@
24.4 +535 1.069
24.5 +554 1.143
24.6 +571 1.299
24.7 +614 1.331
24.8 +634 1.439
24.9 +636 1.477
24.10 +639 1.451
24.11 +730 1.87
24.12 +731 1.903
24.13 +734 2.025
24.14 +735 1.897
24.15 +737 1.938
24.16 +755 1.982
24.17 +760 2.175
24.18 +788 2.207
24.19 +794 2.373
24.20 +795 2.183
24.21 +802 2.206
24.22 +816 2.268
24.23 +819 2.419
24.24 +831 2.409
24.25 +833 2.432
24.26 +838 2.376
24.27 +858 2.511
24.28 +861 2.592
24.29 +874 2.581
24.30 +877 2.585
24.31 +879 2.609
24.32 +886 2.851
24.33 +899 2.731
24.34 +957 3.039
24.35 +982 3.273
24.36 +995 3.267
24.37 +1019 3.489
24.38 +1022 3.462
24.39 +1046 3.622
24.40 +1049 3.629
24.41 +1060 3.837
24.42 +1073 3.861
24.43 +1076 3.853
24.44 +1097 4.068
24.45 +1140 4.22
24.46 +1156 4.367
24.47 +1183 4.528
24.48 +1192 4.608
24.49 +1202 5.595
24.50 +1227 4.938
24.51 +1271 5.279
24.52 +1300 5.488
24.53 +1304 5.437
24.54 +1312 5.72
24.55 +1356 5.976
24.56 +1494 7.134
24.57 +1502 8.15
24.58 +1559 7.822
24.59 +1568 7.824
24.60 +1576 8.457
24.61 +1605 8.123
24.62 +1608 8.801
24.63 +1625 8.968
24.64 +1632 9.29
24.65 +1635 9.113
24.66 +1648 9.986
24.67 +1674 8.959
24.68 +1684 9.119
24.69 +1732 9.504
24.70 +1744 10.306
24.71 +1750 9.97
24.72 +1754 11.228
24.73 +1764 10.573
24.74 +1775 10.007
24.75 +1826 10.534
24.76 +1844 11.526
24.77 +1886 11.179
24.78 +1930 11.844
24.79 +1938 11.87
24.80 +1948 11.886
24.81 +1995 12.518
24.82 +2000 13.394
24.83 +2020 12.788
24.84 +2044 13.162
24.85 +2086 14.446
24.86 +2115 14.052
24.87 +2123 14.273
24.88 +2126 14.565
24.89 +2138 14.486
24.90 +2168 14.635
24.91 +2190 14.84
24.92 +2222 15.311
24.93 +2250 15.622
24.94 +2298 16.504
24.95 +2314 16.704
24.96 +2316 16.47
24.97 +2325 18.018
24.98 +2327 16.582
24.99 +2329 16.898
24.100 +2344 17.201
24.101 +2348 16.955
24.102 +2355 17.22
24.103 +2379 17.399
24.104 +2383 17.421
24.105 +2413 17.892
24.106 +2423 17.98
24.107 +2464 18.903
24.108 +2466 18.911
24.109 +2512 19.285
24.110 +2546 19.636
24.111 +2553 20.608
24.112 +2593 20.407
24.113 +2604 20.587
24.114 +2605 22.925
24.115 +2607 22.02
24.116 +2610 22.058
24.117 +2611 20.714
24.118 +2612 22.399
24.119 +2613 20.741
24.120 +2616 20.734
24.121 +2620 20.816
24.122 +2622 20.836
24.123 +2625 23.002
24.124 +2648 21.074
24.125 +2651 21.203
24.126 +2716 22.383
24.127 +2733 22.638
24.128 +2775 23.708
24.129 +2812 24.236
24.130 +2838 24.318
24.131 +2857 24.726
24.132 +2872 24.967
24.133 +2887 25.232
24.134 +2916 25.613
24.135 +2920 25.956
24.136 +2928 26.125
24.137 +2949 26.15
24.138 +2966 26.491
24.139 +2977 28.792
24.140 +2991 26.936
24.141 +3104 29.15
24.142 +3130 29.567
24.143 +3131 29.599
24.144 +3134 29.537
24.145 +3135 29.58
24.146 +3172 32.619
24.147 +3182 30.799
24.148 +3209 30.968
24.149 +3243 31.605
24.150 +3255 31.917
24.151 +3310 32.849
24.152 +3366 37.254
24.153 +3429 35.516
24.154 +3459 35.948
24.155 +3469 36.118
24.156 +3483 36.311
24.157 +3497 37.724
24.158 +3502 36.693
24.159 +3505 36.826
24.160 +3515 37.302
24.161 +3517 37
24.162 +3522 37.049
24.163 +3523 37.207
24.164 +3526 37.279
24.165 +3529 37.504
24.166 +3560 37.933
24.167 +3563 37.97
24.168 +3566 38.104
24.169 +3574 41.274
24.170 +3582 38.911
24.171 +3584 38.693
24.172 +3618 42.161
24.173 +3619 39.329
24.174 +3623 42.255
24.175 +3703 41.236
24.176 +3711 41.072
24.177 +3819 43.676
24.178 +3829 43.854
24.179 +3839 44.288
24.180 +3870 44.748
24.181 +3906 45.735
24.182 +4025 48.558
24.183 +4083 50.176
24.184 +4156 52.911
24.185 +4194 52.261
24.186 +4220 52.864
24.187 +4244 57.808
24.188 +4283 58.769
24.189 +4353 56.757
24.190 +4382 56.877
24.191 +4389 57.042
24.192 +4421 57.984
24.193 +4446 59.382
24.194 +4521 60.846
24.195 +4531 60.868
24.196 +4532 60.751
24.197 +4568 62.485
24.198 +4573 66.916
24.199 +4582 62.751
24.200 +4604 67.785
24.201 +4629 64.027
24.202 +4710 66.067
24.203 +4713 66.413
24.204 +4797 68.111
24.205 +4879 70.489
24.206 +4970 72.896
24.207 +5059 75.988
24.208 +5126 77.531
24.209 +5241 80.788
24.210 +5254 81.485
24.211 +5271 82.209
24.212 +5318 89.984
24.213 +5346 84.387
24.214 +5455 87.669
24.215 +5492 88.66
24.216 +5608 92.515
24.217 +5735 97.315
24.218 +5775 98.216
24.219 +5777 98.269
24.220 +5795 99.811
24.221 +5845 100.942
24.222 +5859 101.083
24.223 +5877 101.587
24.224 +5945 105.329
24.225 +5974 104.948
24.226 +5976 104.704
24.227 +5983 105.662
24.228 +6261 115.383
24.229 +6606 128.002
24.230 +6713 132.525
24.231 +6778 145.161
24.232 +6883 138.36
24.233 +7349 158.714
24.234 +7453 163.23
24.235 +9114 242.24
24.236 +9174 244.585
24.237 +10081 295.288
25.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
25.2 +++ b/Orig/proteinsIso.txt Tue Nov 22 08:15:16 2016 +0100
25.3 @@ -0,0 +1,234 @@
25.4 +535 0.1026
25.5 +554 0.1098
25.6 +571 0.1199
25.7 +614 0.131
25.8 +634 0.1407
25.9 +636 0.1439
25.10 +639 0.1433
25.11 +730 0.1818
25.12 +731 0.1823
25.13 +734 0.1857
25.14 +735 0.1834
25.15 +737 0.1883
25.16 +755 0.1935
25.17 +760 0.1947
25.18 +788 0.2109
25.19 +794 0.214
25.20 +795 0.2146
25.21 +802 0.2168
25.22 +816 0.225
25.23 +819 0.227
25.24 +831 0.2305
25.25 +833 0.2311
25.26 +838 0.2345
25.27 +858 0.2436
25.28 +861 0.2476
25.29 +874 0.2516
25.30 +877 0.2528
25.31 +879 0.257
25.32 +886 0.2598
25.33 +899 0.2651
25.34 +957 0.2984
25.35 +982 0.316
25.36 +995 0.3251
25.37 +1019 0.3412
25.38 +1022 0.3391
25.39 +1046 0.3555
25.40 +1049 0.3567
25.41 +1060 0.3628
25.42 +1073 0.3769
25.43 +1076 0.3735
25.44 +1097 0.3869
25.45 +1140 0.4171
25.46 +1156 0.4276
25.47 +1183 0.4452
25.48 +1192 0.4535
25.49 +1202 0.4626
25.50 +1227 0.4815
25.51 +1271 0.5108
25.52 +1300 0.5313
25.53 +1304 0.5402
25.54 +1312 0.5451
25.55 +1356 0.5765
25.56 +1494 0.7052
25.57 +1502 0.7017
25.58 +1559 0.7573
25.59 +1568 0.7675
25.60 +1576 0.7678
25.61 +1605 0.7975
25.62 +1608 0.8003
25.63 +1625 0.8217
25.64 +1632 0.8223
25.65 +1635 0.8315
25.66 +1648 0.8444
25.67 +1674 0.8698
25.68 +1684 0.8829
25.69 +1732 0.9302
25.70 +1744 0.9393
25.71 +1750 0.9475
25.72 +1754 0.9523
25.73 +1764 0.9621
25.74 +1775 0.9685
25.75 +1826 1.0346
25.76 +1844 1.0558
25.77 +1886 1.0843
25.78 +1930 1.1393
25.79 +1938 1.1485
25.80 +1948 1.1616
25.81 +1995 1.2225
25.82 +2000 1.2204
25.83 +2020 1.2573
25.84 +2044 1.2795
25.85 +2086 1.324
25.86 +2115 1.3709
25.87 +2123 1.3716
25.88 +2126 1.3823
25.89 +2138 1.4018
25.90 +2168 1.4301
25.91 +2190 1.4717
25.92 +2222 1.5018
25.93 +2250 1.5523
25.94 +2298 1.6168
25.95 +2314 1.6113
25.96 +2316 1.6139
25.97 +2325 1.6484
25.98 +2327 1.6354
25.99 +2329 1.6418
25.100 +2344 1.6727
25.101 +2348 1.7223
25.102 +2355 1.6953
25.103 +2379 1.7201
25.104 +2383 1.7298
25.105 +2413 1.7556
25.106 +2423 1.7728
25.107 +2464 1.8413
25.108 +2466 1.8505
25.109 +2512 1.9063
25.110 +2546 1.9325
25.111 +2553 1.9573
25.112 +2593 2.0054
25.113 +2604 2.0312
25.114 +2605 2.0276
25.115 +2607 2.0539
25.116 +2610 2.1964
25.117 +2611 2.0548
25.118 +2612 2.0526
25.119 +2613 2.053
25.120 +2616 2.0359
25.121 +2620 2.0667
25.122 +2622 2.0891
25.123 +2625 2.0852
25.124 +2648 2.1203
25.125 +2651 2.1229
25.126 +2716 2.2364
25.127 +2733 2.2416
25.128 +2775 2.3391
25.129 +2812 2.3688
25.130 +2838 2.4074
25.131 +2857 2.4468
25.132 +2872 2.4905
25.133 +2887 2.4887
25.134 +2916 2.5366
25.135 +2920 2.5714
25.136 +2928 2.5654
25.137 +2949 2.6144
25.138 +2966 2.6556
25.139 +2977 2.6479
25.140 +2991 2.6759
25.141 +3104 2.9095
25.142 +3130 2.9316
25.143 +3131 2.9374
25.144 +3134 2.9346
25.145 +3135 2.9655
25.146 +3172 3.0049
25.147 +3182 3.0244
25.148 +3209 3.1033
25.149 +3243 3.1742
25.150 +3255 3.2008
25.151 +3310 3.286
25.152 +3366 3.3784
25.153 +3429 3.5129
25.154 +3459 3.5943
25.155 +3469 3.5759
25.156 +3483 3.6419
25.157 +3497 3.6432
25.158 +3502 3.652
25.159 +3505 3.6932
25.160 +3515 3.7099
25.161 +3517 3.7241
25.162 +3522 3.7322
25.163 +3523 3.7347
25.164 +3526 3.7029
25.165 +3529 3.7313
25.166 +3560 3.8172
25.167 +3563 3.7798
25.168 +3566 3.8196
25.169 +3574 3.7936
25.170 +3582 3.8519
25.171 +3584 3.8339
25.172 +3618 3.8882
25.173 +3619 3.9308
25.174 +3623 3.8938
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26.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
26.2 +++ b/VF2PP/ContactMaps.8.txt Tue Nov 22 08:15:16 2016 +0100
26.3 @@ -0,0 +1,90 @@
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27.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
27.2 +++ b/VF2PPLabel/ContactMaps.128.txt Tue Nov 22 08:15:16 2016 +0100
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28.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
28.2 +++ b/VF2PPLabel/ContactMaps.16.txt Tue Nov 22 08:15:16 2016 +0100
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29.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
29.2 +++ b/VF2PPLabel/ContactMaps.256.txt Tue Nov 22 08:15:16 2016 +0100
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30.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
30.2 +++ b/VF2PPLabel/ContactMaps.32.txt Tue Nov 22 08:15:16 2016 +0100
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31.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
31.2 +++ b/VF2PPLabel/ContactMaps.64.txt Tue Nov 22 08:15:16 2016 +0100
31.3 @@ -0,0 +1,90 @@
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32.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
32.2 +++ b/VF2PPLabel/ContactMaps.8.txt Tue Nov 22 08:15:16 2016 +0100
32.3 @@ -0,0 +1,90 @@
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33.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
33.2 +++ b/VF2PPLabel/ContactMaps.txt Tue Nov 22 08:15:16 2016 +0100
33.3 @@ -0,0 +1,90 @@
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34.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
34.2 +++ b/VF2PPLabel/Molecules.16.txt Tue Nov 22 08:15:16 2016 +0100
34.3 @@ -0,0 +1,83 @@
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35.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
35.2 +++ b/VF2PPLabel/Molecules.32.txt Tue Nov 22 08:15:16 2016 +0100
35.3 @@ -0,0 +1,67 @@
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36.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
36.2 +++ b/VF2PPLabel/Molecules.64.txt Tue Nov 22 08:15:16 2016 +0100
36.3 @@ -0,0 +1,35 @@
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37.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
37.2 +++ b/VF2PPLabel/Molecules.8.txt Tue Nov 22 08:15:16 2016 +0100
37.3 @@ -0,0 +1,90 @@
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38.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
38.2 +++ b/VF2PPLabel/Molecules.txt Tue Nov 22 08:15:16 2016 +0100
38.3 @@ -0,0 +1,90 @@
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39.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
39.2 +++ b/VF2PPLabel/Proteins.128.txt Tue Nov 22 08:15:16 2016 +0100
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40.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
40.2 +++ b/VF2PPLabel/Proteins.16.txt Tue Nov 22 08:15:16 2016 +0100
40.3 @@ -0,0 +1,234 @@
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41.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
41.2 +++ b/VF2PPLabel/Proteins.256.txt Tue Nov 22 08:15:16 2016 +0100
41.3 @@ -0,0 +1,234 @@
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42.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
42.2 +++ b/VF2PPLabel/Proteins.32.txt Tue Nov 22 08:15:16 2016 +0100
42.3 @@ -0,0 +1,234 @@
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43.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
43.2 +++ b/VF2PPLabel/Proteins.64.txt Tue Nov 22 08:15:16 2016 +0100
43.3 @@ -0,0 +1,234 @@
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44.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
44.2 +++ b/VF2PPLabel/Proteins.8.txt Tue Nov 22 08:15:16 2016 +0100
44.3 @@ -0,0 +1,234 @@
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45.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
45.2 +++ b/VF2PPLabel/Proteins.txt Tue Nov 22 08:15:16 2016 +0100
45.3 @@ -0,0 +1,234 @@
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46.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
46.2 +++ b/VF2PPLabel/contactMapIso.txt Tue Nov 22 08:15:16 2016 +0100
46.3 @@ -0,0 +1,90 @@
46.4 +99 0.344
46.5 +102 0.385
46.6 +104 0.415
46.7 +105 0.38
46.8 +107 0.366
46.9 +114 0.353
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46.34 +164 0.581
46.35 +169 0.666
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46.40 +190 0.803
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46.54 +228 1.007
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46.56 +252 1.067
46.57 +264 1.151
46.58 +265 1.067
46.59 +271 1.153
46.60 +274 1.279
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46.68 +330 1.49
46.69 +334 1.581
46.70 +336 1.513
46.71 +359 1.689
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46.75 +388 1.684
46.76 +393 1.681
46.77 +395 1.715
46.78 +414 1.797
46.79 +428 2.109
46.80 +457 2.107
46.81 +484 2.438
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46.83 +556 2.314
46.84 +558 2.252
46.85 +564 2.707
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46.89 +611 2.957
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46.91 +694 3.851
46.92 +741 3.578
46.93 +773 2.433
47.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
47.2 +++ b/VF2PPLabel/contactMapsIso.txt Tue Nov 22 08:15:16 2016 +0100
47.3 @@ -0,0 +1,90 @@
47.4 +99 0.0291
47.5 +102 0.0328
47.6 +104 0.0368
47.7 +105 0.0354
47.8 +107 0.0324
47.9 +114 0.0306
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47.17 +133 0.0468
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47.19 +137 0.0474
47.20 +138 0.0486
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47.25 +147 0.0468
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47.30 +157 0.0584
47.31 +158 0.0573
47.32 +159 0.0572
47.33 +162 0.0556
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47.35 +169 0.061
47.36 +181 0.0565
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47.52 +225 0.0857
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47.63 +277 0.1116
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47.65 +286 0.11
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47.76 +393 0.1488
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47.78 +414 0.1563
47.79 +428 0.18
47.80 +457 0.1943
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47.84 +558 0.2014
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47.92 +741 0.3004
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48.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
48.2 +++ b/VF2PPLabel/moleculeIso.txt Tue Nov 22 08:15:16 2016 +0100
48.3 @@ -0,0 +1,91 @@
48.4 +8 0.007
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48.6 +11 0.01
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48.11 +16 0.009
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48.20 +25 0.023
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48.88 +93 0.046
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48.94 +99 0.085
49.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
49.2 +++ b/VF2PPLabel/moleculesIso.txt Tue Nov 22 08:15:16 2016 +0100
49.3 @@ -0,0 +1,89 @@
49.4 +8 0.0019
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49.9 +14 0.00355
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49.13 +18 0.00405
49.14 +19 0.00366
49.15 +20 0.00578
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49.17 +22 0.00552
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49.21 +26 0.00792
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49.25 +30 0.00954
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49.47 +52 0.01659
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49.92 +99 0.03396
50.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
50.2 +++ b/VF2PPLabel/proteinIso.txt Tue Nov 22 08:15:16 2016 +0100
50.3 @@ -0,0 +1,234 @@
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50.9 +636 0.321
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50.11 +730 0.334
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50.14 +735 0.309
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50.16 +755 0.305
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50.20 +795 0.375
50.21 +802 0.333
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50.23 +819 0.387
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50.32 +886 0.42
50.33 +899 0.44
50.34 +957 0.399
50.35 +982 0.475
50.36 +995 0.415
50.37 +1019 3.067
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50.39 +1046 0.477
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50.54 +1312 0.607
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50.56 +1494 0.658
50.57 +1502 1.021
50.58 +1559 0.64
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50.117 +2611 2
50.118 +2612 1.241
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50.198 +4573 2.494
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50.234 +7453 3.998
50.235 +9114 4.116
50.236 +9174 5.284
50.237 +10081 4.765
51.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
51.2 +++ b/VF2PPLabel/proteinsIso.txt Tue Nov 22 08:15:16 2016 +0100
51.3 @@ -0,0 +1,234 @@
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52.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
52.2 +++ b/bibliography.bib Tue Nov 22 08:15:16 2016 +0100
52.3 @@ -0,0 +1,174 @@
52.4 +%Since BibTeX does not appear to have a built in entry
52.5 +%for citing websites (what kind of design is that?),
52.6 +%I decided to stick websites in with articles, and to stick
52.7 +%the URL in the journal field. This is not technically correct.
52.8 +%I did something largely similar with Walter Lewin's video lectures,
52.9 +%(which are wonderful), posted online through MIT OCW. Since a
52.10 +%'proper' way of citing it wasn't built in, I made one up. If
52.11 +%you're doing a bibliography where correct formatting is very
52.12 +%important, you'd do best not to follow my example. Otherwise,
52.13 +%I think it's good enough.
52.14 +
52.15 +@Article{CordellaVentoSymbolRecognition,
52.16 + author = "L.P. Cordella, M. Vento",
52.17 + title = "Symbol recognition in documents: a collection of techniques?",
52.18 + journal = "International Journal on Document Analysis and Recognition Volume 3, Issue 2, pp 73-88",
52.19 + year = 2000,
52.20 + month = "December"
52.21 +}
52.22 +
52.23 +@Article{ProteinDataBank,
52.24 + author = "Protein Data Bank",
52.25 + title = "http://www.rcsb.org/pdb",
52.26 + journal = "",
52.27 + year = 2015,
52.28 + month = "June"
52.29 +}
52.30 +
52.31 +@Article{LEMON,
52.32 + author = "LEMON: Library for Efficient Modeling and Optimization in Networks",
52.33 + title = "https://lemon.cs.elte.hu",
52.34 + journal = "",
52.35 + year = 2015,
52.36 + month = "March"
52.37 +}
52.38 +
52.39 +@Article{QUANTUMBIO,
52.40 + author = "QuantumBio Inc.",
52.41 + title = "http://www.quantumbioinc.com/"
52.42 +}
52.43 +
52.44 +@Article{Content,
52.45 + author = "Vento, Mario and Jiang, Xiaoyi and Foggia, Pasquale",
52.46 + title = "International contest on pattern search in biological databases, http://biograph2014.unisa.it ",
52.47 + journal = "",
52.48 + year = 2015,
52.49 + month = "June"
52.50 +}
52.51 +
52.52 +@Article{VF,
52.53 + author = "L. P. Cordella, P. Foggia, C. Sansone and M. Vento",
52.54 + title = "Performance Evaluation of the VF Graph Matching Algorithm",
52.55 + journal = "Proc. of the 10th ICIAP, IEEE Computer Society Press, pp. 1172-1177",
52.56 + year = 1999,
52.57 + month = ""
52.58 +}
52.59 +
52.60 +@Article{UllmannBit,
52.61 + author = "Julian R. Ullmann",
52.62 + title = "Bit-vector algorithms for binary constraint satisfaction and subgraph isomorphism",
52.63 + journal = "Journal of Experimental Algorithmics (JEA),Volume 15, Article No. 1.6, ACM New York, NY, USA ",
52.64 + year = 2010,
52.65 + month = ""
52.66 +}
52.67 +
52.68 +@Article{Ullmann,
52.69 + author = "J. R. Ullmann",
52.70 + title = "An Algorithm for Subgraph Isomorphism",
52.71 + journal = "Journal of the ACM (JACM), Volume 23 Issue 1, Pages 31-42 ",
52.72 + year = 1976,
52.73 + month = "January"
52.74 +}
52.75 +
52.76 +@Article{SubgraphNPC,
52.77 + author = "Cook, S. A.",
52.78 + title = "The complexity of theorem-proving procedures",
52.79 + journal = "Proc. 3rd ACM Symposium on Theory of Computing, pp. 151-158",
52.80 + year = 1971,
52.81 + month = ""
52.82 +}
52.83 +
52.84 +@Article{VF2,
52.85 + author = "Cordella, L.P. and Foggia, P. and Sansone, C. and Vento, M.",
52.86 + title = "A (sub)graph isomorphism algorithm for matching large graphs ",
52.87 + journal = "Journal of the ACM (JACM) JACM Homepage archive Volume 23 Issue 1, Pages 31-42 ",
52.88 + year = 2004,
52.89 + month = ""
52.90 +}
52.91 +
52.92 +@Article{VF2Plus,
52.93 + author = "Carletti Vincenzo and Pasquale Foggia and Mario Vento",
52.94 + title = "VF2 Plus: An Improved Version of VF2 For Biological Graphs",
52.95 + journal = "Conference: Graph-Based Representations in Pattern Recognition, At Beijing ",
52.96 + year = 2015,
52.97 + month = "May"
52.98 +}
52.99 +
52.100 +@Article{RI,
52.101 + author = "Vincenzo Bonnici and Rosalba Giugno and Alfredo Pulvirenti and Dennis Shasha and Alfredo Ferro",
52.102 + title = "A subgraph isomorphism algorithm and its application to biochemical data",
52.103 + journal = "BMC Bioinformatics. 2013; 14(Suppl 7): S13. ",
52.104 + year = 2013,
52.105 + month = "April"
52.106 +}
52.107 +
52.108 +@Article{Lad,
52.109 + author = "Brendan D. McKay",
52.110 + title = "AllDifferent-based filtering for subgraph isomorphism",
52.111 + journal = "Artificial Intelligence 174,850-864",
52.112 + year = 2010,
52.113 + month = ""
52.114 +}
52.115 +@Article{Nauty,
52.116 + author = "Christine Solnon",
52.117 + title = "Practical Graph Isomorphism",
52.118 + journal = "Congressus Numerantium, vol.30, pp. 45-87",
52.119 + year = 1981,
52.120 + month = ""
52.121 +}
52.122 +
52.123 +@Article{JianzhuangYongFaceIdentification,
52.124 + author = "Jianzhuang Liu and Yong Tsui Lee",
52.125 + title = "A Graph-Based Method for Face Identification from a Single 2D Line Drawing",
52.126 + journal = "IEEE Transactions on Pattern Analysis and Machine Intelligence - Graph Algorithms and Computer Vision archive Volume 23 Issue 10",
52.127 + year = 2001,
52.128 + month = "October"
52.129 +}
52.130 +
52.131 +@Article{HorstBunkeApplications,
52.132 + author = "Horst Bunke",
52.133 + title = "Graph Matching: Theoretical Foundations, Algorithms, and Applications",
52.134 + journal = "In International Conference on Vision Interface, pp. 82-84",
52.135 + year = 2000,
52.136 + month = "May"
52.137 +}
52.138 +
52.139 +@Article{AlexandruApplications,
52.140 + author = "Alexandru T. Balaban",
52.141 + title = "Applications of graph theory in chemistry",
52.142 + journal = "J. Chem. Inf. Comput. Sci., 1985,25(3),pp 334-343",
52.143 + year = 1985,
52.144 + month = "March"
52.145 +}
52.146 +
52.147 +@Article{PlanarGraphIso,
52.148 + author = "J. E. Hopcroft and J. K. Wong",
52.149 + title = "Linear time algorithm for isomorphism of planar graphs",
52.150 + journal = "Proceeding STOC '74 Proceedings of the sixth annual ACM symposium on Theory of computing, Pages 172-184 ",
52.151 + year = 1974,
52.152 + month = "April"
52.153 +}
52.154 +
52.155 +@Article{BondedDegGraphIso,
52.156 + author = "Eugene M. Luks",
52.157 + title = "Isomorphism of Graphs of Bounded Valence Can Be Tested in Polynomial Time",
52.158 + journal = "Journal of Computer and System Sciences, Volume 25, Issue 1, Pages 42-65",
52.159 + year = 1982,
52.160 + month = "August"
52.161 +}
52.162 +
52.163 +@Article{IntervalGraphIso,
52.164 + author = "George S. Lue and Kellogg S. Booth",
52.165 + title = "A Linear Time Algorithm for Deciding Interval Graph Isomorphism",
52.166 + journal = "Journal of the ACM (JACM), Volume 26, Issue 2, Pages 183-195 ",
52.167 + year = 1979,
52.168 + month = "April"
52.169 +}
52.170 +
52.171 +@Article{PermGraphIso,
52.172 + author = "Charles J. Colbourn",
52.173 + title = "On testing isomorphism of permutation graphs",
52.174 + journal = "Networks, Volume 11, Issue 1, Pages 13-21 ",
52.175 + year = 1981,
52.176 + month = "March"
52.177 +}
53.1 --- a/damecco.tex Fri Nov 18 16:45:03 2016 +0100
53.2 +++ b/damecco.tex Tue Nov 22 08:15:16 2016 +0100
53.3 @@ -47,6 +47,34 @@
53.4 %% for the whole article with \linenumbers.
53.5 %% \usepackage{lineno}
53.6
53.7 +\usepackage{amsmath}
53.8 +%% \usepackage[pdftex]{graphicx}
53.9 +
53.10 +\usepackage{pgfplots}
53.11 +\pgfplotsset{width=9cm}
53.12 +\pgfplotsset{compat=1.8}
53.13 +
53.14 +\usepackage{caption}
53.15 +\usepackage{subcaption}
53.16 +
53.17 +\usepackage{algorithm}
53.18 +\usepackage{algpseudocode}
53.19 +\usepackage{tikz}
53.20 +
53.21 +\usepackage{amsthm,amssymb}
53.22 +\renewcommand{\qedsymbol}{\rule{0.7em}{0.7em}}
53.23 +
53.24 +\newtheorem{theorem}{Theorem}[subsection]
53.25 +\newtheorem{corollary}{Corollary}[theorem]
53.26 +\newtheorem{claim}[theorem]{Claim}
53.27 +
53.28 +\newtheorem{definition}{Definition}[subsection]
53.29 +\newtheorem{notation}{Notation}[subsection]
53.30 +\newtheorem{example}{Example}[subsection]
53.31 +\usetikzlibrary{decorations.markings}
53.32 +\let\oldproofname=\proofname
53.33 +%% \renewcommand{\proofname}{\rm\bf{Proof:}}
53.34 +
53.35 \journal{Discrete Applied Mathematics}
53.36
53.37 \begin{document}
53.38 @@ -147,8 +175,1201 @@
53.39 %% \linenumbers
53.40
53.41 %% main text
53.42 -\section{}
53.43 -\label{}
53.44 +\section{Introduction}
53.45 +\label{sec:intro}
53.46 +
53.47 +In the last decades, combinatorial structures, and especially graphs have been considered with ever increasing interest, and applied to the solution of several new and revised questions.
53.48 +The expressiveness, the simplicity and the studiedness of graphs make them practical for modelling and appear constantly in several seemingly independent fields.
53.49 +Bioinformatics and chemistry are amongst the most relevant and most important fields.
53.50 +
53.51 +Complex biological systems arise from the interaction and cooperation of plenty of molecular components. Getting acquainted with such systems at the molecular level has primary importance, since protein-protein interaction, DNA-protein interaction, metabolic interaction, transcription factor binding, neuronal networks, and hormone signaling networks can be understood only this way.
53.52 +
53.53 +For instance, a molecular structure can be considered as a graph, whose nodes correspond to atoms and whose edges to chemical bonds. The secondary structure of a protein can also be represented as a graph, where nodes are associated with aminoacids and the edges with hydrogen bonds. The nodes are often whole molecular components and the edges represent some relationships among them.
53.54 +The similarity and dissimilarity of objects corresponding to nodes are incorporated to the model by \emph{node labels}.
53.55 +Many other chemical and biological structures can easily be modeled in a similar way. Understanding such networks basically requires finding specific subgraphs, which can not avoid the application of graph matching algorithms.
53.56 +
53.57 +Finally, let some of the other real-world fields related to some variants of graph matching be briefly mentioned: pattern recognition and machine vision \cite{HorstBunkeApplications}, symbol recognition \cite{CordellaVentoSymbolRecognition}, face identification \cite{JianzhuangYongFaceIdentification}.
53.58 +\\
53.59 +
53.60 +Subgraph and induced subgraph matching problems are known to be NP-Complete\cite{SubgraphNPC}, while the graph isomorphism problem is one of the few problems in NP neither known to be in P nor NP-Complete. Although polynomial time isomorphism algorithms are known for various graph classes, like trees and planar graphs\cite{PlanarGraphIso}, bounded valence graphs\cite{BondedDegGraphIso}, interval graphs\cite{IntervalGraphIso} or permutation graphs\cite{PermGraphIso}.
53.61 +
53.62 +In the following, some algorithms based on other approaches are summarized, which do not need any restrictions on the graphs. However, an overall polynomial behaviour is not expectable from such an alternative, it may often have good performance, even on a graph class for which polynomial algorithm is known. Note that this summary containing only exact matching algorithms is far not complete, neither does it cover all the recent algorithms.
53.63 +
53.64 +The first practically usable approach was due to \textbf{Ullmann}\cite{Ullmann} which is a commonly used depth-first search based algorithm with a complex heuristic for reducing the number of visited states. A major problem is its $\Theta(n^3)$ space complexity, which makes it impractical in the case of big sparse graphs.
53.65 +
53.66 +In a recent paper, \textbf{Ullmann}\cite{UllmannBit} presents an improved version of this algorithm based on a bit-vector solution for the binary Constraint Satisfaction Problem.
53.67 +
53.68 +The \textbf{Nauty} algorithm\cite{Nauty} transforms the two graphs to a canonical form before starting to check for the isomorphism. It has been considered as one of the fastest graph isomorphism algorithms, although graph categories were shown in which it takes exponentially many steps. This algorithm handles only the graph isomorphism problem.
53.69 +
53.70 +The \textbf{LAD} algorithm\cite{Lad} uses a depth-first search strategy and formulates the matching as a Constraint Satisfaction Problem to prune the search tree. The constraints are that the mapping has to be injective and edge-preserving, hence it is possible to handle new matching types as well.
53.71 +
53.72 +The \textbf{RI} algorithm\cite{RI} and its variations are based on a state space representation. After reordering the nodes of the graphs, it uses some fast executable heuristic checks without using any complex pruning rules. It seems to run really efficiently on graphs coming from biology, and won the International Contest on Pattern Search in Biological Databases\cite{Content}.
53.73 +
53.74 +The currently most commonly used algorithm is the \textbf{VF2}\cite{VF2}, the improved version of VF\cite{VF}, which was designed for solving pattern matching and computer vision problems, and has been one of the best overall algorithms for more than a decade. Although, it can't be up to new specialized algorithms, it is still widely used due to its simplicity and space efficiency. VF2 uses a state space representation and checks some conditions in each state to prune the search tree.
53.75 +
53.76 +Our first graph matching algorithm was the first version of VF2 which recognizes the significance of the node ordering, more opportunities to increase the cutting efficiency and reduce its computational complexity. This project was initiated and sponsored by QuantumBio Inc.\cite{QUANTUMBIO} and the implementation --- along with a source code --- has been published as a part of LEMON\cite{LEMON} open source graph library.
53.77 +
53.78 +This thesis introduces \textbf{VF2++}, a new further improved algorithm for the graph and (induced)subgraph isomorphism problem, which uses efficient cutting rules and determines a node order in which VF2 runs significantly faster on practical inputs.
53.79 +
53.80 +Meanwhile, another variant called \textbf{VF2 Plus}\cite{VF2Plus} has been published. It is considered to be as efficient as the RI algorithm and has a strictly better behavior on large graphs. The main idea of VF2 Plus is to precompute a heuristic node order of the small graph, in which the VF2 works more efficiently.\newline
53.81 +\newline
53.82 +
53.83 +\section{Problem Statement}
53.84 +This section provides a detailed description of the problems to be solved.
53.85 +\subsection{Definitions}
53.86 +
53.87 +Throughout the paper $G_{small}=(V_{small}, E_{small})$ and $G_{large}=(V_{large}, E_{large})$ denote two undirected graphs.
53.88 +\begin{definition}\label{sec:ismorphic}
53.89 +$G_{small}$ and $G_{large}$ are \textbf{isomorphic} if $\exists M: V_{small} \longrightarrow V_{large}$ bijection, for which the following is true:
53.90 +\begin{center}
53.91 +$\forall u,v\in{V_{small}} : (u,v)\in{E_{small}} \Leftrightarrow (M(u),M(v))\in{E_{large}}$
53.92 +\end{center}
53.93 +\end{definition}
53.94 +For the sake of simplicity in this paper subgraphs and induced subgraphs are defined in a more general way than usual:
53.95 +\begin{definition}
53.96 +$G_{small}$ is a \textbf{subgraph} of $G_{large}$ if $\exists I: V_{small}\longrightarrow V_{large}$ injection, for which the following is true:
53.97 +\begin{center}
53.98 +$\forall u,v \in{V_{small}} : (u,v)\in{E_{small}} \Rightarrow (I(u),I(v))\in E_{large}$
53.99 +\end{center}
53.100 +\end{definition}
53.101 +
53.102 +\begin{definition}
53.103 +$G_{small}$ is an \textbf{induced subgraph} of $G_{large}$ if $\exists I: V_{small}\longrightarrow V_{large}$ injection, for which the following is true:
53.104 +\begin{center}
53.105 +$\forall u,v \in{V_{small}} : (u,v)\in{E_{small}} \Leftrightarrow (I(u),I(v))\in E_{large}$
53.106 +\end{center}
53.107 +\end{definition}
53.108 +
53.109 +\begin{definition}
53.110 +$lab: (V_{small}\cup V_{large}) \longrightarrow K$ is a \textbf{node label function}, where K is an arbitrary set. The elements in K are the \textbf{node labels}. Two nodes, u and v are said to be \textbf{equivalent}, if $lab(u)=lab(v)$.
53.111 +\end{definition}
53.112 +
53.113 +When node labels are also given, the matched nodes must have the same labels.
53.114 +For example, the node labeled isomorphism is phrased by
53.115 +\begin{definition}
53.116 +$G_{small}$ and $G_{large}$ are \textbf{isomorphic by the node label function lab} if $\exists M: V_{small} \longrightarrow V_{large}$ bijection, for which the following is true:
53.117 +\begin{center}
53.118 +$(\forall u,v\in{V_{small}} : (u,v)\in{E_{small}} \Leftrightarrow (M(u),M(v))\in{E_{large}})$
53.119 + and $(\forall u\in{V_{small}} : lab(u)=lab(M(u)))$
53.120 +\end{center}
53.121 +\end{definition}
53.122 +
53.123 +The other two definitions can be extended in the same way.
53.124 +
53.125 +Note that edge label function can be defined similarly to node label function, and all the definitions can be extended with additional conditions, but it is out of the scope of this work.
53.126 +
53.127 +The equivalence of two nodes is usually defined by another relation, $\\R\subseteq (V_{small}\cup V_{large})^2$. This overlaps with the definition given above if R is an equivalence relation, which does not mean restriction in biological and chemical applications.
53.128 +
53.129 +\subsection{Common problems}\label{sec:CommProb}
53.130 +
53.131 +The focus of this paper is on two extensively studied topics, the subgraph isomorphism and its variations. However, the following problems also appear in many applications.
53.132 +
53.133 +The \textbf{subgraph matching problem} is the following: is $G_{small}$ isomorphic to any subgraph of $G_{large}$ by a given node label?
53.134 +
53.135 +The \textbf{induced subgraph matching problem} asks the same about the existence of an induced subgraph.
53.136 +
53.137 +The \textbf{graph isomorphism problem} can be defined as induced subgraph matching problem where the sizes of the two graphs are equal.
53.138 +
53.139 +In addition to existence, it may be needed to show such a subgraph, or it may be necessary to list all of them.
53.140 +
53.141 +It should be noted that some authors misleadingly refer to the term \emph{subgraph isomorphism problem} as an \emph{induced subgraph isomorphism problem}.
53.142 +
53.143 +The following sections give the descriptions of VF2, VF2++, VF2 Plus and a particular comparison.
53.144 +
53.145 +\section{The VF2 Algorithm}
53.146 +This algorithm is the basis of both the VF2++ and the VF2 Plus.
53.147 +VF2 is able to handle all the variations mentioned in \textbf{Section \ref{sec:CommProb})}.
53.148 +Although it can also handle directed graphs, for the sake of simplicity, only the undirected case will be discussed.
53.149 +
53.150 +
53.151 +\subsection{Common notations}
53.152 +\indent
53.153 +Assume $G_{small}$ is searched in $G_{large}$.
53.154 +The following definitions and notations will be used throughout the whole paper.
53.155 +\begin{definition}
53.156 +A set $M\subseteq V_{small}\times V_{large}$ is called \textbf{mapping}, if no node of $V_{small}$ or of $V_{large}$ appears in more than one pair in M.
53.157 +That is, M uniquely associates some of the nodes in $V_{small}$ with some nodes of $V_{large}$ and vice versa.
53.158 +\end{definition}
53.159 +
53.160 +\begin{definition}
53.161 +Mapping M \textbf{covers} a node v, if there exists a pair in M, which contains v.
53.162 +\end{definition}
53.163 +
53.164 +\begin{definition}
53.165 +A mapping $M$ is $\mathbf{whole\ mapping}$, if $M$ covers all the nodes in $V_{small}$.
53.166 +\end{definition}
53.167 +
53.168 +\begin{notation}
53.169 +Let $\mathbf{M_{small}(s)} := \{u\in V_{small} : \exists v\in V_{large}: (u,v)\in M(s)\}$ and $\mathbf{M_{large}(s)} := \{v\in V_{large} : \exists u\in V_{small}: (u,v)\in M(s)\}$.
53.170 +\end{notation}
53.171 +
53.172 +\begin{notation}
53.173 +Let $\mathbf{Pair(M,v)}$ be the pair of $v$ in $M$, if such a node exist, otherwise $\mathbf{Pair(M,v)}$ is undefined. For a mapping $M$ and $v\in V_{small}\cup V_{large}$.
53.174 +\end{notation}
53.175 +
53.176 +Note that if $\mathbf{Pair(M,v)}$ exists, then it is unique
53.177 +
53.178 +The definitions of the isomorphism types can be rephrased on the existence of a special whole mapping $M$, since it represents a bijection. For example
53.179 +\begin{center}
53.180 +$M\subseteq V_{small}\times V_{large}$ represents an induced subgraph isomorphism $\Leftrightarrow$ $M$ is whole mapping and $\forall u,v \in{V_{small}} : (u,v)\in{E_{small}} \Leftrightarrow (Pair(M,u),Pair(M,v))\in E_{large}$.
53.181 +\end{center}
53.182 +
53.183 +\begin{definition}
53.184 +A set of whole mappings is called \textbf{problem type}.
53.185 +\end{definition}
53.186 +Throughout the paper, $\mathbf{PT}$ denotes a generic problem type which can be substituted by any problem type.
53.187 +
53.188 +A whole mapping $W\mathbf{\ is\ of\ type\ PT}$, if $W\in PT$. Using this notations, VF2 searches a whole mapping $W$ of type $PT$.
53.189 +
53.190 +For example the problem type of graph isomorphism problem is the following.
53.191 +A whole mapping $W$ is in $\mathbf{ISO}$, iff the bijection represented by $W$ satisfies \textbf{Definition \ref{sec:ismorphic})}.
53.192 +The subgraph- and induced subgraph matching problems can be formalized in a similar way. Let their problem types be denoted as $\mathbf{SUB}$ and $\mathbf{IND}$.
53.193 +
53.194 +\begin{definition}
53.195 +\label{expPT}
53.196 +$PT$ is an \textbf{expanding problem type} if $\ \forall\ W\in PT:\ \forall u_1,u_2\in V_{small}:\ (u_1,u_2)\in E_{small}\Rightarrow (Pair(W,u_1),Pair(W,u_2))\in E_{large}$, that is each edge of $G_{small}$ has to be mapped to an edge of $G_{large}$ for each mapping in $PT$.
53.197 +\end{definition}
53.198 +
53.199 +Note that $ISO$, $SUB$ and $IND$ are expanding problem types.
53.200 +
53.201 +This paper deals with the three problem types mentioned above only, but
53.202 +the following generic definitions make it possible to handle other types as well.
53.203 +Although it may be challenging to find a proper consistency function and an efficient
53.204 +cutting function.
53.205 +
53.206 +\begin{definition}
53.207 +Let M be a mapping. A logical function $\mathbf{Cons_{PT}}$ is a \textbf{consistency function by } $\mathbf{PT}$, if the following holds. If there exists whole mapping $W$ of type PT for which $M\subseteq W$, then $Cons_{PT}(M)$ is true.
53.208 +\end{definition}
53.209 +
53.210 +\begin{definition}
53.211 +Let M be a mapping. A logical function $\mathbf{Cut_{PT}}$ is a \textbf{cutting function by } $\mathbf{PT}$, if the following holds. $\mathbf{Cut_{PT}(M)}$ is false if $M$ can be extended to a whole mapping W of type PT.
53.212 +\end{definition}
53.213 +
53.214 +\begin{definition}
53.215 +$M$ is said to be \textbf{consistent mapping by} $\mathbf{PT}$, if $Cons_{PT}(M)$ is true.
53.216 +\end{definition}
53.217 +
53.218 +$Cons_{PT}$ and $Cut_{PT}$ will often be used in the following form.
53.219 +\begin{notation}
53.220 +Let $\mathbf{Cons_{PT}(p, M)}:=Cons_{PT}(M\cup\{p\})$ and $\mathbf{Cut_{PT}(p, M)}:=Cut_{PT}(M\cup\{p\})$, where $p\in{V_{small}\!\times\!V_{large}}$ and $M\cup\{p\}$ is mapping.
53.221 +\end{notation}
53.222 +
53.223 +$Cons_{PT}$ will be used to check the consistency of the already covered nodes, while $Cut_{PT}$ is for looking ahead to recognize if no whole consistent mapping can contain the current mapping.
53.224 +
53.225 +\subsection{Overview of the algorithm}
53.226 +VF2 uses a state space representation of mappings, $Cons_{PT}$ for excluding inconsistency with the problem type and $Cut_{PT}$ for pruning the search tree.
53.227 +Each state $s$ of the matching process can be associated with a mapping $M(s)$.
53.228 +
53.229 +\textbf{Algorithm~\ref{alg:VF2Pseu})} is a high level description of the VF2 matching algorithm.
53.230 +
53.231 +
53.232 +\begin{algorithm}
53.233 +\algtext*{EndIf}%ne nyomtasson end if-et
53.234 +\algtext*{EndFor}%ne nyomtasson ..
53.235 +\algtext*{EndProcedure}%ne nyomtasson ..
53.236 +\caption{\hspace{0.5cm}$A\ high\ level\ description\ of\ VF2$}\label{alg:VF2Pseu}
53.237 +\begin{algorithmic}[1]
53.238 +
53.239 +\Procedure{VF2}{State $s$, ProblemType $PT$}
53.240 + \If{$M(s$) covers $V_{small}$}
53.241 + \State Output($M(s)$)
53.242 + \Else
53.243 +
53.244 + \State Compute the set $P(s)$ of the pairs candidate for inclusion in $M(s)$
53.245 + \ForAll{$p\in{P(s)}$}
53.246 + \If{Cons$_{PT}$($p, M(s)$) $\wedge$ $\neg$Cut$_{PT}$($p, M(s)$)}
53.247 + \State Compute the nascent state $\tilde{s}$ by adding $p$ to $M(s)$
53.248 + \State \textbf{call} VF2($\tilde{s}$, $PT$)
53.249 + \EndIf
53.250 + \EndFor
53.251 + \EndIf
53.252 +\EndProcedure
53.253 +\end{algorithmic}
53.254 +\end{algorithm}
53.255 +
53.256 +
53.257 +The initial state $s_0$ is associated with $M(s_0)=\emptyset$, i.e. it starts with an empty mapping.
53.258 +
53.259 +For each state $s$, the algorithm computes $P(s)$, the set of candidate node pairs for adding to the current state $s$.
53.260 +
53.261 +For each pair $p$ in $P(s)$, $Cons_{PT}(p,M(s))$ and $Cut_{PT}(p,M(s))$ are evaluated. If $Cons_{PT}(p,M(s))$ is true and $Cut_{PT}(p,M(s))$ is false, the successor state $\tilde{s}=s\cup \{p\}$ is computed, and the whole process is recursively applied to $\tilde{s}$. Otherwise, $\tilde{s}$ is not consistent by $PT$ or it can be proved that $s$ can not be extended to a whole mapping.
53.262 +
53.263 +In order to make sure of the correctness see
53.264 +\begin{claim}
53.265 +Through consistent mappings, only consistent whole mappings can be reached, and all of the whole mappings are reachable through consistent mappings.
53.266 +\end{claim}
53.267 +
53.268 +Note that a state may be reached in many different ways, since the order of insertions into M does not influence the nascent mapping. In fact, the number of different ways which lead to the same state can be exponentially large. If $G_{small}$ and $G_{large}$ are circles with n nodes and n different node labels, there exists exactly one graph isomorphism between them, but it will be reached in $n!$ different ways.
53.269 +
53.270 +However, one may observe
53.271 +
53.272 +\begin{claim}
53.273 +\label{claim:claimTotOrd}
53.274 +Let $\prec$ an arbitrary total ordering relation on $V_{small}$.
53.275 +If the algorithm ignores each $p=(u,v) \in P(s)$, for which
53.276 +\begin{center}
53.277 +$\exists (\hat{u},\hat{v})\in P(s): \hat{u} \prec u$,
53.278 +\end{center}
53.279 +then no state can be reached more than ones and each state associated with a whole mapping remains reachable.
53.280 +\end{claim}
53.281 +
53.282 +Note that the cornerstone of the improvements to VF2 is a proper choice of a total ordering.
53.283 +
53.284 +\subsection{The candidate set P(s)}
53.285 +\label{candidateComputingVF2}
53.286 +$P(s)$ is the set of the candidate pairs for inclusion in $M(s)$.
53.287 +Suppose that $PT$ is an expanding problem type, see \textbf{Definition~\ref{expPT})}.
53.288 +
53.289 +\begin{notation}
53.290 +Let $\mathbf{T_{small}(s)}:=\{u \in V_{small} : u$ is not covered by $M(s)\wedge\exists \tilde{u}\in{V_{small}: (u,\tilde{u})\in E_{small}} \wedge \tilde{u}$ is covered by $M(s)\}$, and \\ $\mathbf{T_{large}(s)}\!:=\!\{v \in\!V_{large}\!:\!v$ is not covered by $M(s)\wedge\!\exists\tilde{v}\!\in\!{V_{large}\!:\!(v,\tilde{v})\in\!E_{large}} \wedge \tilde{v}$ is covered by $M(s)\}$
53.291 +\end{notation}
53.292 +
53.293 +The set $P(s)$ includes the pairs of uncovered neighbours of covered nodes and if there is not such a node pair, all the pairs containing two uncovered nodes are added. Formally, let
53.294 +\[
53.295 + P(s)\!=\!
53.296 + \begin{cases}
53.297 + T_{small}(s)\times T_{large}(s)&\hspace{-0.15cm}\text{if } T_{small}(s)\!\neq\!\emptyset\!\wedge\!T_{large}(s)\!\neq \emptyset,\\
53.298 + (V_{small}\!\setminus\!M_{small}(s))\!\times\!(V_{large}\!\setminus\!M_{large}(s)) &\hspace{-0.15cm}otherwise.
53.299 + \end{cases}
53.300 +\]
53.301 +
53.302 +\subsection{Consistency}
53.303 +This section defines the consistency functions for the different problem types mentioned in \textbf{Section \ref{sec:CommProb})}.
53.304 +\begin{notation}
53.305 +Let $\mathbf{\Gamma_{small} (u)}:=\{\tilde{u}\in V_{small} : (u,\tilde{u})\in E_{small}\}$\\
53.306 +Let $\mathbf{\Gamma_{large} (v)}:=\{\tilde{v}\in V_{large} : (v,\tilde{v})\in E_{large}\}$
53.307 +\end{notation}
53.308 +Suppose $p=(u,v)$, where $u\in V_{small}$ and $v\in V_{large}$,
53.309 +$s$ is a state of the matching procedure,
53.310 +$M(s)$ is consistent mapping by $PT$ and $lab(u)=lab(v)$.
53.311 +$Cons_{PT}(p,M(s))$ checks whether including pair $p$ into $M(s)$ leads to a consistent mapping by $PT$.
53.312 +
53.313 +\subsubsection{Induced subgraph isomorphism}
53.314 +$M(s)\cup \{(u,v)\}$ is a consistent mapping by $IND$ $\Leftrightarrow (\forall \tilde{u}\in M_{small}: (u,\tilde{u})\in E_{small} \Leftrightarrow (v,Pair(M(s),\tilde{u}))\in E_{large})$.\newline
53.315 +The following formulation gives an efficient way of calculating $Cons_{IND}$.
53.316 +\begin{claim}
53.317 +$Cons_{IND}((u,v),M(s)):=(\forall \tilde{v}\in \Gamma_{large}(v) \ \cap\ M_{large}(s):\\(Pair(M(s),\tilde{v}),u)\in E_{small})\wedge
53.318 +(\forall \tilde{u}\in \Gamma_{small}(u) \ \cap\ M_{small}(s):(v,Pair(M(s),\tilde{u}))\in E_{large})$ is a consistency function in the case of $IND$.
53.319 +\end{claim}
53.320 +
53.321 +\subsubsection{Graph isomorphism}
53.322 +$M(s)\cup \{(u,v)\}$ is a consistent mapping by $ISO$ $\Leftrightarrow$ $M(s)\cup \{(u,v)\}$ is a consistent mapping by $IND$.
53.323 +\begin{claim}
53.324 +$Cons_{ISO}((u,v),M(s))$ is a consistency function by $ISO$ if and only if it is a consistency function by $IND$.
53.325 +\end{claim}
53.326 +\subsubsection{Subgraph isomorphism}
53.327 +$M(s)\cup \{(u,v)\}$ is a consistent mapping by $SUB$ $\Leftrightarrow (\forall \tilde{u}\in M_{small}:\\(u,\tilde{u})\in E_{small} \Rightarrow (v,Pair(M(s),\tilde{u}))\in E_{large})$.
53.328 +\newline
53.329 +The following formulation gives an efficient way of calculating $Cons_{SUB}$.
53.330 +\begin{claim}
53.331 +$Cons_{SUB}((u,v),M(s)):=
53.332 +(\forall \tilde{u}\in \Gamma_{small}(u) \ \cap\ M_{small}(s):\\(v,Pair(M(s),\tilde{u}))\in E_{large})$ is a consistency function by $SUB$.
53.333 +\end{claim}
53.334 +
53.335 +\subsection{Cutting rules}
53.336 +$Cut_{PT}(p,M(s))$ is defined by a collection of efficiently verifiable conditions. The requirement is that $Cut_{PT}(p,M(s))$ can be true only if it is impossible to extended $M(s)\cup \{p\}$ to a whole mapping.
53.337 +\begin{notation}
53.338 +
53.339 +Let $\mathbf{\tilde{T}_{small}}(s):=(V_{small}\backslash M_{small}(s))\backslash T_{small}(s)$, and \\ $\mathbf{\tilde{T}_{large}}(s):=(V_{large}\backslash M_{large}(s))\backslash T_{large}(s)$.
53.340 +\end{notation}
53.341 +\subsubsection{Induced subgraph isomorphism}
53.342 +\begin{claim}
53.343 +$Cut_{IND}((u,v),M(s)):= |\Gamma_{large} (v)\ \cap\ T_{large}(s)| < |\Gamma_{small} (u)\ \cap\ T_{small}(s)| \vee |\Gamma_{large}(v)\cap \tilde{T}_{large}(s)| < |\Gamma_{small}(u)\cap \tilde{T}_{small}(s)|$ is a cutting function by $IND$.
53.344 +\end{claim}
53.345 +\subsubsection{Graph isomorphism}
53.346 +Note that the cutting function of induced subgraph isomorphism defined above is a cutting function by $ISO$, too, however it is less efficient than the following while their computational complexity is the same.
53.347 +\begin{claim}
53.348 +$Cut_{ISO}((u,v),M(s)):= |\Gamma_{large} (v)\ \cap\ T_{large}(s)| \neq |\Gamma_{small} (u)\ \cap\ T_{small}(s)| \vee |\Gamma_{large}(v)\cap \tilde{T}_{large}(s)| \neq |\Gamma_{small}(u)\cap \tilde{T}_{small}(s)|$ is a cutting function by $ISO$.
53.349 +\end{claim}
53.350 +
53.351 +\subsubsection{Subgraph isomorphism}
53.352 +\begin{claim}
53.353 +$Cut_{SUB}((u,v),M(s)):= |\Gamma_{large} (v)\ \cap\ T_{large}(s)| < |\Gamma_{small} (u)\ \cap\ T_{small}(s)|$ is a cutting function by $SUB$.
53.354 +\end{claim}
53.355 +Note that there is a significant difference between induced and non-induced subgraph isomorphism:
53.356 +
53.357 +\begin{claim}
53.358 +\label{claimSUB}
53.359 +$Cut_{SUB}'((u,v),M(s)):= |\Gamma_{large} (v)\ \cap\ T_{large}(s)| < |\Gamma_{small} (u)\ \cap\ T_{small}(s)| \vee |\Gamma_{large}(v)\cap \tilde{T}_{large}(s)| < |\Gamma_{small}(u)\cap \tilde{T}_{small}(s)|$ is \textbf{not} a cutting function by $SUB$.
53.360 +\end{claim}
53.361 +\begin{proof}$ $\\
53.362 +\vspace*{-0.5cm}
53.363 +
53.364 +\begin{figure}
53.365 +\begin{center}
53.366 +\begin{tikzpicture}
53.367 + [scale=.8,auto=left,every node/.style={circle,fill=black!15}]
53.368 + \node[rectangle,fill=black!15] at (4,6) {$G_{small}$};
53.369 + \node (u4) at (2.5,10) {$u_4$};
53.370 + \node (u3) at (5.5,10) {$u_3$};
53.371 + \node (u1) at (2.5,7) {$u_1$};
53.372 + \node (u2) at (5.5,7) {$u_2$};
53.373 +
53.374 + \node[rectangle,fill=black!30] at (13.5,6) {$G_{large}$};
53.375 + \node[fill=black!30] (v4) at (12,10) {$v_4$};
53.376 + \node[fill=black!30] (v3) at (15,10) {$v_3$};
53.377 + \node[fill=black!30] (v1) at (12,7) {$v_1$};
53.378 + \node[fill=black!30] (v2) at (15,7) {$v_2$};
53.379 +
53.380 +
53.381 + \foreach \from/\to in {u1/u2,u2/u3,u3/u4,u4/u1}
53.382 + \draw (\from) -- (\to);
53.383 + \foreach \from/\to in {v1/v2,v2/v3,v3/v4,v4/v1,v1/v3}
53.384 + \draw (\from) -- (\to);
53.385 +% \draw[dashed] (\from) -- (\to);
53.386 +\end{tikzpicture}
53.387 +\caption{Graphs for the proof of \textbf{Claim \ref{claimSUB}}} \label{fig:proofSUB}
53.388 +\end{center}
53.389 +\end{figure}
53.390 +Let the two graphs of \textbf{Figure \ref{fig:proofSUB})} be the input graphs.
53.391 +Suppose the total ordering relation is $u_1 \prec u_2 \prec u_3 \prec u_4$,$M(s)\!=\{(u_1,v_1)\}$, and VF2 tries to add $(u_2,v_2)\in P(s)$.\newline
53.392 +$Cons_{SUB}((u_2,v_2),M(s))=true$, so $M\cup \{(u_2,v_2)\}$ is consistent by $SUB$. The cutting function $Cut_{SUB}((u_2,v_2),M(s))$ is false, so it does not let cut the tree.\newline
53.393 +On the other hand $Cut_{SUB}'((u_2,v_2),M(s))$ is true, since\\$0=|\Gamma_{large}(v_2)\cap \tilde{T}_{large}(s)|<|\Gamma_{small}(u_2)\cap \tilde{T}_{small}(s)|=1$ is true, but still the tree can not be pruned, because otherwise the $\{(u_1,v_1)(u_2,v_2)(u_3,v_3)(u_4,v_4)\}$ mapping can not be found.
53.394 +\end{proof}
53.395 +
53.396 +\newpage
53.397 +\section{The VF2++ Algorithm}
53.398 +Although any total ordering relation makes the search space of VF2 a tree, its
53.399 +choice turns out to dramatically influence the number of visited states. The goal is to determine an efficient one as quickly as possible.
53.400 +
53.401 +The main reason for VF2++' superiority over VF2 is twofold. Firstly, taking into account the structure and the node labeling of the graph, VF2++ determines a state order in which most of the unfruitful branches of the search space can be pruned immediately. Secondly, introducing more efficient --- nevertheless still easier to compute --- cutting rules reduces the chance of going astray even further.
53.402 +
53.403 +In addition to the usual subgraph isomorphism, specialized versions for induced subgraph isomorphism and for graph isomorphism have been designed. VF2++ has gained a runtime improvement of one order of magnitude respecting induced subgraph isomorphism and a better asymptotical behaviour in the case of graph isomorphism problem.
53.404 +
53.405 +Note that a weaker version of the cutting rules and the more efficient candidate
53.406 +set calculating were described in \cite{VF2Plus}, too.
53.407 +
53.408 +It should be noted that all the methods described in this section are extendable to handle directed graphs and edge labels as well.
53.409 +
53.410 +The basic ideas and the detailed description of VF2++ are provided in the following.
53.411 +
53.412 +\subsection{Preparations}
53.413 +\begin{claim}
53.414 +\label{claim:claimCoverFromLeft}
53.415 +The total ordering relation uniquely determines a node order, in which the nodes of $V_{small}$ will be covered by VF2. From the point of view of the matching procedure, this means, that always the same node of $G_{small}$ will be covered on the d-th level.
53.416 +\end{claim}
53.417 +\begin{proof}
53.418 +In order to make the search space a tree, the pairs in $\{(u,v)\in P(s) : \exists \hat{u} : \hat{u}\prec u\}$ are excluded from $P(s)$.
53.419 +\newline
53.420 +Let $\tilde{P}(s):=P(s)\backslash \{(u,v)\in P(s) : \exists \hat{u} : \hat{u}\prec u\}$
53.421 +\newline
53.422 +The relation $\prec$ is a total ordering, so $\exists!\ \tilde{u} : \forall\ (u,v)\in \tilde{P}(s): u=\tilde u$. Since a pair form $\tilde{P}(s)$ is chosen for including into $M(s)$, it is obvious, that only $\tilde{u}$ can be covered in $V_{small}$. Actually, $\tilde{u}$ is the smallest element in $T_{small}(s)$ (or in $V_{small}\backslash M_{small}(s)$, if $T_{small}(s)$ were empty), and $T_{small}(s)$ depends only on the covered nodes of $G_{small}$.
53.423 +\newline
53.424 +Simple induction on $d$ shows that the set of covered nodes of $G_{small}$ is unique, if $d$ is given, so $\tilde{u}$ is unique if $d$ is given.
53.425 +\end{proof}
53.426 +
53.427 +\begin{definition}
53.428 +An order $(u_{\sigma(1)},u_{\sigma(2)},..,u_{\sigma(|V_{small}|)})$ of $V_{small}$ is \textbf{matching order}, if exists $\prec$ total ordering relation, s.t. the VF2 with $\prec$ on the d-th level finds pair for $u_{\sigma(d)}$ for all $d\in\{1,..,|V_{small}|\}$.
53.429 +\end{definition}
53.430 +
53.431 +\begin{claim}\label{claim:MOclaim}
53.432 +A total ordering is matching order, iff the nodes of every component form an interval in the node sequence, and every node connects to a previous node in its component except the first node of the component. The order of the components is arbitrary.
53.433 +\\Formally spoken, an order $(u_{\sigma(1)},u_{\sigma(2)},..,u_{\sigma(|V_{small}|)})$ of $V_{small}$ is matching order $\Leftrightarrow$ $\forall G'_{small}=(V'_{small},E'_{small})\ component\ of\ G_{small}: \forall i: (\exists j : j<i\wedge u_{\sigma(j)},u_{\sigma(i)}\in V'_{small})\Rightarrow \exists k : k < i \wedge (\forall l: k\leq l\leq i \Rightarrow u_{l}\in V'_{small}) \wedge (u_{\sigma{(k)}},u_{\sigma{(i)}})\in E'_{small}$, where $i,j,k,l\in \{1,..,|V_{small}|\}$\newline
53.434 +\end{claim}
53.435 +\begin{proof}
53.436 +Suppose a matching order is given. It has to be shown that the node sequence has a structure described above.\\
53.437 +Let $G'_{small}=(V'_{small},E'_{small})$ be an arbitrary component and $i$ an arbitrary index.
53.438 +\newline
53.439 +$(\exists j : j<i\wedge u_{\sigma(j)},u_{\sigma(i)}\in V'_{small})\Rightarrow u_{\sigma(i)}$ is not the first covered node in $G_{small}\ \Rightarrow u_{\sigma(i)}$ is connected to a covered node $u_{\sigma(k)}$ where $k<i$, since $u_{\sigma(i)}\in T_{small}(s)$ for some $s$ and $T_{small}(s)\subseteq{V'_{small}}$ contains only nodes connected to at least one covered node. It's easy to see, that $\forall l: k\leq l\leq i \Rightarrow u_{l}\in V'_{small}$, since $T_{small}(s)$ contains only nodes connected to some covered ones, while it is not empty, but if it were empty, then $u_{\sigma(k)}$ and $u_{\sigma(i)}$ were not in the same component.
53.440 +
53.441 +Now, let us show that if a node sequence has the special structure described above, it has to be matching order.\\ $(u_{\sigma(1)},u_{\sigma(2)},..,u_{\sigma(|V_{small}|)})$ is a total ordering, which determines the matching order $(u_{\sigma(1)},u_{\sigma(2)},..,u_{\sigma(|V_{small}|)})$.
53.442 +\end{proof}
53.443 +
53.444 +To summing up, a total ordering always uniquely determines a matching order, and every matching order can be determined by a total ordering, however, more than one different total orderings may determine the same matching order.
53.445 +\subsection{Idea behind the algorithm}
53.446 +The goal is to find a matching order in which the algorithm is able to recognize inconsistency or prune the infeasible branches on the highest levels and goes deep only if it is needed.
53.447 +
53.448 +\begin{notation}
53.449 +Let $\mathbf{Conn_{H}(u)}:=|\Gamma_{small}(u)\cap H\}|$, that is the number of neighbours of u which are in H, where $u\in V_{small} $ and $H\subseteq V_{small}$.
53.450 +\end{notation}
53.451 +
53.452 +The principal question is the following. Suppose a state $s$ is given. For which node of $T_{small}(s)$ is the hardest to find a consistent pair in $G_{large}$? The more covered neighbours a node in $T_{small}(s)$ has --- i.e. the largest $Conn_{M_{small}(s)}$ it has ---, the more rarely satisfiable consistency constraints for its pair are given.
53.453 +
53.454 +In biology, most of the graphs are sparse, thus several nodes in $T_{small}(s)$ may have the same $Conn_{M_{small}(s)}$, which makes reasonable to define a secondary and a tertiary order between them.
53.455 +The observation above proves itself to be as determining, that the secondary ordering prefers nodes with the most uncovered neighbours among which have the same $Conn_{M_{small}(s)}$ to increase $Conn_{M_{small}(s)}$ of uncovered nodes so much, as possible.
53.456 +The tertiary ordering prefers nodes having the rarest uncovered labels.
53.457 +
53.458 +Note that the secondary ordering is the same as the ordering by $deg$, which is a static data in front of the above used.
53.459 +
53.460 +These rules can easily result in a matching order which contains the nodes of a long path successively, whose nodes may have low $Conn$ and is easily matchable into $G_{large}$. To avoid that, a BFS order is used, which provides the shortest possible paths.
53.461 +\newline
53.462 +
53.463 +In the following, some examples on which the VF2 may be slow are described, although they are easily solvable by using a proper matching order.
53.464 +
53.465 +\begin{example}
53.466 +Suppose $G_{small}$ can be mapped into $G_{large}$ in many ways without node labels. Let $u\in V_{small}$ and $v\in V_{large}$.
53.467 +\newline
53.468 +$lab(u):=black$
53.469 +\newline
53.470 +$lab(v):=black$
53.471 +\newline
53.472 +$lab(\tilde{u}):=red \ \forall \tilde{u}\in (V_{small}\backslash \{u\})$
53.473 +\newline
53.474 +$lab(\tilde{v}):=red \ \forall \tilde{v}\in (V_{large}\backslash \{v\})$
53.475 +\newline
53.476 +
53.477 +Now, any mapping by the node label $lab$ must contain $(u,v)$, since $u$ is black and no node in $V_{large}$ has a black label except $v$. If unfortunately $u$ were the last node which will get covered, VF2 would check only in the last steps, whether $u$ can be matched to $v$.
53.478 +\newline
53.479 +However, had $u$ been the first matched node, u would have been matched immediately to v, so all the mappings would have been precluded in which node labels can not correspond.
53.480 +\end{example}
53.481 +
53.482 +\begin{example}
53.483 +Suppose there is no node label given, $G_{small}$ is a small graph and can not be mapped into $G_{large}$ and $u\in V_{small}$.
53.484 +\newline
53.485 +Let $G'_{small}:=(V_{small}\cup \{u'_{1},u'_{2},..,u'_{k}\},E_{small}\cup \{(u,u'_{1}),(u'_{1},u'_{2}),..,(u'_{k-1},u'_{k})\})$, that is, $G'_{small}$ is $G_{small}\cup \{ a\ k$ long path, which is disjoint from $G_{small}$ and one of its starting points is connected to $u\in V_{small}\}$.
53.486 +\newline
53.487 +Is there a subgraph of $G_{large}$, which is isomorph with $G'_{small}$?
53.488 +\newline
53.489 +If unfortunately the nodes of the path were the first $k$ nodes in the matching order, the algorithm would iterate through all the possible k long paths in $G_{large}$, and it would recognize that no path can be extended to $G'_{small}$.
53.490 +\newline
53.491 +However, had it started by the matching of $G_{small}$, it would not have matched any nodes of the path.
53.492 +\end{example}
53.493 +
53.494 +These examples may look artificial, but the same problems also appear in real-world examples, even though in a less obvious way.
53.495 +
53.496 +\subsection{Total ordering}
53.497 +Instead of the total ordering relation, the matching order will be searched directly.
53.498 +\begin{notation}
53.499 +Let \textbf{F$_\mathcal{M}$(l)}$:=|\{v\in V_{large} : l=lab(v)\}|-|\{u\in V_{small}\backslash \mathcal{M} : l=lab(u)\}|$ , where $l$ is a label and $\mathcal{M}\subseteq V_{small}$.
53.500 +\end{notation}
53.501 +
53.502 +\begin{definition}Let $\mathbf{arg\ max}_{f}(S) :=\{u : u\in S \wedge f(u)=max_{v\in S}\{f(v)\}\}$ and $\mathbf{arg\ min}_{f}(S) := arg\ max_{-f}(S)$, where $S$ is a finite set and $f:S\longrightarrow \mathbb{R}$.
53.503 +\end{definition}
53.504 +
53.505 +\begin{algorithm}
53.506 +\algtext*{EndIf}%ne nyomtasson end if-et
53.507 +\algtext*{EndFor}%ne nyomtasson ..
53.508 +\algtext*{EndProcedure}%ne nyomtasson ..
53.509 +\algtext*{EndWhile}
53.510 +\caption{\hspace{0.5cm}$The\ method\ of\ VF2++\ for\ determining\ the\ node\ order$}\label{alg:VF2PPPseu}
53.511 +\begin{algorithmic}[1]
53.512 +\Procedure{VF2++order}{}
53.513 + \State $\mathcal{M}$ := $\emptyset$ \Comment{matching order}
53.514 + \While{$V_{small}\backslash \mathcal{M} \neq\emptyset$}
53.515 + \State $r\in$ arg max$_{deg}$ (arg min$_{F_\mathcal{M}\circ lab}(V_{small}\backslash \mathcal{M})$)\label{alg:findMin}
53.516 + \State Compute $T$, a BFS tree with root node $r$.
53.517 + \For{$d=0,1,...,depth(T)$}
53.518 + \State $V_d$:=nodes of the $d$-th level
53.519 + \State Process $V_d$ \Comment{See Algorithm \ref{alg:VF2PPProcess1}) and \ref{alg:VF2PPProcess2})}
53.520 + \EndFor
53.521 + \EndWhile
53.522 +\EndProcedure
53.523 +\end{algorithmic}
53.524 +\end{algorithm}
53.525 +
53.526 +\begin{algorithm}
53.527 +\algtext*{EndIf}%ne nyomtasson end if-et
53.528 +\algtext*{EndFor}%ne nyomtasson ..
53.529 +\algtext*{EndProcedure}%ne nyomtasson ..
53.530 +\algtext*{EndWhile}
53.531 +\caption{\hspace{.5cm}$A\ method\ for\ processing\ a\ level\ of\ the\ BFS\ tree$}\label{alg:VF2PPProcess1}
53.532 +\begin{algorithmic}[1]
53.533 +\Procedure{VF2++ProcessLevel1}{$V_{d}$}
53.534 + \While{$V_d\neq\emptyset$}
53.535 + \State $m\in$ arg min$_{F_\mathcal{M}\circ\ lab}($ arg max$_{deg}($arg max$_{Conn_{\mathcal{M}}}(V_{d})))$
53.536 + \State $V_d:=V_d\backslash m$
53.537 + \State Append node $m$ to the end of $\mathcal{M}$
53.538 + \State Refresh $F_\mathcal{M}$
53.539 + \EndWhile
53.540 +\EndProcedure
53.541 +\end{algorithmic}
53.542 +\end{algorithm}
53.543 +
53.544 +\begin{algorithm}
53.545 +\algtext*{EndIf}%ne nyomtasson end if-et
53.546 +\algtext*{EndFor}%ne nyomtasson ..
53.547 +\algtext*{EndProcedure}%ne nyomtasson ..
53.548 +\algtext*{EndWhile}
53.549 +\caption{\hspace{0.5cm}$Another\ method\ for\ processing\ a\ level\ of\ the\ BFS\ tree$}\label{alg:VF2PPProcess2}
53.550 +\begin{algorithmic}[1]
53.551 +\Procedure{VF2++ProcessLevel2}{$V_{d}$}
53.552 + \State Sort $V_d$ in descending lex. order by $(Conn_{\mathcal{M}},deg,-F_\mathcal{M})$
53.553 + \State Append the sorted $V_d$ to the end of $\mathcal{M}$
53.554 + \State Refresh $F_\mathcal{M}$
53.555 +\EndProcedure
53.556 +\end{algorithmic}
53.557 +\end{algorithm}
53.558 +\textbf{Algorithm~\ref{alg:VF2PPPseu})} is a high level description of the matching order procedure of VF2++. It computes a BFS tree for each component in ascending order of their rarest $lab$ and largest $deg$, whose root vertex is the component's minimal node. \textbf{Algorithm \ref{alg:VF2PPProcess1})} and \textbf{\ref{alg:VF2PPProcess2})} are two different methods to process a level of the BFS tree.
53.559 +
53.560 +After sorting the nodes of the current level in descending lexicographic order by $(Conn_{\mathcal{M}},deg,-F_\mathcal{M})$, \textbf{Algorithm \ref{alg:VF2PPProcess2})} appends them simultaneously to the matching order $\mathcal{M}$ and refreshes $F_\mathcal{M}$ just once, whereas \textbf{Algorithm \ref{alg:VF2PPProcess1})} appends the nodes separately to $\mathcal{M}$ and refreshes $F_\mathcal{M}$ immediately, so it provides up-to-date label information and may result in a more efficient matching order.
53.561 +
53.562 +\textbf{Claim~\ref{claim:MOclaim})} shows that \textbf{Algorithm \ref{alg:VF2PPPseu})} provides a matching order.
53.563 +
53.564 +
53.565 +\subsection{Cutting rules}
53.566 +\label{VF2PPCuttingRules}
53.567 +This section presents the cutting rules of VF2++, which are improved by using extra information coming from the node labels.
53.568 +\begin{notation}
53.569 +Let $\mathbf{\Gamma_{small}^{l}(u)}:=\{\tilde{u} : lab(\tilde{u})=l \wedge \tilde{u}\in \Gamma_{small} (u)\}$ and $\mathbf{\Gamma_{large}^{l}(v)}:=\{\tilde{v} : lab(\tilde{v})=l \wedge \tilde{v}\in \Gamma_{large} (v)\}$, where $u\in V_{small}$, $v\in V_{large}$ and $l$ is a label.
53.570 +\end{notation}
53.571 +
53.572 +\subsubsection{Induced subgraph isomorphism}
53.573 +\begin{claim}
53.574 +\[LabCut_{IND}((u,v),M(s))\!:=\!\!\!\!\!\bigvee_{l\ is\ label}\!\!\!\!\!\!\!|\Gamma_{large}^{l} (v) \cap T_{large}(s)|\!<\!|\Gamma_{small}^{l}(u)\cap T_{small}(s)|\ \vee\]\[\bigvee_{l\ is\ label} \newline |\Gamma_{large}^{l}(v)\cap \tilde{T}_{large}(s)| < |\Gamma_{small}^{l}(u)\cap \tilde{T}_{small}(s)|\] is a cutting function by IND.
53.575 +\end{claim}
53.576 +\begin{proof}
53.577 +It has to be shown, that $LabCut_{IND}((u,v),M(s))=true\Rightarrow$ the mapping can not be extended to a whole mapping.\\
53.578 +$LabCut_{IND}((u,v),M(s))=true,$ iff the following holds. $\\\exists l: |\Gamma_{large}^{l} (v)\ \cap\ T_{large}(s)| < |\Gamma_{small}^{l} (u)\ \cap\ T_{small}(s)| \vee |\Gamma_{large}^{l}(v)\cap \tilde{T}_{large}(s)| < |\Gamma_{small}^{l}(u)\cap \tilde{T}_{small}(s)|$.
53.579 +
53.580 +Suppose that $|\Gamma_{large}^{l} (v)\ \cap\ T_{large}(s)| < |\Gamma_{small}^{l} (u)\ \cap\ T_{small}(s)|$. Each node of $\Gamma_{small}^{l} (u)\ \cap\ T_{small}(s)$ has to be matched to a node in $\Gamma_{large}^{l} (v)\ \cap\ T_{large}(s)$, so $\Gamma_{large}^{l} (v)\ \cap\ T_{large}(s)$ can not be smaller than $\Gamma_{small}^{l} (u)\ \cap\ T_{small}(s)$. That is why $M(s)$ can not be extended to a whole mapping.
53.581 +
53.582 +Otherwise $|\Gamma_{large}^{l}(v)\cap \tilde{T}_{large}(s)| < |\Gamma_{small}^{l}(u)\cap \tilde{T}_{small}(s)|$ has to be true. Similarly, each node of $\Gamma_{large}^{l}(v)\cap \tilde{T}_{large}(s)$ has to be matched to a node in $\Gamma_{small}^{l}(u)\cap \tilde{T}_{small}(s)$, i.e. $\Gamma_{large}^{l}(v)\cap \tilde{T}_{large}(s)$ can not be smaller than $\Gamma_{small}^{l}(u)\cap \tilde{T}_{small}(s)$, if $M(s)$ is extendible.
53.583 +\end{proof}
53.584 +The following claims can be proven similarly.
53.585 +\subsubsection{Graph isomorphism}
53.586 +\begin{claim}
53.587 +\[LabCut_{ISO}((u,v),M(s))\!:=\!\!\!\!\!\bigvee_{l\ is\ label}\!\!\!\!\!\!\!|\Gamma_{large}^{l} (v) \cap T_{large}(s)|\!\neq\!|\Gamma_{small}^{l}(u)\cap T_{small}(s)|\ \vee\]\[\bigvee_{l\ is\ label} \newline |\Gamma_{large}^{l}(v)\cap \tilde{T}_{large}(s)| \neq |\Gamma_{small}^{l}(u)\cap \tilde{T}_{small}(s)|\] is a cutting function by ISO.
53.588 +\end{claim}
53.589 +
53.590 +\subsubsection{Subgraph isomorphism}
53.591 +\begin{claim}
53.592 +\[LabCut_{IND}((u,v),M(s))\!:=\!\!\!\!\!\bigvee_{l\ is\ label}\!\!\!\!\!\!\!|\Gamma_{large}^{l} (v) \cap T_{large}(s)|\!<\!|\Gamma_{small}^{l}(u)\cap T_{small}(s)|\] is a cutting function by SUB.
53.593 +\end{claim}
53.594 +
53.595 +
53.596 +
53.597 +\subsection{Implementation details}
53.598 +This section provides a detailed summary of an efficient implementation of VF2++.
53.599 +\subsubsection{Storing a mapping}
53.600 +After fixing an arbitrary node order ($u_0, u_1, .., u_{|G_{small}|-1}$) of $G_{small}$, an array $M$ is usable to store the current mapping in the following way.
53.601 +\[
53.602 + M[i] =
53.603 + \begin{cases}
53.604 + v & if\ (u_i,v)\ is\ in\ the\ mapping\\
53.605 + INVALID & if\ no\ node\ has\ been\ mapped\ to\ u_i.
53.606 + \end{cases}
53.607 +\]
53.608 +Where $i\in\{0,1, ..,|G_{small}|-1\}$, $v\in V_{large}$ and $INVALID$ means "no node".
53.609 +\subsubsection{Avoiding the recurrence}
53.610 +Exploring the state space was described in a recursive fashion using sets (see \textbf{Algorithm~\ref{alg:VF2Pseu})}), which makes the algorithm easy to understand but does not show directly an efficient way of implementation. The following approach avoiding recursion and using lookup tables instead of sets is not only fast but has linear space complexity as well.
53.611 +
53.612 +The recursion of \textbf{Algorithm~\ref{alg:VF2Pseu})} can be realized as a while loop, which has a loop counter $depth$ denoting the all-time depth of the recursion.
53.613 +Fixing a matching order, let $M$ denote the array storing the all-time mapping.
53.614 +The initial state is associated with the empty mapping, which means that $\forall i: M[i]=INVALID$ and $depth=0$.
53.615 +In case of a recursive call, $depth$ has to be incremented, while in case of a return, it has to be decremented.
53.616 +Based on \textbf{Claim~\ref{claim:claimCoverFromLeft})}, $M$ is $INVALID$ from index $depth$+1 and not $INVALID$ before $depth$, i.e. $\forall i: i < depth \Rightarrow M[i]\neq INVALID$ and $\forall i: i > depth \Rightarrow M[i]= INVALID$. $M[depth]$ changes while the state is being processed, but the property is held before both stepping back to a predecessor state and exploring a successor state.
53.617 +
53.618 +The necessary part of the candidate set is easily maintainable or computable by following \textbf{Section~\ref{candidateComputingVF2})}. A much faster method has been designed for biological- and sparse graphs, see the next section for details.
53.619 +
53.620 +\subsubsection{Calculating the candidates for a node}
53.621 +Being aware of \textbf{Claim~\ref{claim:claimCoverFromLeft})}, the task is not to maintain the candidate set, but to generate the candidate nodes in $G_{large}$ for a given node $u\in V_{small}$.
53.622 +In case of an expanding problem type and $M$ mapping, if a node $v\in V_{large}$ is a potential pair of $u\in V_{small}$, then $\forall u'\in V_{small} : (u,u')\in E_{small}\ and\ u'\ is\ covered\ by\ M\ \Rightarrow (v,Pair(M,u'))\in E_{large}$. That is, each covered neighbour of $u$ has to be mapped to a covered neighbour of $v$.
53.623 +
53.624 +Having said that, an algorithm running in $\Theta(deg)$ time is describable if there exists a covered node in the component containing $u$. In this case choose a covered neighbour $u'$ of $u$ arbitrarily --- such a node exists based on \textbf{Claim~\ref{claim:MOclaim})}. With all the candidates of $u$ being among the uncovered neighbours of $Pair(M,u')$, there are solely $deg(Pair(M,u'))$ nodes to check.
53.625 +
53.626 +An easy trick is to choose an $u'$, for which $|\{uncovered\ neighbours\ $ $of\ Pair(M,u')\}|$ is the smallest possible.
53.627 +
53.628 +Note that if $u$ is the first node of its component, then all the uncovered nodes of $G_{large}$ are candidates, so giving a sublinear method is impossible.
53.629 +
53.630 +
53.631 +\subsubsection{Determining the node order}
53.632 +This section describes how the node order preprocessing method of VF2++ can efficiently be implemented.
53.633 +
53.634 +For using lookup tables, the node labels are associated with the numbers $\{0,1,..,|K|-1\}$, where $K$ is the set of the labels. It enables $F_\mathcal{M}$ to be stored in an array, for which $F_\mathcal{M}[i]=F_\mathcal{M}(i)$ where $i=0,1,..,|K|-1$. At first, $\mathcal{M}=\emptyset$, so $F_\mathcal{M}[i]$ is the number of nodes in $V_{small}$ having label i, which is easy to compute in $\Theta(|V_{small}|)$ steps.
53.635 +
53.636 +$\mathcal{M}\subseteq V_{small}$ can be represented as an array of size $|V_{small}|$.
53.637 +
53.638 +The BFS tree is computed by using a FIFO data structure which is usually implemented as a linked list, but one can avoid it by using the array $\mathcal{M}$ itself. $\mathcal{M}$ contains all the nodes seen before, a pointer shows where the first node of the FIFO is, and another one shows where the next explored node has to be inserted. So the nodes of each level of the BFS tree can be processed by \textbf{Algorithm \ref{alg:VF2PPProcess1})} and \textbf{\ref{alg:VF2PPProcess2})} in place by swapping nodes.
53.639 +
53.640 +After a node $u$ gets to the next place of the node order, $F_\mathcal{M}[lab[u]]$ has to be decreased by one, because there is one less covered node in $V_{large}$ with label $lab(u)$, that is why min selection sort is preferred which gives the elements from left to right in descending order, see \textbf{Algorithm \ref{alg:VF2PPProcess1})}.
53.641 +
53.642 +Note that using a $\Theta(n^2)$ sort absolutely does not slow down the procedure on biological (and on sparse) graphs, since they have few nodes on a level. If a level had a large number of nodes, \textbf{Algorithm \ref{alg:VF2PPProcess2})} would seem to be a better choice with a $\Theta(nlog(n))$ or Bucket sort, but it may reduce the efficiency of the matching procedure, since $F_\mathcal{M}(i)$ can not be immediately refreshed, so it is unable to provide up-to-date label information.
53.643 +
53.644 +Note that the \textit{while loop} of \textbf{Algorithm \ref{alg:VF2PPPseu})} takes one iteration per graph component and the graphs in biology are mostly connected.
53.645 +\subsubsection{Cutting rules}
53.646 +In \textbf{Section \ref{VF2PPCuttingRules})}, the cutting rules were described using the sets $T_{small}$, $T_{large}$, $\tilde T_{small}$ and $\tilde T_{large}$, which are dependent on the all-time mapping (i.e. on the all-time state). The aim is to check the labeled cutting rules of VF2++ in $\Theta(deg)$ time.
53.647 +
53.648 +Firstly, suppose that these four sets are given in such a way, that checking whether a node is in a certain set takes constant time, e.g. they are given by their 0-1 characteristic vectors. Let $L$ be an initially zero integer lookup table of size $|K|$. After incrementing $L[lab(u')]$ for all $u'\in \Gamma_{small}(u) \cap T_{small}(s)$ and decrementing $L[lab(v')]$ for all $v'\in\Gamma_{large} (v) \cap T_{large}(s)$, the first part of the cutting rules is checkable in $\Theta(deg)$ time by considering the proper signs of $L$. Setting $L$ to zero takes $\Theta(deg)$ time again, which makes it possible to use the same table through the whole algorithm.
53.649 +The second part of the cutting rules can be verified using the same method with $\tilde T_{small}$ and $\tilde T_{large}$ instead of $T_{small}$ and $T_{large}$. Thus, the overall complexity is $\Theta(deg)$.
53.650 +
53.651 +An other integer lookup table storing the number of covered neighbours of each node in $G_{large}$ gives all the information about the sets $T_{large}$ and $\tilde T_{large}$, which is maintainable in $\Theta(deg)$ time when a pair is added or substracted by incrementing or decrementing the proper indices. A further improvement is that the values of $L[lab(u')]$ in case of checking $u$ is dependent only on $u$, i.e. on the size of the mapping, so for each $u\in V_{small}$ an array of pairs (label, number of such labels) can be stored to skip the maintaining operations. Note that these arrays are at most of size $deg$. Skipping this trick, the number of covered neighbours has to be stored for each node of $G_{small}$ as well to get the sets $T_{small}$ and $\tilde T_{small}$.
53.652 +
53.653 +Using similar tricks, the consistency function can be evaluated in $\Theta(deg)$ steps, as well.
53.654 +
53.655 +\section{The VF2 Plus Algorithm}
53.656 +The VF2 Plus algorithm is a recently improved version of VF2. It was compared with the state of the art algorithms in \cite{VF2Plus} and has proven itself to be competitive with RI, the best algorithm on biological graphs.
53.657 +\\
53.658 +A short summary of VF2 Plus follows, which uses the notation and the conventions of the original paper.
53.659 +
53.660 +\subsection{Ordering procedure}
53.661 +VF2 Plus uses a sorting procedure that prefers nodes in $V_{small}$ with the lowest probability to find a pair in $V_{small}$ and the highest number of connections with the nodes already sorted by the algorithm.
53.662 +
53.663 +\begin{definition}
53.664 +$(u,v)$ is a \textbf{feasible pair}, if $lab(u)=lab(v)$ and $deg(u)\leq deg(v)$, where $u\in{V_{small}}$ and $ v\in{V_{large}}$.
53.665 +\end{definition}
53.666 +$P_{lab}(L):=$ a priori probability to find a node with label $L$ in $V_{large}$
53.667 +\newline
53.668 +$P_{deg}(d):=$ a priori probability to find a node with degree $d$ in $V_{large}$
53.669 +\newline
53.670 +$P(u):=P_{lab}(L)*\bigcup_{d'>d}P_{deg}(d')$\\
53.671 +$M$ is the set of already sorted nodes, $T$ is the set of nodes candidate to be selected, and $degreeM$ of a node is the number of its neighbours in $M$.
53.672 +\begin{algorithm}
53.673 +\algtext*{EndIf}%ne nyomtasson end if-et
53.674 +\algtext*{EndFor}%ne nyomtasson ..
53.675 +\algtext*{EndProcedure}%ne nyomtasson ..
53.676 +\algtext*{EndWhile}
53.677 +\caption{}\label{alg:VF2PlusPseu}
53.678 +\begin{algorithmic}[1]
53.679 +\Procedure{VF2 Plus order}{}
53.680 + \State Select the node with the lowest $P$.
53.681 + \If {more nodes share the same $P$}
53.682 + \State select the one with maximum degree
53.683 + \EndIf
53.684 + \If {more nodes share the same $P$ and have the max degree}
53.685 + \State select the first
53.686 + \EndIf
53.687 + \State Put the selected node in the set $M$. \label{alg:putIn}
53.688 + \State Put all its unsorted neighbours in the set $T$.
53.689 + \If {$M\neq V_{small}$}
53.690 + \State From set $T$ select the node with maximum $degreeM$.
53.691 + \If {more nodes have maximum $degreeM$}
53.692 + \State Select the one with the lowest $P$
53.693 + \EndIf
53.694 + \If {more nodes have maximum $degreeM$ and $P$}
53.695 + \State Select the first.
53.696 + \EndIf
53.697 + \State \textbf{goto \ref{alg:putIn}.}
53.698 + \EndIf
53.699 +\EndProcedure
53.700 +\end{algorithmic}
53.701 +\end{algorithm}
53.702 +
53.703 +Using these notations, \textbf{Algorithm~\ref{alg:VF2PlusPseu})} provides the description of the sorting procedure.
53.704 +
53.705 +Note that $P(u)$ is not the exact probability of finding a consistent pair for $u$ by choosing a node of $V_{large}$ randomly, since $P_{lab}$ and $P_{deg}$ are not independent, though calculating the real probability would take quadratic time, which may be reduced by using fittingly lookup tables.
53.706 +
53.707 +\newpage
53.708 +\section{Experimental results}
53.709 +This section compares the performance of VF2++ and VF2 Plus. Both algorithms have run faster with orders of magnitude than VF2, thus its inclusion was not reasonable.
53.710 +\subsection{Biological graphs}
53.711 +The tests have been executed on a recent biological dataset created for the International Contest on Pattern Search in Biological Databases\cite{Content}, which has been constructed of Molecule, Protein and Contact Map graphs extracted from the Protein Data Bank\cite{ProteinDataBank}.
53.712 +
53.713 +The molecule dataset contains small graphs with less than 100 nodes and an average degree of less than 3. The protein dataset contains graphs having 500-10 000 nodes and an average degree of 4, while the contact map dataset contains graphs with 150-800 nodes and an average degree of 20.
53.714 +\\
53.715 +
53.716 +In the following, the induced subgraph isomorphism and the graph isomorphism will be examined.
53.717 +
53.718 +\subsubsection{Induced subgraph isomorphism}
53.719 +This dataset contains a set of graph pairs, and \textbf{all} the induced subgraph ismorphisms have to be found between them. \textbf{Figure \ref{fig:INDProt}), \ref{fig:INDContact}),} and \textbf{ \ref{fig:INDMolecule})} show the solution time of the problems in the problem set.
53.720 +
53.721 +\begin{figure}[H]
53.722 +\begin{center}
53.723 +\begin{tikzpicture}
53.724 + \begin{axis}[title=Proteins IND,xlabel={target size},ylabel={time (ms)},legend entries={VF2 Plus,VF2++},grid
53.725 + =major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \thinspace}]
53.726 + %\addplot+[only marks] table {proteinsOrig.txt};
53.727 + \addplot[mark=*,mark size=1.2pt,color=blue] table {Orig/Proteins.256.txt};
53.728 + \addplot[mark=triangle*,mark size=1.8pt,color=red] table {VF2PPLabel/Proteins.256.txt};
53.729 + \end{axis}
53.730 + \end{tikzpicture}
53.731 +\end{center}
53.732 +\vspace*{-0.8cm}
53.733 +\caption{Both the algorithms have linear behaviour on protein graphs. VF2++ is more than 10 times faster than VF2 Plus.} \label{fig:INDProt}
53.734 +\end{figure}
53.735 +
53.736 +\begin{figure}[H]
53.737 +\begin{center}
53.738 +\begin{tikzpicture}
53.739 +\begin{axis}[title=Contact Maps IND,xlabel={target size},ylabel={time (ms)},legend entries={VF2 Plus,VF2++},grid
53.740 +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \thinspace}]
53.741 +%\addplot+[only marks] table {proteinsOrig.txt};
53.742 +\addplot table {Orig/ContactMaps.128.txt};
53.743 +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {VF2PPLabel/ContactMaps.128.txt};
53.744 +\end{axis}
53.745 +\end{tikzpicture}
53.746 +\end{center}
53.747 +\vspace*{-0.8cm}
53.748 +\caption{On Contact Maps, VF2++ runs in near constant time, while VF2 Plus has a near linear behaviour.} \label{fig:INDContact}
53.749 +\end{figure}
53.750 +
53.751 +\begin{figure}[H]
53.752 +\begin{center}
53.753 +\begin{tikzpicture}
53.754 +\begin{axis}[title=Molecules IND,xlabel={target size},ylabel={time (ms)},legend entries={VF2 Plus,VF2++},grid
53.755 +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \thinspace}]
53.756 +%\addplot+[only marks] table {proteinsOrig.txt};
53.757 +\addplot table {Orig/Molecules.32.txt};
53.758 +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {VF2PPLabel/Molecules.32.txt};
53.759 +\end{axis}
53.760 +\end{tikzpicture}
53.761 +\end{center}
53.762 +\vspace*{-0.8cm}
53.763 +\caption{In the case of Molecules, the algorithms seem to have a similar behaviour, but VF2++ is almost two times faster even on such small graphs.} \label{fig:INDMolecule}
53.764 +\end{figure}
53.765 +
53.766 +
53.767 +\subsubsection{Graph ismorphism}
53.768 +In this experiment, the nodes of each graph in the database have been shuffled and an isomorphism between the shuffled and the original graph has been searched. For runtime results, see \textbf{Figure \ref{fig:ISOProt}), \ref{fig:ISOContact}),} and \textbf{\ref{fig:ISOMolecule})}.
53.769 +\begin{figure}[H]
53.770 +\begin{center}
53.771 +\begin{tikzpicture}
53.772 +\begin{axis}[title=Proteins ISO,xlabel={target size},ylabel={time (ms)},legend entries={VF2 Plus,VF2++},grid
53.773 +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \thinspace}]
53.774 +%\addplot+[only marks] table {proteinsOrig.txt};
53.775 +\addplot table {Orig/proteinsIso.txt};
53.776 +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {VF2PPLabel/proteinsIso.txt};
53.777 +\end{axis}
53.778 +\end{tikzpicture}
53.779 +\end{center}
53.780 +\vspace*{-0.8cm}
53.781 +\caption{On protein graphs, VF2 Plus has a super linear time complexity, while VF2++ runs in near constant time. The difference is about two order of magnitude on large graphs.}\label{fig:ISOProt}
53.782 +\end{figure}
53.783 +
53.784 +\begin{figure}[H]
53.785 +\begin{center}
53.786 +\begin{tikzpicture}
53.787 +\begin{axis}[title=Contact Maps ISO,xlabel={target size},ylabel={time (ms)},legend entries={VF2 Plus,VF2++},grid
53.788 +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \thinspace}]
53.789 +%\addplot+[only marks] table {proteinsOrig.txt};
53.790 +\addplot table {Orig/contactMapsIso.txt};
53.791 +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {VF2PPLabel/contactMapsIso.txt};
53.792 +\end{axis}
53.793 +\end{tikzpicture}
53.794 +\end{center}
53.795 +\vspace*{-0.8cm}
53.796 +\caption{The results are closer to each other on Contact Maps, but VF2++ still performs consistently better.}\label{fig:ISOContact}
53.797 +\end{figure}
53.798 +
53.799 +\begin{figure}[H]
53.800 +\begin{center}
53.801 +\begin{tikzpicture}
53.802 +\begin{axis}[title=Molecules ISO,xlabel={target size},ylabel={time (ms)},legend entries={VF2 Plus,VF2++},grid
53.803 +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \thinspace}]
53.804 +%\addplot+[only marks] table {proteinsOrig.txt};
53.805 +\addplot table {Orig/moleculesIso.txt};
53.806 +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {VF2PPLabel/moleculesIso.txt};
53.807 +\end{axis}
53.808 +\end{tikzpicture}
53.809 +\end{center}
53.810 +\vspace*{-0.8cm}
53.811 +\caption{In the case of Molecules, there is not such a significant difference, but VF2++ seems to be faster as the number of nodes increases.}\label{fig:ISOMolecule}
53.812 +\end{figure}
53.813 +
53.814 +
53.815 +\subsection{Random graphs}
53.816 +This section compares VF2++ with VF2 Plus on random graphs of a large size. The node labels are uniformly distributed.
53.817 +Let $\delta$ denote the average degree.
53.818 +For the parameters of problems solved in the experiments, please see the top of each chart.
53.819 +\subsubsection{Graph isomorphism}
53.820 +To evaluate the efficiency of the algorithms in the case of graph isomorphism, connected graphs of less than 20 000 nodes have been considered. Generating a random graph and shuffling its nodes, an isomorphism had to be found. \textbf{Figure \ref{fig:randISO5}), \ref{fig:randISO10}), \ref{fig:randISO15}), \ref{fig:randISO35}), \ref{fig:randISO45}),} and \textbf{\ref{fig:randISO100}) } show the runtime results on graph sets of various density.
53.821 +
53.822 +\begin{figure}[H]
53.823 +\begin{center}
53.824 +\begin{tikzpicture}
53.825 +\begin{axis}[title={Random ISO, $\delta = 5$},xlabel={target size},ylabel={time (ms)},legend entries={VF2 Plus,VF2++},grid
53.826 +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \thinspace}]
53.827 +%\addplot+[only marks] table {proteinsOrig.txt};
53.828 +\addplot table {randGraph/iso/vf2pIso5_1.txt};
53.829 +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/iso/vf2ppIso5_1.txt};
53.830 +\end{axis}
53.831 +\end{tikzpicture}
53.832 +\end{center}
53.833 +\vspace*{-0.8cm}
53.834 +\caption{}\label{fig:randISO5}
53.835 +\end{figure}
53.836 +
53.837 +\begin{figure}[H]
53.838 +\begin{center}
53.839 +\begin{tikzpicture}
53.840 +\begin{axis}[title={Random ISO, $\delta = 10$},xlabel={target size},ylabel={time (ms)},legend entries={VF2 Plus,VF2++},grid
53.841 +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \thinspace}]
53.842 +%\addplot+[only marks] table {proteinsOrig.txt};
53.843 +\addplot table {randGraph/iso/vf2pIso10_1.txt};
53.844 +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/iso/vf2ppIso10_1.txt};
53.845 +\end{axis}
53.846 +\end{tikzpicture}
53.847 +\end{center}
53.848 +\vspace*{-0.8cm}
53.849 +\caption{}\label{fig:randISO10}
53.850 +\end{figure}
53.851 +
53.852 +\begin{figure}[H]
53.853 +\begin{center}
53.854 +\begin{tikzpicture}
53.855 +\begin{axis}[title={Random ISO, $\delta = 15$},xlabel={target size},ylabel={time (ms)},legend entries={VF2 Plus,VF2++},grid
53.856 +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \thinspace}]
53.857 +%\addplot+[only marks] table {proteinsOrig.txt};
53.858 +\addplot table {randGraph/iso/vf2pIso15_1.txt};
53.859 +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/iso/vf2ppIso15_1.txt};
53.860 +\end{axis}
53.861 +\end{tikzpicture}
53.862 +\end{center}
53.863 +\vspace*{-0.8cm}
53.864 +\caption{}\label{fig:randISO15}
53.865 +\end{figure}
53.866 +
53.867 +\begin{figure}[H]
53.868 +\begin{center}
53.869 +\begin{tikzpicture}
53.870 +\begin{axis}[title={Random ISO, $\delta = 35$},xlabel={target size},ylabel={time (ms)},legend entries={VF2 Plus,VF2++},grid
53.871 +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \thinspace}]
53.872 +%\addplot+[only marks] table {proteinsOrig.txt};
53.873 +\addplot table {randGraph/iso/vf2pIso35_1.txt};
53.874 +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/iso/vf2ppIso35_1.txt};
53.875 +\end{axis}
53.876 +\end{tikzpicture}
53.877 +\end{center}
53.878 +\vspace*{-0.8cm}
53.879 +\caption{}\label{fig:randISO35}
53.880 +\end{figure}
53.881 +
53.882 +\begin{figure}[H]
53.883 +\begin{center}
53.884 +\begin{tikzpicture}
53.885 +\begin{axis}[title={Random ISO, $\delta = 45$},xlabel={target size},ylabel={time (ms)},legend entries={VF2 Plus,VF2++},grid
53.886 +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \thinspace}]
53.887 +%\addplot+[only marks] table {proteinsOrig.txt};
53.888 +\addplot table {randGraph/iso/vf2pIso45_1.txt};
53.889 +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/iso/vf2ppIso45_1.txt};
53.890 +\end{axis}
53.891 +\end{tikzpicture}
53.892 +\end{center}
53.893 +\vspace*{-0.8cm}
53.894 +\caption{}\label{fig:randISO45}
53.895 +\end{figure}
53.896 +
53.897 +\begin{figure}[H]
53.898 +\begin{center}
53.899 +\begin{tikzpicture}
53.900 +\begin{axis}[title={Random ISO, $\delta = 100$},xlabel={target size},ylabel={time (ms)},legend entries={VF2 Plus,VF2++},grid
53.901 +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \thinspace}]
53.902 +%\addplot+[only marks] table {proteinsOrig.txt};
53.903 +\addplot table {randGraph/iso/vf2pIso100_1.txt};
53.904 +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/iso/vf2ppIso100_1.txt};
53.905 +\end{axis}
53.906 +\end{tikzpicture}
53.907 +\end{center}
53.908 +\vspace*{-0.8cm}
53.909 +\caption{}\label{fig:randISO100}
53.910 +\end{figure}
53.911 +
53.912 +
53.913 +Considering the graph isomorphism problem, VF2++ consistently outperforms its rival especially on sparse graphs. The reason for the slightly super linear behaviour of VF2++ on denser graphs is the larger number of nodes in the BFS tree constructed in \textbf{Algorithm \ref{alg:VF2PPPseu})}.
53.914 +
53.915 +\subsubsection{Induced subgraph isomorphism}
53.916 +This section provides a comparison of VF2++ and VF2 Plus in the case of induced subgraph isomorphism. In addition to the size of the large graph, that of the small graph dramatically influences the hardness of a given problem too, so the overall picture is provided by examining small graphs of various size.
53.917 +
53.918 +For each chart, a number $0<\rho< 1$ has been fixed and the following has been executed 150 times. Generating a large graph $G_{large}$, choose 10 of its induced subgraphs having $\rho\ |V_{large}|$ nodes, and for all the 10 subgraphs find a mapping by using both the graph matching algorithms.
53.919 +The $\delta = 5, 10, 35$ and $\rho = 0.05, 0.1, 0.3, 0.6, 0.8, 0.95$ cases have been examined (see \textbf{Figure \ref{fig:randIND5}), \ref{fig:randIND10})} and \textbf{\ref{fig:randIND35})}), and for each $\delta$, a cumulative chart is given as well, which excludes $\rho = 0.05$ and $0.1$ for the sake of perspicuity (see \textbf{Figure \ref{fig:randIND5Sum}), \ref{fig:randIND10Sum})} and \textbf{\ref{fig:randIND35Sum})}).
53.920 +
53.921 +
53.922 +
53.923 +
53.924 +
53.925 +\begin{figure}[H]
53.926 +\vspace*{-0.8cm}
53.927 +\begin{subfigure}[b]{0.55\textwidth}
53.928 +\begin{center}
53.929 +\begin{tikzpicture}
53.930 +\begin{axis}[title={Random IND, $\delta = 5$, $\rho = 0.05$},width=7.2cm,height=6cm,xlabel={target size},ylabel={time (ms)},ylabel near ticks,legend entries={VF2 Plus,VF2++},grid
53.931 +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \space}]
53.932 +%\addplot+[only marks] table {proteinsOrig.txt};
53.933 +\addplot table {randGraph/ind/vf2pInd5_0.05.txt};
53.934 +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd5_0.05.txt};
53.935 +\end{axis}
53.936 +\end{tikzpicture}
53.937 +\end{center}
53.938 + \end{subfigure}
53.939 + \begin{subfigure}[b]{0.55\textwidth}
53.940 +\begin{center}
53.941 +\begin{tikzpicture}
53.942 +\begin{axis}[title={Random IND, $\delta = 5$, $\rho = 0.1$},width=7.2cm,height=6cm,xlabel={target size},ylabel={time (ms)},ylabel near ticks,legend entries={VF2 Plus,VF2++},grid
53.943 +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \space}]
53.944 +%\addplot+[only marks] table {proteinsOrig.txt};
53.945 +\addplot table {randGraph/ind/vf2pInd5_0.1.txt};
53.946 +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd5_0.1.txt};
53.947 +\end{axis}
53.948 +\end{tikzpicture}
53.949 +\end{center}
53.950 +\end{subfigure}
53.951 +\hspace{1cm}
53.952 +\begin{subfigure}[b]{0.55\textwidth}
53.953 +\begin{center}
53.954 +\begin{tikzpicture}
53.955 +\begin{axis}[title={Random IND, $\delta = 5$, $\rho = 0.3$},width=7.2cm,height=6cm,xlabel={target size},ylabel={time (ms)},ylabel near ticks,legend entries={VF2 Plus,VF2++},grid
53.956 +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \space}]
53.957 +%\addplot+[only marks] table {proteinsOrig.txt};
53.958 +\addplot table {randGraph/ind/vf2pInd5_0.3.txt};
53.959 +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd5_0.3.txt};
53.960 +\end{axis}
53.961 +\end{tikzpicture}
53.962 +\end{center}
53.963 + \end{subfigure}
53.964 + \begin{subfigure}[b]{0.55\textwidth}
53.965 +\begin{center}
53.966 +\begin{tikzpicture}
53.967 +\begin{axis}[title={Random IND, $\delta = 5$, $\rho = 0.6$},width=7.2cm,height=6cm,xlabel={target size},ylabel={time (ms)},ylabel near ticks,legend entries={VF2 Plus,VF2++},grid
53.968 +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \space}]
53.969 +%\addplot+[only marks] table {proteinsOrig.txt};
53.970 +\addplot table {randGraph/ind/vf2pInd5_0.6.txt};
53.971 +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd5_0.6.txt};
53.972 +\end{axis}
53.973 +\end{tikzpicture}
53.974 +\end{center}
53.975 +\end{subfigure}
53.976 +\begin{subfigure}[b]{0.55\textwidth}
53.977 +
53.978 +\begin{tikzpicture}
53.979 +\begin{axis}[title={Random IND, $\delta = 5$, $\rho = 0.8$},width=7.2cm,height=6cm,xlabel={target size},ylabel={time (ms)},ylabel near ticks,legend entries={VF2 Plus,VF2++},grid
53.980 +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \space}]
53.981 +%\addplot+[only marks] table {proteinsOrig.txt};
53.982 +\addplot table {randGraph/ind/vf2pInd5_0.8.txt};
53.983 +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd5_0.8.txt};
53.984 +\end{axis}
53.985 +\end{tikzpicture}
53.986 + \end{subfigure}
53.987 + \begin{subfigure}[b]{0.55\textwidth}
53.988 +\begin{tikzpicture}
53.989 +\begin{axis}[title={Random IND, $\delta = 5$, $\rho = 0.95$},width=7.2cm,height=6cm,xlabel={target size},ylabel={time (ms)},ylabel near ticks,legend entries={VF2 Plus,VF2++},grid
53.990 +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \thinspace}]
53.991 +%\addplot+[only marks] table {proteinsOrig.txt};
53.992 +\addplot table {randGraph/ind/vf2pInd5_0.95.txt};
53.993 +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd5_0.95.txt};
53.994 +\end{axis}
53.995 +\end{tikzpicture}
53.996 +\end{subfigure}
53.997 +\vspace*{-0.8cm}
53.998 +\caption{IND on graphs having an average degree of 5.}\label{fig:randIND5}
53.999 +\end{figure}
53.1000 +
53.1001 +\begin{figure}[H]
53.1002 +\begin{center}
53.1003 +\begin{tikzpicture}
53.1004 +\begin{axis}[title={Rand IND Summary, $\delta = 5$, $\rho = 0.3, 0.6, 0.8, 0.95$},height=17cm,width=16cm,xlabel={target size},ylabel={time (ms)},legend entries={VF2 Plus,VF2++},line width=0.8pt,grid
53.1005 +=major,mark size=1pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \thinspace}]
53.1006 +%\addplot+[only marks] table {proteinsOrig.txt};
53.1007 +\addplot[mark=*,mark size=1.5pt,color=blue] table {randGraph/ind/vf2pInd5_0.3.txt};
53.1008 +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd5_0.3.txt};
53.1009 +\addplot[mark=*,mark size=1.5pt,color=blue] table {randGraph/ind/vf2pInd5_0.6.txt};
53.1010 +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd5_0.6.txt};
53.1011 +\addplot[mark=*,mark size=1.5pt,color=blue] table {randGraph/ind/vf2pInd5_0.8.txt};
53.1012 +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd5_0.8.txt};
53.1013 +\addplot[mark=*,mark size=1.5pt,color=blue] table {randGraph/ind/vf2pInd5_0.95.txt};
53.1014 +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd5_0.95.txt};
53.1015 +\end{axis}
53.1016 +\end{tikzpicture}
53.1017 +\end{center}
53.1018 +\vspace*{-0.8cm}
53.1019 +\caption{Cummulative chart for $\delta=5$.}\label{fig:randIND5Sum}
53.1020 +\end{figure}
53.1021 +
53.1022 +
53.1023 +
53.1024 +\begin{figure}[H]
53.1025 +\vspace*{-0.8cm}
53.1026 +\begin{subfigure}[b]{0.55\textwidth}
53.1027 +\begin{center}
53.1028 +\begin{tikzpicture}
53.1029 +\begin{axis}[title={Random IND, $\delta = 10$, $\rho = 0.05$},width=7.2cm,height=6cm,xlabel={target size},ylabel={time (ms)},ylabel near ticks,legend entries={VF2 Plus,VF2++},grid
53.1030 +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \space}]
53.1031 +%\addplot+[only marks] table {proteinsOrig.txt};
53.1032 +\addplot table {randGraph/ind/vf2pInd10_0.05.txt};
53.1033 +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd10_0.05.txt};
53.1034 +\end{axis}
53.1035 +\end{tikzpicture}
53.1036 +\end{center}
53.1037 + \end{subfigure}
53.1038 + \begin{subfigure}[b]{0.55\textwidth}
53.1039 +\begin{center}
53.1040 +\begin{tikzpicture}
53.1041 +\begin{axis}[title={Random IND, $\delta = 10$, $\rho = 0.1$},width=7.2cm,height=6cm,xlabel={target size},ylabel={time (ms)},ylabel near ticks,legend entries={VF2 Plus,VF2++},grid
53.1042 +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \space}]
53.1043 +%\addplot+[only marks] table {proteinsOrig.txt};
53.1044 +\addplot table {randGraph/ind/vf2pInd10_0.1.txt};
53.1045 +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd10_0.1.txt};
53.1046 +\end{axis}
53.1047 +\end{tikzpicture}
53.1048 +\end{center}
53.1049 +\end{subfigure}
53.1050 +\hspace{1cm}
53.1051 +\begin{subfigure}[b]{0.55\textwidth}
53.1052 +\begin{center}
53.1053 +\begin{tikzpicture}
53.1054 +\begin{axis}[title={Random IND, $\delta = 10$, $\rho = 0.3$},width=7.2cm,height=6cm,xlabel={target size},ylabel={time (ms)},ylabel near ticks,legend entries={VF2 Plus,VF2++},grid
53.1055 +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \space}]
53.1056 +%\addplot+[only marks] table {proteinsOrig.txt};
53.1057 +\addplot table {randGraph/ind/vf2pInd10_0.3.txt};
53.1058 +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd10_0.3.txt};
53.1059 +\end{axis}
53.1060 +\end{tikzpicture}
53.1061 +\end{center}
53.1062 + \end{subfigure}
53.1063 + \begin{subfigure}[b]{0.55\textwidth}
53.1064 +\begin{center}
53.1065 +\begin{tikzpicture}
53.1066 +\begin{axis}[title={Random IND, $\delta = 10$, $\rho = 0.6$},width=7.2cm,height=6cm,xlabel={target size},ylabel={time (ms)},ylabel near ticks,legend entries={VF2 Plus,VF2++},grid
53.1067 +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \space}]
53.1068 +%\addplot+[only marks] table {proteinsOrig.txt};
53.1069 +\addplot table {randGraph/ind/vf2pInd10_0.6.txt};
53.1070 +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd10_0.6.txt};
53.1071 +\end{axis}
53.1072 +\end{tikzpicture}
53.1073 +\end{center}
53.1074 +\end{subfigure}
53.1075 +\begin{subfigure}[b]{0.55\textwidth}
53.1076 +
53.1077 +\begin{tikzpicture}
53.1078 +\begin{axis}[title={Random IND, $\delta = 10$, $\rho = 0.8$},width=7.2cm,height=6cm,xlabel={target size},ylabel={time (ms)},ylabel near ticks,legend entries={VF2 Plus,VF2++},grid
53.1079 +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \space}]
53.1080 +%\addplot+[only marks] table {proteinsOrig.txt};
53.1081 +\addplot table {randGraph/ind/vf2pInd10_0.8.txt};
53.1082 +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd10_0.8.txt};
53.1083 +\end{axis}
53.1084 +\end{tikzpicture}
53.1085 + \end{subfigure}
53.1086 + \begin{subfigure}[b]{0.55\textwidth}
53.1087 +\begin{tikzpicture}
53.1088 +\begin{axis}[title={Random IND, $\delta = 10$, $\rho = 0.95$},width=7.2cm,height=6cm,xlabel={target size},ylabel={time (ms)},ylabel near ticks,legend entries={VF2 Plus,VF2++},grid
53.1089 +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \thinspace}]
53.1090 +%\addplot+[only marks] table {proteinsOrig.txt};
53.1091 +\addplot table {randGraph/ind/vf2pInd10_0.95.txt};
53.1092 +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd10_0.95.txt};
53.1093 +\end{axis}
53.1094 +\end{tikzpicture}
53.1095 +\end{subfigure}
53.1096 +\vspace*{-0.8cm}
53.1097 +\caption{IND on graphs having an average degree of 10.}\label{fig:randIND10}
53.1098 +\end{figure}
53.1099 +
53.1100 +\begin{figure}[H]
53.1101 +\begin{center}
53.1102 +\begin{tikzpicture}
53.1103 +\begin{axis}[title={Rand IND Summary, $\delta = 10$, $\rho = 0.3, 0.6, 0.8, 0.95$},height=17cm,width=16cm,xlabel={target size},ylabel={time (ms)},legend entries={VF2 Plus,VF2++},line width=0.8pt,grid
53.1104 +=major,mark size=1pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \thinspace}]
53.1105 +%\addplot+[only marks] table {proteinsOrig.txt};
53.1106 +\addplot[mark=*,mark size=1.5pt,color=blue] table {randGraph/ind/vf2pInd10_0.3.txt};
53.1107 +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd10_0.3.txt};
53.1108 +\addplot[mark=*,mark size=1.5pt,color=blue] table {randGraph/ind/vf2pInd10_0.6.txt};
53.1109 +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd10_0.6.txt};
53.1110 +\addplot[mark=*,mark size=1.5pt,color=blue] table {randGraph/ind/vf2pInd10_0.8.txt};
53.1111 +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd10_0.8.txt};
53.1112 +\addplot[mark=*,mark size=1.5pt,color=blue] table {randGraph/ind/vf2pInd10_0.95.txt};
53.1113 +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd10_0.95.txt};
53.1114 +\end{axis}
53.1115 +\end{tikzpicture}
53.1116 +\end{center}
53.1117 +\vspace*{-0.8cm}
53.1118 +\caption{Cummulative chart for $\delta=10$.}\label{fig:randIND10Sum}
53.1119 +\end{figure}
53.1120 +
53.1121 +
53.1122 +
53.1123 +\begin{figure}[H]
53.1124 +\vspace*{-0.8cm}
53.1125 +\begin{subfigure}[b]{0.55\textwidth}
53.1126 +\begin{center}
53.1127 +\begin{tikzpicture}
53.1128 +\begin{axis}[title={Random IND, $\delta = 35$, $\rho = 0.05$},width=7.2cm,height=6cm,xlabel={target size},ylabel={time (ms)},ylabel near ticks,legend entries={VF2 Plus,VF2++},grid
53.1129 +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \space}]
53.1130 +%\addplot+[only marks] table {proteinsOrig.txt};
53.1131 +\addplot table {randGraph/ind/vf2pInd35_0.05.txt};
53.1132 +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd35_0.05.txt};
53.1133 +\end{axis}
53.1134 +\end{tikzpicture}
53.1135 +\end{center}
53.1136 + \end{subfigure}
53.1137 + \begin{subfigure}[b]{0.55\textwidth}
53.1138 +\begin{center}
53.1139 +\begin{tikzpicture}
53.1140 +\begin{axis}[title={Random IND, $\delta = 35$, $\rho = 0.1$},width=7.2cm,height=6cm,xlabel={target size},ylabel={time (ms)},ylabel near ticks,legend entries={VF2 Plus,VF2++},grid
53.1141 +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \space}]
53.1142 +%\addplot+[only marks] table {proteinsOrig.txt};
53.1143 +\addplot table {randGraph/ind/vf2pInd35_0.1.txt};
53.1144 +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd35_0.1.txt};
53.1145 +\end{axis}
53.1146 +\end{tikzpicture}
53.1147 +\end{center}
53.1148 +\end{subfigure}
53.1149 +\hspace{1cm}
53.1150 +\begin{subfigure}[b]{0.55\textwidth}
53.1151 +\begin{center}
53.1152 +\begin{tikzpicture}
53.1153 +\begin{axis}[title={Random IND, $\delta = 35$, $\rho = 0.3$},width=7.2cm,height=6cm,xlabel={target size},ylabel={time (ms)},ylabel near ticks,legend entries={VF2 Plus,VF2++},grid
53.1154 +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \space}]
53.1155 +%\addplot+[only marks] table {proteinsOrig.txt};
53.1156 +\addplot table {randGraph/ind/vf2pInd35_0.3.txt};
53.1157 +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd35_0.3.txt};
53.1158 +\end{axis}
53.1159 +\end{tikzpicture}
53.1160 +\end{center}
53.1161 + \end{subfigure}
53.1162 + \begin{subfigure}[b]{0.55\textwidth}
53.1163 +\begin{center}
53.1164 +\begin{tikzpicture}
53.1165 +\begin{axis}[title={Random IND, $\delta = 35$, $\rho = 0.6$},width=7.2cm,height=6cm,xlabel={target size},ylabel={time (ms)},ylabel near ticks,legend entries={VF2 Plus,VF2++},grid
53.1166 +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \space}]
53.1167 +%\addplot+[only marks] table {proteinsOrig.txt};
53.1168 +\addplot table {randGraph/ind/vf2pInd35_0.6.txt};
53.1169 +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd35_0.6.txt};
53.1170 +\end{axis}
53.1171 +\end{tikzpicture}
53.1172 +\end{center}
53.1173 +\end{subfigure}
53.1174 +\begin{subfigure}[b]{0.55\textwidth}
53.1175 +
53.1176 +\begin{tikzpicture}
53.1177 +\begin{axis}[title={Random IND, $\delta = 35$, $\rho = 0.8$},width=7.2cm,height=6cm,xlabel={target size},ylabel={time (ms)},ylabel near ticks,legend entries={VF2 Plus,VF2++},grid
53.1178 +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \space}]
53.1179 +%\addplot+[only marks] table {proteinsOrig.txt};
53.1180 +\addplot table {randGraph/ind/vf2pInd35_0.8.txt};
53.1181 +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd35_0.8.txt};
53.1182 +\end{axis}
53.1183 +\end{tikzpicture}
53.1184 + \end{subfigure}
53.1185 + \begin{subfigure}[b]{0.55\textwidth}
53.1186 +\begin{tikzpicture}
53.1187 +\begin{axis}[title={Random IND, $\delta = 35$, $\rho = 0.95$},width=7.2cm,height=6cm,xlabel={target size},ylabel={time (ms)},ylabel near ticks,legend entries={VF2 Plus,VF2++},grid
53.1188 +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \thinspace}]
53.1189 +%\addplot+[only marks] table {proteinsOrig.txt};
53.1190 +\addplot table {randGraph/ind/vf2pInd35_0.95.txt};
53.1191 +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd35_0.95.txt};
53.1192 +\end{axis}
53.1193 +\end{tikzpicture}
53.1194 +\end{subfigure}
53.1195 +\vspace*{-0.8cm}
53.1196 +\caption{IND on graphs having an average degree of 35.}\label{fig:randIND35}
53.1197 +\end{figure}
53.1198 +
53.1199 +\begin{figure}[H]
53.1200 +\begin{center}
53.1201 +\begin{tikzpicture}
53.1202 +\begin{axis}[title={Rand IND Summary, $\delta = 35$, $\rho = 0.3, 0.6, 0.8, 0.95$},height=17cm,width=16cm,xlabel={target size},ylabel={time (ms)},legend entries={VF2 Plus,VF2++},line width=0.8pt,grid
53.1203 +=major,mark size=1pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \thinspace}]
53.1204 +%\addplot+[only marks] table {proteinsOrig.txt};
53.1205 +\addplot[mark=*,mark size=1.5pt,color=blue] table {randGraph/ind/vf2pInd35_0.3.txt};
53.1206 +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd35_0.3.txt};
53.1207 +\addplot[mark=*,mark size=1.5pt,color=blue] table {randGraph/ind/vf2pInd35_0.6.txt};
53.1208 +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd35_0.6.txt};
53.1209 +\addplot[mark=*,mark size=1.5pt,color=blue] table {randGraph/ind/vf2pInd35_0.8.txt};
53.1210 +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd35_0.8.txt};
53.1211 +\addplot[mark=*,mark size=1.5pt,color=blue] table {randGraph/ind/vf2pInd35_0.95.txt};
53.1212 +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd35_0.95.txt};
53.1213 +\end{axis}
53.1214 +\end{tikzpicture}
53.1215 +\end{center}
53.1216 +\vspace*{-0.8cm}
53.1217 +\caption{Cummulative chart for $\delta=35$.}\label{fig:randIND35Sum}
53.1218 +\end{figure}
53.1219 +
53.1220 +Based on these experiments, VF2++ is faster than VF2 Plus and able to handle really large graphs in milliseconds. Note that when $IND$ was considered and the small graphs had proportionally few nodes ($\rho = 0.05$, or $\rho = 0.1$), then VF2 Plus produced some inefficient node orders(e.g. see the $\delta=10$ case on \textbf{Figure \ref{fig:randIND10})}). If these examples had been excluded, the charts would have seemed to be similar to the other ones.
53.1221 +Unsurprisingly, as denser graphs are considered, both VF2++ and VF2 Plus slow slightly down, but remain practically usable even on graphs having 10 000 nodes.
53.1222 +
53.1223 +
53.1224 +
53.1225 +
53.1226 +\newpage
53.1227 +\section{Conclusion}
53.1228 +In this thesis, after providing a short summary of the recent algorithms, a new graph matching algorithm based on VF2, called VF2++, has been presented and analyzed from a practical viewpoint.
53.1229 +
53.1230 +Recognizing the importance of the node order and determining an efficient one, VF2++ is able to match graphs of thousands of nodes in near practically linear time including preprocessing. In addition to the proper order, VF2++ uses more efficient consistency and cutting rules which are easy to compute and make the algorithm able to prune most of the unfruitful branches without going astray.
53.1231 +
53.1232 +In order to show the efficiency of the new method, it has been compared to VF2 Plus, which is the best concurrent algorithm based on \cite{VF2Plus}.
53.1233 +
53.1234 +The experiments show that VF2++ consistently outperforms VF2 Plus on biological graphs. It seems to be asymptotically faster on protein and on contact map graphs in the case of induced subgraph isomorphism, while in the case of graph isomorphism, it has definitely better asymptotic behaviour on protein graphs.
53.1235 +
53.1236 +Regarding random sparse graphs, not only has VF2++ proved itself to be faster than VF2 Plus, but it has a practically linear behaviour both in the case of induced subgraph- and graph isomorphism, as well.
53.1237 +
53.1238 +
53.1239
53.1240 %% The Appendices part is started with the command \appendix;
53.1241 %% appendix sections are then done as normal sections
53.1242 @@ -160,20 +1381,21 @@
53.1243 %% If you have bibdatabase file and want bibtex to generate the
53.1244 %% bibitems, please use
53.1245 %%
53.1246 -%% \bibliographystyle{elsarticle-num}
53.1247 -%% \bibliography{<your bibdatabase>}
53.1248 +\bibliographystyle{elsarticle-num}
53.1249 +\bibliography{bibliography}
53.1250
53.1251 %% else use the following coding to input the bibitems directly in the
53.1252 %% TeX file.
53.1253
53.1254 -\begin{thebibliography}{00}
53.1255 +%% \begin{thebibliography}{00}
53.1256
53.1257 -%% \bibitem{label}
53.1258 -%% Text of bibliographic item
53.1259 +%% %% \bibitem{label}
53.1260 +%% %% Text of bibliographic item
53.1261
53.1262 -\bibitem{}
53.1263 +%% \bibitem{}
53.1264
53.1265 -\end{thebibliography}
53.1266 +%% \end{thebibliography}
53.1267 +
53.1268 \end{document}
53.1269 \endinput
53.1270 %%
54.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
54.2 +++ b/proteins.txt Tue Nov 22 08:15:16 2016 +0100
54.3 @@ -0,0 +1,6 @@
54.4 +1 10
54.5 +2 11
54.6 +3 15
54.7 +4 7
54.8 +1 7
54.9 +1000 6
55.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
55.2 +++ b/proteinsBfs.txt Tue Nov 22 08:15:16 2016 +0100
55.3 @@ -0,0 +1,234 @@
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56.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
56.2 +++ b/proteinsOrig.txt Tue Nov 22 08:15:16 2016 +0100
56.3 @@ -0,0 +1,234 @@
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57.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
57.2 +++ b/proteinsOrigDeg.txt Tue Nov 22 08:15:16 2016 +0100
57.3 @@ -0,0 +1,234 @@
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58.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
58.2 +++ b/randGraph/ind/vf2pInd10_0.05.txt Tue Nov 22 08:15:16 2016 +0100
58.3 @@ -0,0 +1,50 @@
58.4 +354 0.0506667
58.5 +446 0.0713333
58.6 +631 0.122
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58.8 +842 0.192667
58.9 +1296 0.379333
58.10 +1327 0.402333
58.11 +1495 0.486667
58.12 +1713 0.614333
58.13 +2027 0.811667
58.14 +2106 0.911667
58.15 +2901 1.551
58.16 +2967 1.64133
58.17 +3169 1.91167
58.18 +3279 2.42867
58.19 +3320 1.978
58.20 +3510 2.36667
58.21 +4337 3.20533
58.22 +4535 3.66033
58.23 +4841 4.295
58.24 +5020 4.30967
58.25 +5111 4.54233
58.26 +5163 4.843
58.27 +5359 5.03067
58.28 +5542 5.158
58.29 +5547 5.323
58.30 +5562 5.51767
58.31 +5631 7.10833
58.32 +5877 5.91167
58.33 +6101 10.3863
58.34 +6295 6.685
58.35 +6584 7.42
58.36 +6594 13.237
58.37 +6788 22.985
58.38 +6849 10.606
58.39 +7173 26.576
58.40 +7279 10.1943
58.41 +7361 8.77533
58.42 +7384 11.583
58.43 +7461 10.739
58.44 +7467 16.974
58.45 +8091 11.3097
58.46 +8451 11.6927
58.47 +8827 481.983
58.48 +9031 14.1033
58.49 +9134 262.166
58.50 +9427 793.866
58.51 +9639 14.7117
58.52 +9720 37.7563
58.53 +9943 45.4893
59.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
59.2 +++ b/randGraph/ind/vf2pInd10_0.1.txt Tue Nov 22 08:15:16 2016 +0100
59.3 @@ -0,0 +1,48 @@
59.4 +70 0.0116667
59.5 +377 0.0986667
59.6 +390 0.108
59.7 +788 0.311
59.8 +1086 0.513333
59.9 +1922 1.40767
59.10 +2057 1.694
59.11 +2071 1.62867
59.12 +2169 1.81033
59.13 +2406 2.13267
59.14 +2958 3.30467
59.15 +3323 5.543
59.16 +3631 4.47733
59.17 +4224 6.646
59.18 +4380 7.38133
59.19 +4427 6.581
59.20 +4663 7.362
59.21 +4942 10.6473
59.22 +5118 8.73467
59.23 +5183 8.83933
59.24 +5204 11.3987
59.25 +5368 13.5937
59.26 +5704 11.7243
59.27 +5826 14.3633
59.28 +5859 11.0753
59.29 +5925 11.8847
59.30 +5948 12.146
59.31 +5961 13.188
59.32 +6052 12.8867
59.33 +6472 13.8913
59.34 +6871 17.5173
59.35 +7015 17.576
59.36 +7039 261.191
59.37 +7221 18.2443
59.38 +7238 17.9493
59.39 +7776 22.4403
59.40 +7806 23.5107
59.41 +8478 23.7227
59.42 +8679 26.0337
59.43 +8966 32.3923
59.44 +9251 46.147
59.45 +9336 364.076
59.46 +9499 47.7787
59.47 +9583 31.54
59.48 +9585 33.146
59.49 +9649 312.417
59.50 +9816 2676
59.51 +9932 86.0923
60.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
60.2 +++ b/randGraph/ind/vf2pInd10_0.3.txt Tue Nov 22 08:15:16 2016 +0100
60.3 @@ -0,0 +1,50 @@
60.4 +326 0.229
60.5 +399 0.312
60.6 +476 0.405667
60.7 +1069 1.47167
60.8 +1389 2.27967
60.9 +1415 2.38233
60.10 +1654 3.13433
60.11 +1909 4.10367
60.12 +1961 4.20967
60.13 +2638 7.205
60.14 +3005 9.56467
60.15 +3045 9.63067
60.16 +3144 9.97967
60.17 +3229 10.535
60.18 +3275 11.0397
60.19 +3803 14.7707
60.20 +4459 19.1907
60.21 +4816 22.2127
60.22 +4988 24.2533
60.23 +5097 24.7153
60.24 +5424 27.775
60.25 +5465 28.1747
60.26 +5634 29.9557
60.27 +5645 30.4397
60.28 +6022 34.0627
60.29 +6094 34.7597
60.30 +6159 35.3867
60.31 +6291 36.908
60.32 +6361 37.735
60.33 +6638 40.9127
60.34 +6677 41.3867
60.35 +6838 43.3643
60.36 +7043 45.8147
60.37 +7531 53.295
60.38 +7566 53.4237
60.39 +7623 53.3137
60.40 +7647 53.7857
60.41 +7822 56.64
60.42 +7994 58.8747
60.43 +8003 58.516
60.44 +8475 65.5097
60.45 +8996 73.5733
60.46 +9118 75.6233
60.47 +9256 78.6417
60.48 +9702 88.3083
60.49 +9750 86.066
60.50 +9753 87.0243
60.51 +9787 86.666
60.52 +9848 87.6113
60.53 +9849 87.6647
61.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
61.2 +++ b/randGraph/ind/vf2pInd10_0.6.txt Tue Nov 22 08:15:16 2016 +0100
61.3 @@ -0,0 +1,50 @@
61.4 +379 0.659
61.5 +431 0.756333
61.6 +665 1.45333
61.7 +718 1.63233
61.8 +826 2.04067
61.9 +1251 4.01467
61.10 +1279 4.19167
61.11 +1443 5.15233
61.12 +1500 5.49767
61.13 +2193 10.6097
61.14 +2231 10.9357
61.15 +2280 11.4147
61.16 +2632 14.7793
61.17 +2687 15.294
61.18 +2907 18.394
61.19 +3053 19.4757
61.20 +3091 19.9403
61.21 +3437 24.123
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61.23 +3976 31.5933
61.24 +4702 43.1907
61.25 +4751 43.979
61.26 +5349 54.99
61.27 +5350 55.042
61.28 +5353 55.103
61.29 +5722 64.9803
61.30 +5764 63.4153
61.31 +5844 64.99
61.32 +5946 67.22
61.33 +5969 68.138
61.34 +5993 68.2493
61.35 +6013 68.6733
61.36 +6323 76.4853
61.37 +6883 88.9913
61.38 +6917 89.7673
61.39 +7218 97.8563
61.40 +7383 101.738
61.41 +7663 108.983
61.42 +8038 120.23
61.43 +8119 122.024
61.44 +8133 122.049
61.45 +8165 123.306
61.46 +8204 124.93
61.47 +8694 138.633
61.48 +8805 142.476
61.49 +9262 157.005
61.50 +9334 159.177
61.51 +9569 167.333
61.52 +9811 176.083
61.53 +9876 178.658
62.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
62.2 +++ b/randGraph/ind/vf2pInd10_0.8.txt Tue Nov 22 08:15:16 2016 +0100
62.3 @@ -0,0 +1,50 @@
62.4 +22 0.031
62.5 +155 0.288
62.6 +176 0.334333
62.7 +911 3.326
62.8 +1065 4.422
62.9 +1107 4.559
62.10 +1526 7.77567
62.11 +1623 8.72367
62.12 +1782 10.411
62.13 +1863 10.9807
62.14 +1911 11.4317
62.15 +2165 14.2203
62.16 +2192 14.5207
62.17 +2217 14.8703
62.18 +2247 15.8033
62.19 +3050 26.4777
62.20 +3167 28.3113
62.21 +3398 32.1737
62.22 +3627 36.211
62.23 +4092 45.6143
62.24 +4240 50.091
62.25 +4346 50.883
62.26 +4615 56.5223
62.27 +4673 58.2247
62.28 +4708 58.7723
62.29 +4739 59.827
62.30 +4818 61.338
62.31 +5058 67.4893
62.32 +5222 73.828
62.33 +5414 76.2007
62.34 +5868 92.0477
62.35 +5946 90.9583
62.36 +6061 97.862
62.37 +6263 101.006
62.38 +6611 111.016
62.39 +6699 114.082
62.40 +6799 118.224
62.41 +6915 121.078
62.42 +7100 127.908
62.43 +7136 128.96
62.44 +7218 131.624
62.45 +7369 136.738
62.46 +7736 150.767
62.47 +8219 170.233
62.48 +8377 174.837
62.49 +8482 178.895
62.50 +8632 184.569
62.51 +8886 195.908
62.52 +9351 215.669
62.53 +9771 234.352
63.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
63.2 +++ b/randGraph/ind/vf2pInd10_0.95.txt Tue Nov 22 08:15:16 2016 +0100
63.3 @@ -0,0 +1,39 @@
63.4 +550 1.89167
63.5 +628 2.30633
63.6 +660 2.44667
63.7 +1192 6.26733
63.8 +1663 11.008
63.9 +1749 12.0033
63.10 +2082 16.0437
63.11 +2355 20.0497
63.12 +2375 20.4247
63.13 +2946 31.0957
63.14 +3499 40.91
63.15 +3567 42.3037
63.16 +3774 47.1067
63.17 +4189 57.4717
63.18 +4205 57.1887
63.19 +4214 57.5237
63.20 +4414 62.6257
63.21 +4430 65.425
63.22 +4481 66.262
63.23 +4500 64.8567
63.24 +4861 75.0313
63.25 +5162 83.857
63.26 +5650 99.221
63.27 +5724 104.668
63.28 +5795 107.559
63.29 +5944 109.123
63.30 +6006 114.083
63.31 +6057 113.29
63.32 +6462 127.495
63.33 +6536 130.378
63.34 +6668 135.983
63.35 +6996 149.48
63.36 +7594 173.137
63.37 +7936 188.337
63.38 +8716 229
63.39 +9142 250.944
63.40 +9164 248.217
63.41 +9435 263.048
63.42 +9465 264.229
64.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
64.2 +++ b/randGraph/ind/vf2pInd15_0.05.txt Tue Nov 22 08:15:16 2016 +0100
64.3 @@ -0,0 +1,29 @@
64.4 +216 0.03
64.5 +407 0.0723333
64.6 +538 0.105667
64.7 +927 0.258
64.8 +1009 0.279667
64.9 +1277 0.448333
64.10 +1668 0.648
64.11 +2391 1.245
64.12 +3229 2.398
64.13 +3754 3.21067
64.14 +3953 2.97767
64.15 +4254 3.59933
64.16 +4525 3.90133
64.17 +4780 4.31133
64.18 +5281 5.78067
64.19 +5861 7.20833
64.20 +6317 7.559
64.21 +6434 7.90267
64.22 +6766 7.867
64.23 +6881 8.46033
64.24 +7047 8.52233
64.25 +7316 11.2903
64.26 +7396 10.8353
64.27 +8696 25.6483
64.28 +9051 186.357
64.29 +9075 51.8077
64.30 +9377 41.2747
64.31 +9573 19.3877
64.32 +9687 27.1643
65.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
65.2 +++ b/randGraph/ind/vf2pInd15_0.1.txt Tue Nov 22 08:15:16 2016 +0100
65.3 @@ -0,0 +1,30 @@
65.4 +344 0.105333
65.5 +1434 1.012
65.6 +2070 1.78833
65.7 +2175 2.12467
65.8 +2480 2.36267
65.9 +2685 2.756
65.10 +3150 3.74
65.11 +3611 4.70533
65.12 +3814 5.17033
65.13 +4085 6.02267
65.14 +4089 5.83567
65.15 +4565 7.13067
65.16 +4742 7.711
65.17 +4778 8.84367
65.18 +5395 11.3127
65.19 +5587 17.3593
65.20 +5598 11.304
65.21 +5741 11.2983
65.22 +6033 12.098
65.23 +6357 15.332
65.24 +6401 14.5977
65.25 +6581 17.253
65.26 +6593 14.869
65.27 +6869 16.6577
65.28 +7079 19.5183
65.29 +7280 17.8077
65.30 +7882 24.0283
65.31 +8847 26.2117
65.32 +9039 30.609
65.33 +9728 33.457
66.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
66.2 +++ b/randGraph/ind/vf2pInd15_0.2.txt Tue Nov 22 08:15:16 2016 +0100
66.3 @@ -0,0 +1,30 @@
66.4 +32 0.0143333
66.5 +471 0.318333
66.6 +1247 1.378
66.7 +2258 3.965
66.8 +2333 4.39633
66.9 +2339 4.09667
66.10 +2344 4.3
66.11 +2642 5.072
66.12 +2756 5.529
66.13 +3207 7.45267
66.14 +3593 8.88933
66.15 +4212 12.124
66.16 +5038 16.622
66.17 +5085 16.935
66.18 +5543 19.945
66.19 +5801 23.4663
66.20 +5861 23.88
66.21 +6042 23.8307
66.22 +6385 26.9897
66.23 +6959 31.8693
66.24 +7205 32.7863
66.25 +8042 40.4443
66.26 +8173 41.864
66.27 +9063 51.899
66.28 +9074 52.9547
66.29 +9736 58.821
66.30 +9770 59.7663
66.31 +9805 59.3963
66.32 +9833 64.1883
66.33 +9914 63.187
67.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
67.2 +++ b/randGraph/ind/vf2pInd15_0.3.txt Tue Nov 22 08:15:16 2016 +0100
67.3 @@ -0,0 +1,30 @@
67.4 +262 0.207667
67.5 +598 0.655
67.6 +694 0.811333
67.7 +722 0.862
67.8 +752 0.915667
67.9 +1116 1.74333
67.10 +1224 2.021
67.11 +1293 2.24967
67.12 +1314 2.264
67.13 +1836 4.03533
67.14 +3127 10.3897
67.15 +3213 10.9287
67.16 +3390 12.0497
67.17 +3613 13.491
67.18 +4832 23.1793
67.19 +5319 27.7917
67.20 +5330 28.0533
67.21 +5379 28.2763
67.22 +5394 28.4367
67.23 +5456 29.08
67.24 +6354 38.7267
67.25 +6579 41.3643
67.26 +6889 45.599
67.27 +7096 47.646
67.28 +8039 60.601
67.29 +8436 66.923
67.30 +8686 71.225
67.31 +9377 81.8663
67.32 +9455 82.5453
67.33 +9608 85.13
68.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
68.2 +++ b/randGraph/ind/vf2pInd15_0.4.txt Tue Nov 22 08:15:16 2016 +0100
68.3 @@ -0,0 +1,30 @@
68.4 +30 0.023
68.5 +315 0.366667
68.6 +439 0.567667
68.7 +717 1.18
68.8 +1550 4.17967
68.9 +1859 5.77533
68.10 +1867 5.795
68.11 +1868 5.83467
68.12 +2687 11.057
68.13 +2734 11.48
68.14 +3357 15.9177
68.15 +3709 19.078
68.16 +5091 34.3267
68.17 +5510 39.729
68.18 +5514 40.0913
68.19 +5654 42.2553
68.20 +5732 44.3647
68.21 +6082 49.7163
68.22 +6549 57.0103
68.23 +6816 61.572
68.24 +6920 62.6347
68.25 +7859 78.1077
68.26 +8363 87.6337
68.27 +8757 97.692
68.28 +8854 97.8513
68.29 +8868 98.074
68.30 +9036 105.005
68.31 +9515 115.057
68.32 +9686 118.096
68.33 +9782 119.309
69.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
69.2 +++ b/randGraph/ind/vf2pInd15_0.5.txt Tue Nov 22 08:15:16 2016 +0100
69.3 @@ -0,0 +1,30 @@
69.4 +313 0.5
69.5 +542 1.03933
69.6 +623 1.258
69.7 +1531 5.05733
69.8 +1753 6.40633
69.9 +1856 7.12267
69.10 +1903 7.40533
69.11 +2022 8.23633
69.12 +2064 8.50633
69.13 +3364 20.2557
69.14 +3982 27.4747
69.15 +4041 29.015
69.16 +5037 42.9267
69.17 +6089 60.5607
69.18 +6443 67.2083
69.19 +6638 70.898
69.20 +6892 76.147
69.21 +7005 79.9043
69.22 +7133 81.553
69.23 +7764 97.4677
69.24 +7826 96.7693
69.25 +7839 99.9547
69.26 +7916 99.192
69.27 +8055 104.416
69.28 +8075 103.3
69.29 +8403 111.229
69.30 +8627 119.201
69.31 +8701 118.668
69.32 +9207 134.604
69.33 +9907 152.308
70.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
70.2 +++ b/randGraph/ind/vf2pInd15_0.6.txt Tue Nov 22 08:15:16 2016 +0100
70.3 @@ -0,0 +1,30 @@
70.4 +136 0.212333
70.5 +1037 3.34567
70.6 +1104 3.686
70.7 +1198 4.20867
70.8 +1204 4.26333
70.9 +1400 5.47233
70.10 +1533 6.26367
70.11 +1669 7.33233
70.12 +1723 7.70567
70.13 +2230 11.9047
70.14 +3701 29.3703
70.15 +4063 35.149
70.16 +4258 37.7457
70.17 +4476 41.4307
70.18 +5205 54.5803
70.19 +5304 57.3897
70.20 +5718 65.1257
70.21 +6123 73.9937
70.22 +6141 74.5327
70.23 +6192 75.7483
70.24 +6711 88.3167
70.25 +6973 95.5403
70.26 +7075 97.3207
70.27 +7294 103.253
70.28 +7653 113.638
70.29 +7869 118.855
70.30 +7878 119.142
70.31 +8668 142.724
70.32 +8726 146.63
70.33 +9523 171.031
71.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
71.2 +++ b/randGraph/ind/vf2pInd15_0.7.txt Tue Nov 22 08:15:16 2016 +0100
71.3 @@ -0,0 +1,49 @@
71.4 +115 0.218
71.5 +197 0.397667
71.6 +364 0.887
71.7 +392 1.00033
71.8 +914 3.32667
71.9 +950 3.571
71.10 +1556 7.707
71.11 +1561 7.97633
71.12 +1702 9.129
71.13 +1740 9.286
71.14 +1834 10.1287
71.15 +1982 11.714
71.16 +1998 11.8017
71.17 +2023 12.0333
71.18 +2092 12.644
71.19 +2187 13.815
71.20 +2297 14.967
71.21 +2314 15.2977
71.22 +2399 16.366
71.23 +2450 16.9767
71.24 +2534 17.7803
71.25 +2570 18.3007
71.26 +2666 19.3383
71.27 +2712 20.1653
71.28 +2725 20.0667
71.29 +2729 20.199
71.30 +2730 20.3637
71.31 +2895 22.468
71.32 +2920 22.8947
71.33 +2929 22.7347
71.34 +2997 23.8593
71.35 +3117 26.0253
71.36 +3149 26.2507
71.37 +3182 26.712
71.38 +3451 30.7967
71.39 +3464 31.0397
71.40 +3537 32.0727
71.41 +3564 32.3637
71.42 +3594 32.905
71.43 +3825 37.2367
71.44 +3892 38.0317
71.45 +3972 39.4953
71.46 +4198 44.1937
71.47 +4243 44.3837
71.48 +4281 45.103
71.49 +4327 46.527
71.50 +4776 55.3963
71.51 +4882 57.6943
71.52 +4938 59.1703
72.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
72.2 +++ b/randGraph/ind/vf2pInd15_0.8.txt Tue Nov 22 08:15:16 2016 +0100
72.3 @@ -0,0 +1,30 @@
72.4 +57 0.116333
72.5 +864 3.572
72.6 +1355 7.141
72.7 +1402 7.623
72.8 +1580 9.21167
72.9 +2161 15.619
72.10 +2295 17.2717
72.11 +2368 18.1977
72.12 +2585 21.1427
72.13 +2720 23.1157
72.14 +3126 29.6183
72.15 +3151 30.101
72.16 +4218 50.8937
72.17 +4886 66.1663
72.18 +5022 69.6627
72.19 +5239 75.1977
72.20 +5344 77.9527
72.21 +5361 78.2973
72.22 +5375 78.8197
72.23 +5819 91.158
72.24 +5930 94.378
72.25 +6273 104.835
72.26 +6477 111.401
72.27 +6947 126.704
72.28 +7598 150.256
72.29 +8269 175.992
72.30 +8736 195.773
72.31 +8868 200.751
72.32 +8972 205.386
72.33 +9725 239.337
73.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
73.2 +++ b/randGraph/ind/vf2pInd15_0.95.txt Tue Nov 22 08:15:16 2016 +0100
73.3 @@ -0,0 +1,30 @@
73.4 +28 0.0666667
73.5 +322 1.093
73.6 +449 1.70833
73.7 +561 2.337
73.8 +848 4.31167
73.9 +1519 10.761
73.10 +2125 18.4883
73.11 +2211 19.9937
73.12 +2218 20.0537
73.13 +2232 20.2807
73.14 +2533 25.162
73.15 +2622 26.9303
73.16 +2892 31.6663
73.17 +3128 36.231
73.18 +3156 36.8187
73.19 +3379 41.4687
73.20 +3513 44.631
73.21 +3807 51.283
73.22 +4061 57.5013
73.23 +4750 76.1557
73.24 +4767 76.6803
73.25 +4768 76.782
73.26 +5893 112.892
73.27 +6203 124.273
73.28 +8331 214.711
73.29 +8433 220.537
73.30 +8547 225.248
73.31 +8935 245.129
73.32 +9613 281.541
73.33 +9627 282.218
74.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
74.2 +++ b/randGraph/ind/vf2pInd32_0.05.txt Tue Nov 22 08:15:16 2016 +0100
74.3 @@ -0,0 +1,30 @@
74.4 +216 0.0483333
74.5 +407 0.115333
74.6 +538 0.164667
74.7 +927 0.400667
74.8 +1009 0.462333
74.9 +1277 0.626333
74.10 +1668 0.952
74.11 +2391 2.458
74.12 +3229 2.67367
74.13 +3754 3.469
74.14 +3953 4.03
74.15 +4254 6.02067
74.16 +4525 4.731
74.17 +4780 5.11633
74.18 +5281 8.87567
74.19 +5861 7.817
74.20 +6317 8.872
74.21 +6434 9.72067
74.22 +6766 18.1203
74.23 +6881 10.3367
74.24 +7047 10.7203
74.25 +7316 11.1883
74.26 +7396 12.2917
74.27 +8696 18.3437
74.28 +8897 59.4683
74.29 +9051 16.5487
74.30 +9075 18.851
74.31 +9377 40.446
74.32 +9573 26.39
74.33 +9687 22.042
75.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
75.2 +++ b/randGraph/ind/vf2pInd32_0.1.txt Tue Nov 22 08:15:16 2016 +0100
75.3 @@ -0,0 +1,30 @@
75.4 +344 0.143667
75.5 +1434 1.20933
75.6 +2070 2.17767
75.7 +2175 2.31733
75.8 +2480 2.87733
75.9 +2685 3.26067
75.10 +3150 4.33467
75.11 +3611 5.43433
75.12 +3814 5.951
75.13 +4085 6.683
75.14 +4089 6.89333
75.15 +4565 8.10867
75.16 +4742 8.58733
75.17 +4778 8.7
75.18 +5395 11.0133
75.19 +5587 11.8337
75.20 +5598 11.5987
75.21 +5741 12.2517
75.22 +6033 13.1647
75.23 +6357 15.0947
75.24 +6401 15.002
75.25 +6581 15.561
75.26 +6593 15.532
75.27 +6869 16.6487
75.28 +7079 17.639
75.29 +7280 19.2467
75.30 +7882 21.8247
75.31 +8847 26.6603
75.32 +9039 27.7043
75.33 +9728 32.275
76.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
76.2 +++ b/randGraph/ind/vf2pInd32_0.3.txt Tue Nov 22 08:15:16 2016 +0100
76.3 @@ -0,0 +1,30 @@
76.4 +262 0.309333
76.5 +598 0.919333
76.6 +694 1.15533
76.7 +722 1.219
76.8 +752 1.30767
76.9 +1116 2.30967
76.10 +1224 2.66167
76.11 +1293 2.898
76.12 +1314 2.959
76.13 +1836 5.00933
76.14 +3127 12.1287
76.15 +3213 12.703
76.16 +3390 13.9413
76.17 +3613 15.6223
76.18 +4832 25.7873
76.19 +5319 30.644
76.20 +5330 30.6967
76.21 +5379 31.2433
76.22 +5394 31.361
76.23 +5456 32.0347
76.24 +6354 42.1693
76.25 +6579 44.9697
76.26 +6889 48.9497
76.27 +7096 51.723
76.28 +8039 65.0997
76.29 +8436 71.3323
76.30 +8686 75.3243
76.31 +9377 86.862
76.32 +9455 88.1637
76.33 +9608 90.9447
77.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
77.2 +++ b/randGraph/ind/vf2pInd32_0.6.txt Tue Nov 22 08:15:16 2016 +0100
77.3 @@ -0,0 +1,30 @@
77.4 +136 0.363333
77.5 +1037 4.887
77.6 +1104 5.14633
77.7 +1198 5.79467
77.8 +1204 5.89733
77.9 +1400 7.36467
77.10 +1533 8.404
77.11 +1669 9.62233
77.12 +1723 10.1267
77.13 +2230 15.155
77.14 +3701 34.8173
77.15 +4063 42.5703
77.16 +4258 43.9077
77.17 +4476 47.9777
77.18 +5205 62.309
77.19 +5304 64.399
77.20 +5718 73.5017
77.21 +6123 83.217
77.22 +6141 83.6287
77.23 +6192 85.131
77.24 +6711 97.8737
77.25 +6973 104.891
77.26 +7075 107.821
77.27 +7294 114.304
77.28 +7653 123.88
77.29 +7869 132.351
77.30 +7878 130.796
77.31 +8668 156.333
77.32 +8726 158.004
77.33 +9523 185.501
78.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
78.2 +++ b/randGraph/ind/vf2pInd32_0.8.txt Tue Nov 22 08:15:16 2016 +0100
78.3 @@ -0,0 +1,30 @@
78.4 +248 1.11367
78.5 +439 2.172
78.6 +443 2.244
78.7 +1328 9.922
78.8 +1471 11.5383
78.9 +2396 24.0727
78.10 +2454 25.0777
78.11 +2968 33.7757
78.12 +2997 34.6437
78.13 +3149 37.539
78.14 +3328 40.9153
78.15 +3472 43.9497
78.16 +3850 52.1813
78.17 +4055 56.9343
78.18 +5013 81.4517
78.19 +5059 83.3097
78.20 +5663 99.998
78.21 +5728 102.454
78.22 +6077 115.204
78.23 +6448 126.072
78.24 +6510 126.617
78.25 +6532 128.346
78.26 +7140 149.279
78.27 +7297 155.374
78.28 +7470 162.685
78.29 +7628 168.062
78.30 +8250 195.35
78.31 +8333 199.957
78.32 +8834 221.676
78.33 +9980 280.124
79.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
79.2 +++ b/randGraph/ind/vf2pInd32_0.95.txt Tue Nov 22 08:15:16 2016 +0100
79.3 @@ -0,0 +1,30 @@
79.4 +28 0.092
79.5 +322 1.87367
79.6 +449 2.88
79.7 +561 3.849
79.8 +848 6.687
79.9 +1519 15.3643
79.10 +2125 24.726
79.11 +2211 26.4567
79.12 +2218 27.1653
79.13 +2232 26.7347
79.14 +2533 32.6447
79.15 +2622 33.9373
79.16 +2892 40.2703
79.17 +3128 46.1757
79.18 +3156 46.894
79.19 +3379 51.711
79.20 +3513 55.5957
79.21 +3807 61.8537
79.22 +4061 70.1517
79.23 +4750 89.438
79.24 +4767 89.9523
79.25 +4768 91.593
79.26 +5893 129.177
79.27 +6203 141.726
79.28 +8331 241.685
79.29 +8433 253.332
79.30 +8547 252.701
79.31 +8935 275.213
79.32 +9613 317.321
79.33 +9627 310.94
80.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
80.2 +++ b/randGraph/ind/vf2pInd35_0.05.txt Tue Nov 22 08:15:16 2016 +0100
80.3 @@ -0,0 +1,47 @@
80.4 +33 0.004
80.5 +684 0.255333
80.6 +1057 0.492
80.7 +1125 0.534667
80.8 +1578 0.919333
80.9 +1860 1.234
80.10 +1867 1.21633
80.11 +1990 1.263
80.12 +2036 1.30067
80.13 +2595 1.94167
80.14 +2607 1.921
80.15 +2756 3.03567
80.16 +3185 2.718
80.17 +3275 2.70667
80.18 +3362 3.14733
80.19 +3466 3.34367
80.20 +3492 3.081
80.21 +3564 3.14767
80.22 +3656 3.34333
80.23 +3834 3.45633
80.24 +4345 4.58533
80.25 +4605 4.69967
80.26 +4740 5.30433
80.27 +4921 6.00767
80.28 +4941 5.772
80.29 +4993 5.941
80.30 +5513 7.71433
80.31 +5767 7.69133
80.32 +6134 9.76067
80.33 +6410 9.202
80.34 +6534 10.31
80.35 +6904 9.506
80.36 +7010 11.4237
80.37 +7067 11.7407
80.38 +7231 11.6627
80.39 +7314 13.9493
80.40 +7525 12.132
80.41 +7538 12.1363
80.42 +7660 12.3557
80.43 +8422 14.5693
80.44 +8696 81.2783
80.45 +8885 17.4843
80.46 +8916 19.0267
80.47 +8919 16.0893
80.48 +9337 19.1553
80.49 +9878 31.0497
80.50 +9999 19.6727
81.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
81.2 +++ b/randGraph/ind/vf2pInd35_0.1.txt Tue Nov 22 08:15:16 2016 +0100
81.3 @@ -0,0 +1,50 @@
81.4 +302 0.128333
81.5 +450 0.220667
81.6 +697 0.416
81.7 +744 0.467667
81.8 +805 0.529333
81.9 +938 0.661333
81.10 +963 0.705
81.11 +1043 0.775333
81.12 +1240 1.00267
81.13 +1797 1.827
81.14 +2018 2.12733
81.15 +2124 2.32367
81.16 +2435 2.88867
81.17 +2810 3.647
81.18 +3368 4.925
81.19 +3401 5.17133
81.20 +3531 5.43567
81.21 +3674 5.749
81.22 +3734 5.86533
81.23 +4359 7.56333
81.24 +4403 7.76633
81.25 +4406 7.923
81.26 +4466 7.878
81.27 +4579 8.57667
81.28 +4685 8.60433
81.29 +4940 9.43967
81.30 +5007 9.67867
81.31 +5312 10.7183
81.32 +5471 11.3373
81.33 +5564 11.6963
81.34 +5962 13.1663
81.35 +6114 14.087
81.36 +6677 16.2267
81.37 +7019 17.8927
81.38 +7128 18.3563
81.39 +7468 19.748
81.40 +7488 20.108
81.41 +7636 21.851
81.42 +7668 21.656
81.43 +7823 21.697
81.44 +8143 23.279
81.45 +8156 23.6977
81.46 +8333 24.531
81.47 +8572 25.7437
81.48 +8961 27.7373
81.49 +9009 28.0237
81.50 +9443 30.8153
81.51 +9784 32.6847
81.52 +9868 33.4577
81.53 +9959 33.867
82.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
82.2 +++ b/randGraph/ind/vf2pInd35_0.3.txt Tue Nov 22 08:15:16 2016 +0100
82.3 @@ -0,0 +1,50 @@
82.4 +326 0.440667
82.5 +399 0.560333
82.6 +476 0.708667
82.7 +1069 2.27567
82.8 +1389 3.39033
82.9 +1415 3.508
82.10 +1654 4.47167
82.11 +1909 5.48733
82.12 +1961 5.76733
82.13 +2638 9.36833
82.14 +3005 11.6703
82.15 +3045 12.1313
82.16 +3144 12.635
82.17 +3229 13.213
82.18 +3275 13.6237
82.19 +3803 17.1753
82.20 +4459 23.0647
82.21 +4816 26.3773
82.22 +4988 27.9623
82.23 +5097 28.998
82.24 +5424 32.519
82.25 +5465 32.8147
82.26 +5634 34.8827
82.27 +5645 34.648
82.28 +6022 39.1953
82.29 +6094 40.01
82.30 +6159 40.5813
82.31 +6291 42.185
82.32 +6361 43.3303
82.33 +6638 47.9383
82.34 +6677 47.2617
82.35 +6838 48.9613
82.36 +7043 52.919
82.37 +7531 58.8037
82.38 +7566 59.2117
82.39 +7623 59.7657
82.40 +7647 60.3467
82.41 +7822 62.6957
82.42 +7994 65.3453
82.43 +8003 65.7403
82.44 +8475 73.3503
82.45 +8996 85.0193
82.46 +9118 83.9353
82.47 +9256 87.1167
82.48 +9702 93.593
82.49 +9750 94.753
82.50 +9753 94.6723
82.51 +9787 95.2873
82.52 +9848 96.5613
82.53 +9849 99.8623
83.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
83.2 +++ b/randGraph/ind/vf2pInd35_0.6.txt Tue Nov 22 08:15:16 2016 +0100
83.3 @@ -0,0 +1,50 @@
83.4 +379 1.304
83.5 +431 1.55767
83.6 +665 2.743
83.7 +718 2.95
83.8 +826 3.58
83.9 +1251 6.52667
83.10 +1279 6.73867
83.11 +1443 8.25133
83.12 +1500 8.65767
83.13 +2193 15.2023
83.14 +2231 15.5933
83.15 +2280 16.1503
83.16 +2632 20.2637
83.17 +2687 21.059
83.18 +2907 23.782
83.19 +3053 25.886
83.20 +3091 26.434
83.21 +3437 31.5983
83.22 +3587 34.2333
83.23 +3976 39.9213
83.24 +4702 53.383
83.25 +4751 54.564
83.26 +5349 67.4433
83.27 +5350 67.262
83.28 +5353 67.505
83.29 +5722 74.9237
83.30 +5764 76.005
83.31 +5844 77.7597
83.32 +5946 80.0777
83.33 +5969 80.8947
83.34 +5993 81.0297
83.35 +6013 81.995
83.36 +6323 89.7763
83.37 +6883 104.332
83.38 +6917 105.1
83.39 +7218 113.653
83.40 +7383 120.002
83.41 +7663 127.383
83.42 +8038 138.808
83.43 +8119 140.824
83.44 +8133 140.531
83.45 +8165 142.339
83.46 +8204 144.491
83.47 +8694 158.751
83.48 +8805 164.991
83.49 +9262 180.528
83.50 +9334 182.775
83.51 +9569 189.227
83.52 +9811 198.952
83.53 +9876 202.732
84.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
84.2 +++ b/randGraph/ind/vf2pInd35_0.8.txt Tue Nov 22 08:15:16 2016 +0100
84.3 @@ -0,0 +1,50 @@
84.4 +22 0.045
84.5 +155 0.700333
84.6 +176 0.784333
84.7 +911 6.20033
84.8 +1065 7.59467
84.9 +1107 8.02367
84.10 +1526 13.3993
84.11 +1623 13.9763
84.12 +1782 15.9207
84.13 +1863 16.9817
84.14 +1911 17.7677
84.15 +2165 22.11
84.16 +2192 22.0237
84.17 +2217 22.17
84.18 +2247 22.46
84.19 +3050 36.9757
84.20 +3167 39.482
84.21 +3398 45.8423
84.22 +3627 47.983
84.23 +4092 60.179
84.24 +4240 62.5573
84.25 +4346 65.6493
84.26 +4615 72.9677
84.27 +4673 75.9043
84.28 +4708 76.1693
84.29 +4739 75.6093
84.30 +4818 78.779
84.31 +5058 85.1967
84.32 +5222 88.906
84.33 +5414 95.2437
84.34 +5868 109.191
84.35 +5946 116.128
84.36 +6061 115.374
84.37 +6263 122.258
84.38 +6611 136.819
84.39 +6699 138.081
84.40 +6799 141.604
84.41 +6915 146.624
84.42 +7100 152.956
84.43 +7136 154.626
84.44 +7218 157.368
84.45 +7369 167.007
84.46 +7736 180.202
84.47 +8219 199.971
84.48 +8377 210.541
84.49 +8482 214.688
84.50 +8632 217.674
84.51 +8886 234.622
84.52 +9351 251.814
84.53 +9771 284.116
85.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
85.2 +++ b/randGraph/ind/vf2pInd35_0.95.txt Tue Nov 22 08:15:16 2016 +0100
85.3 @@ -0,0 +1,49 @@
85.4 +20 0.049
85.5 +550 3.94
85.6 +628 4.72633
85.7 +660 5.01467
85.8 +1192 11.1367
85.9 +1663 18.0227
85.10 +1749 20.1853
85.11 +1768 19.665
85.12 +2082 24.8423
85.13 +2355 31.2503
85.14 +2375 30.2203
85.15 +2617 35.2503
85.16 +2946 43.0043
85.17 +3499 56.9613
85.18 +3567 58.4403
85.19 +3774 67.8443
85.20 +3923 68.45
85.21 +4002 71.0137
85.22 +4189 75.521
85.23 +4205 79.518
85.24 +4214 77.788
85.25 +4414 81.9137
85.26 +4430 82.603
85.27 +4481 84.3963
85.28 +4500 86.8007
85.29 +4706 92.226
85.30 +4861 97.081
85.31 +5162 107.331
85.32 +5650 126.66
85.33 +5724 128.453
85.34 +5795 130.164
85.35 +5938 138.769
85.36 +5944 137.439
85.37 +6006 140.665
85.38 +6057 147.517
85.39 +6462 161.468
85.40 +6536 162.08
85.41 +6668 170.276
85.42 +6996 188.282
85.43 +7594 213.816
85.44 +7936 230.678
85.45 +8716 269.054
85.46 +8871 271.437
85.47 +9032 283.079
85.48 +9142 292.064
85.49 +9164 311.828
85.50 +9435 308.418
85.51 +9465 320.903
85.52 +9792 341.184
86.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
86.2 +++ b/randGraph/ind/vf2pInd5_0.05.txt Tue Nov 22 08:15:16 2016 +0100
86.3 @@ -0,0 +1,50 @@
86.4 +354 0.042
86.5 +446 0.061
86.6 +631 0.104333
86.7 +668 0.109667
86.8 +842 0.164667
86.9 +1296 0.334
86.10 +1327 0.354
86.11 +1495 0.424333
86.12 +1713 0.554667
86.13 +2027 0.735333
86.14 +2106 0.802667
86.15 +2901 1.459
86.16 +2967 1.494
86.17 +3169 1.79833
86.18 +3279 1.942
86.19 +3320 1.84067
86.20 +3510 2.026
86.21 +4337 3.03467
86.22 +4535 3.29267
86.23 +4841 3.747
86.24 +5020 4.00767
86.25 +5111 4.14067
86.26 +5163 4.54733
86.27 +5359 4.85367
86.28 +5542 4.838
86.29 +5547 4.852
86.30 +5562 4.86767
86.31 +5631 4.97333
86.32 +5877 5.41667
86.33 +6101 5.932
86.34 +6295 6.19133
86.35 +6584 6.75567
86.36 +6594 6.82033
86.37 +6788 7.783
86.38 +6849 7.599
86.39 +7173 7.93133
86.40 +7279 8.18833
86.41 +7361 8.36933
86.42 +7384 8.44133
86.43 +7461 8.64567
86.44 +7467 8.59167
86.45 +8091 10.7283
86.46 +8451 10.8843
86.47 +8827 11.8837
86.48 +9031 12.6497
86.49 +9134 12.8663
86.50 +9427 13.48
86.51 +9639 14.1477
86.52 +9720 14.3067
86.53 +9943 15.8237
87.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
87.2 +++ b/randGraph/ind/vf2pInd5_0.1.txt Tue Nov 22 08:15:16 2016 +0100
87.3 @@ -0,0 +1,48 @@
87.4 +70 0.00966667
87.5 +377 0.0856667
87.6 +390 0.0933333
87.7 +788 0.266
87.8 +1086 0.467667
87.9 +1922 1.274
87.10 +2057 1.459
87.11 +2071 1.478
87.12 +2169 1.685
87.13 +2406 1.94833
87.14 +2958 2.95933
87.15 +3323 3.57
87.16 +3631 4.21967
87.17 +4224 6.24633
87.18 +4380 6.09633
87.19 +4427 6.37467
87.20 +4663 7.27233
87.21 +4942 7.83867
87.22 +5118 8.14933
87.23 +5183 8.36267
87.24 +5204 8.44867
87.25 +5368 8.984
87.26 +5704 10.0917
87.27 +5826 10.6517
87.28 +5859 10.569
87.29 +5925 10.901
87.30 +5948 11.0407
87.31 +5961 11.743
87.32 +6052 11.4887
87.33 +6472 15.3517
87.34 +6871 14.5153
87.35 +7015 18.134
87.36 +7039 15.1717
87.37 +7221 17.014
87.38 +7238 17.9207
87.39 +7776 18.545
87.40 +7806 18.5313
87.41 +8478 22.436
87.42 +8679 22.8333
87.43 +8966 24.7707
87.44 +9251 25.9987
87.45 +9336 27.0017
87.46 +9499 29.525
87.47 +9583 27.9523
87.48 +9585 34.779
87.49 +9649 77.52
87.50 +9816 30.638
87.51 +9932 30.6913
88.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
88.2 +++ b/randGraph/ind/vf2pInd5_0.3.txt Tue Nov 22 08:15:16 2016 +0100
88.3 @@ -0,0 +1,49 @@
88.4 +326 0.199
88.5 +399 0.267333
88.6 +476 0.353333
88.7 +1069 1.362
88.8 +1389 2.15067
88.9 +1415 2.15767
88.10 +1654 2.86733
88.11 +1909 3.84733
88.12 +1961 3.961
88.13 +2638 6.87267
88.14 +3005 8.70933
88.15 +3045 8.90867
88.16 +3144 9.68167
88.17 +3229 9.952
88.18 +3275 10.5163
88.19 +3803 13.5883
88.20 +4459 19.0313
88.21 +4816 21.531
88.22 +4988 23.239
88.23 +5097 24.4823
88.24 +5424 28.021
88.25 +5465 27.2663
88.26 +5634 29.8707
88.27 +5645 29.3437
88.28 +6022 33
88.29 +6094 33.717
88.30 +6159 35.262
88.31 +6291 36.5397
88.32 +6361 37.286
88.33 +6638 40.641
88.34 +6677 41.6727
88.35 +6838 42.07
88.36 +7043 45.2177
88.37 +7531 52.6313
88.38 +7566 53.4523
88.39 +7623 52.2727
88.40 +7647 52.8467
88.41 +7822 71.1413
88.42 +7994 59.2187
88.43 +8003 59.824
88.44 +8475 65.147
88.45 +8996 73.607
88.46 +9118 76.1177
88.47 +9256 81.6237
88.48 +9750 87.1797
88.49 +9753 91.3717
88.50 +9787 91.8217
88.51 +9848 86.729
88.52 +9849 87.4
89.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
89.2 +++ b/randGraph/ind/vf2pInd5_0.6.txt Tue Nov 22 08:15:16 2016 +0100
89.3 @@ -0,0 +1,50 @@
89.4 +379 0.494667
89.5 +431 0.605667
89.6 +665 1.189
89.7 +718 1.341
89.8 +826 1.708
89.9 +1251 3.50233
89.10 +1279 3.631
89.11 +1443 4.50667
89.12 +1500 4.84533
89.13 +2193 9.664
89.14 +2231 10.0093
89.15 +2280 10.4237
89.16 +2632 13.5977
89.17 +2687 14.1333
89.18 +2907 16.407
89.19 +3053 18.0413
89.20 +3091 18.4913
89.21 +3437 22.572
89.22 +3587 24.5313
89.23 +3976 29.7623
89.24 +4702 41
89.25 +4751 41.851
89.26 +5349 52.6303
89.27 +5350 52.5897
89.28 +5353 52.5687
89.29 +5722 59.825
89.30 +5764 60.897
89.31 +5844 62.9263
89.32 +5946 64.5117
89.33 +5969 64.9087
89.34 +5993 65.7193
89.35 +6013 65.8873
89.36 +6323 72.5703
89.37 +6883 85.8193
89.38 +6917 86.475
89.39 +7218 93.9543
89.40 +7383 98.1227
89.41 +7663 105.505
89.42 +8038 115.815
89.43 +8119 117.976
89.44 +8133 118.325
89.45 +8165 119.396
89.46 +8204 120.515
89.47 +8694 134.9
89.48 +8805 138.845
89.49 +9262 152.516
89.50 +9334 156.444
89.51 +9569 164.193
89.52 +9811 170.963
89.53 +9876 172.89
90.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
90.2 +++ b/randGraph/ind/vf2pInd5_0.8.txt Tue Nov 22 08:15:16 2016 +0100
90.3 @@ -0,0 +1,50 @@
90.4 +22 0.0186667
90.5 +155 0.201333
90.6 +176 0.222667
90.7 +911 2.74667
90.8 +1065 3.56667
90.9 +1107 3.82
90.10 +1526 6.75067
90.11 +1623 7.55467
90.12 +1782 8.96033
90.13 +1863 9.66833
90.14 +1911 10.1757
90.15 +2165 12.923
90.16 +2192 13.0273
90.17 +2217 13.338
90.18 +2247 13.6367
90.19 +3050 24.2777
90.20 +3167 26.036
90.21 +3398 29.746
90.22 +3627 33.5413
90.23 +4092 42.2407
90.24 +4240 45.1543
90.25 +4346 48.223
90.26 +4615 53.2423
90.27 +4673 54.4743
90.28 +4708 55.3837
90.29 +4739 56.768
90.30 +4818 57.6567
90.31 +5058 63.4033
90.32 +5222 67.2153
90.33 +5414 72.2027
90.34 +5868 84.217
90.35 +5946 86.786
90.36 +6061 89.7137
90.37 +6263 95.571
90.38 +6611 106.224
90.39 +6699 108.918
90.40 +6799 112.146
90.41 +6915 115.882
90.42 +7100 122.037
90.43 +7136 123.168
90.44 +7218 126.608
90.45 +7369 131.009
90.46 +7736 143.94
90.47 +8219 161.959
90.48 +8377 167.891
90.49 +8482 172.541
90.50 +8632 179.712
90.51 +8886 188.364
90.52 +9351 208.154
90.53 +9771 227.267
91.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
91.2 +++ b/randGraph/ind/vf2pInd5_0.95.txt Tue Nov 22 08:15:16 2016 +0100
91.3 @@ -0,0 +1,49 @@
91.4 +20 0.02
91.5 +550 1.45267
91.6 +628 1.793
91.7 +660 1.923
91.8 +1192 5.2
91.9 +1663 9.43367
91.10 +1749 10.3543
91.11 +1768 10.5797
91.12 +2082 14.1837
91.13 +2355 17.9237
91.14 +2375 18.6123
91.15 +2617 21.9483
91.16 +2946 27.5247
91.17 +3499 37.583
91.18 +3567 38.9343
91.19 +3774 43.5577
91.20 +3923 46.8577
91.21 +4002 48.6863
91.22 +4189 52.968
91.23 +4205 53.1783
91.24 +4214 53.656
91.25 +4414 58.3077
91.26 +4430 58.6777
91.27 +4481 59.972
91.28 +4500 60.5043
91.29 +4706 66.548
91.30 +4861 70.1353
91.31 +5162 78.8657
91.32 +5650 96.785
91.33 +5724 95.6887
91.34 +5795 98.4357
91.35 +5938 103.126
91.36 +5944 103.095
91.37 +6006 105.139
91.38 +6057 107.132
91.39 +6462 121.897
91.40 +6536 123.841
91.41 +6668 131.597
91.42 +6996 141.533
91.43 +7594 166.466
91.44 +7936 180.745
91.45 +8716 216.966
91.46 +8871 223.681
91.47 +9032 232.128
91.48 +9142 237.983
91.49 +9164 239.712
91.50 +9435 252.498
91.51 +9465 255.209
91.52 +9792 272.201
92.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
92.2 +++ b/randGraph/ind/vf2pInd7_0.05.txt Tue Nov 22 08:15:16 2016 +0100
92.3 @@ -0,0 +1,30 @@
92.4 +78 0.00633333
92.5 +104 0.00966667
92.6 +132 0.012
92.7 +142 0.0156667
92.8 +173 0.0166667
92.9 +196 0.0213333
92.10 +218 0.0236667
92.11 +276 0.0323333
92.12 +293 0.0353333
92.13 +296 0.0353333
92.14 +366 0.049
92.15 +423 0.0603333
92.16 +458 0.067
92.17 +460 0.0693333
92.18 +486 0.073
92.19 +561 0.095
92.20 +575 0.0953333
92.21 +641 0.118333
92.22 +792 0.153333
92.23 +870 0.178
92.24 +922 0.206667
92.25 +992 0.236333
92.26 +1114 0.267667
92.27 +1166 0.292667
92.28 +1202 0.311667
92.29 +1608 0.505333
92.30 +1737 0.573667
92.31 +2022 0.802333
92.32 +2837 1.41533
92.33 +3271 1.84667
93.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
93.2 +++ b/randGraph/ind/vf2pInd7_0.1.txt Tue Nov 22 08:15:16 2016 +0100
93.3 @@ -0,0 +1,30 @@
93.4 +47 0.006
93.5 +85 0.013
93.6 +150 0.0256667
93.7 +192 0.0356667
93.8 +200 0.037
93.9 +206 0.038
93.10 +273 0.0563333
93.11 +302 0.0693333
93.12 +387 0.092
93.13 +394 0.0936667
93.14 +536 0.153333
93.15 +790 0.282667
93.16 +816 0.311
93.17 +851 0.331333
93.18 +925 0.374
93.19 +1085 0.483333
93.20 +1107 0.496667
93.21 +1135 0.516667
93.22 +1460 0.847
93.23 +1488 0.869667
93.24 +1954 1.39833
93.25 +2054 1.51533
93.26 +2170 1.729
93.27 +2190 1.67733
93.28 +2253 2.786
93.29 +2257 1.77167
93.30 +2311 1.90167
93.31 +2390 2.02967
93.32 +2463 2.083
93.33 +3366 3.765
94.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
94.2 +++ b/randGraph/ind/vf2pInd7_0.3.txt Tue Nov 22 08:15:16 2016 +0100
94.3 @@ -0,0 +1,28 @@
94.4 +20 0.00666667
94.5 +47 0.0176667
94.6 +65 0.0256667
94.7 +82 0.0366667
94.8 +85 0.0346667
94.9 +101 0.0453333
94.10 +141 0.0666667
94.11 +213 0.116667
94.12 +228 0.123333
94.13 +272 0.162667
94.14 +277 0.172333
94.15 +355 0.238333
94.16 +491 0.382333
94.17 +587 0.505
94.18 +654 0.601667
94.19 +698 0.673
94.20 +716 0.703333
94.21 +763 0.779
94.22 +851 0.970333
94.23 +1032 1.29067
94.24 +1095 1.421
94.25 +1603 2.78933
94.26 +1969 4.05133
94.27 +2176 4.86067
94.28 +2412 5.86433
94.29 +2587 6.67967
94.30 +2958 8.63433
94.31 +4266 17.2513
95.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
95.2 +++ b/randGraph/ind/vf2pInd7_0.6.txt Tue Nov 22 08:15:16 2016 +0100
95.3 @@ -0,0 +1,29 @@
95.4 +20 0.0176667
95.5 +25 0.0193333
95.6 +43 0.0386667
95.7 +56 0.0516667
95.8 +227 0.266667
95.9 +397 0.587333
95.10 +448 0.698
95.11 +474 0.769
95.12 +487 0.795
95.13 +509 0.849333
95.14 +630 1.18
95.15 +727 1.50333
95.16 +732 1.52167
95.17 +777 1.68267
95.18 +826 1.83933
95.19 +899 2.108
95.20 +1034 2.67733
95.21 +1078 2.85433
95.22 +1107 3.00767
95.23 +1146 3.19433
95.24 +1282 3.90833
95.25 +1393 4.439
95.26 +1410 4.676
95.27 +1484 5.00367
95.28 +1600 5.71033
95.29 +2053 8.91533
95.30 +2076 9.08333
95.31 +2201 10.1147
95.32 +2813 15.9103
96.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
96.2 +++ b/randGraph/ind/vf2pInd7_0.8.txt Tue Nov 22 08:15:16 2016 +0100
96.3 @@ -0,0 +1,30 @@
96.4 +37 0.0423333
96.5 +70 0.0863333
96.6 +157 0.238
96.7 +199 0.322
96.8 +271 0.468
96.9 +376 0.770333
96.10 +379 0.77
96.11 +380 0.774333
96.12 +522 1.24
96.13 +554 1.35333
96.14 +595 1.523
96.15 +634 1.652
96.16 +718 2.018
96.17 +721 2.04067
96.18 +730 2.07067
96.19 +928 3.07467
96.20 +1046 3.753
96.21 +1088 3.99333
96.22 +1156 4.42567
96.23 +1357 5.83733
96.24 +1443 6.48867
96.25 +1564 7.486
96.26 +1579 7.59033
96.27 +1780 9.42333
96.28 +1860 10.198
96.29 +1876 10.3787
96.30 +2079 12.4097
96.31 +2866 22.3363
96.32 +3318 29.3473
96.33 +4068 43.058
97.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
97.2 +++ b/randGraph/ind/vf2pInd7_0.95.txt Tue Nov 22 08:15:16 2016 +0100
97.3 @@ -0,0 +1,28 @@
97.4 +20 0.0326667
97.5 +59 0.0956667
97.6 +80 0.136333
97.7 +108 0.180667
97.8 +166 0.342667
97.9 +176 0.353667
97.10 +222 0.465333
97.11 +444 1.221
97.12 +624 1.98067
97.13 +631 2.01
97.14 +669 2.198
97.15 +757 2.66133
97.16 +876 3.37067
97.17 +891 3.45033
97.18 +1124 5.10967
97.19 +1193 5.64333
97.20 +1217 5.91667
97.21 +1345 6.93567
97.22 +1449 7.88367
97.23 +1603 9.44067
97.24 +1655 9.98667
97.25 +2079 15.527
97.26 +2129 15.7007
97.27 +2567 21.9067
97.28 +2716 24.4203
97.29 +3103 31.128
97.30 +3657 42.242
97.31 +4945 74.4017
98.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
98.2 +++ b/randGraph/ind/vf2ppInd10_0.05.txt Tue Nov 22 08:15:16 2016 +0100
98.3 @@ -0,0 +1,50 @@
98.4 +354 0.0213333
98.5 +446 0.029
98.6 +631 0.044
98.7 +668 0.043
98.8 +842 0.062
98.9 +1296 0.0866667
98.10 +1327 0.0963333
98.11 +1495 0.105
98.12 +1713 0.128
98.13 +2027 0.129333
98.14 +2106 0.172
98.15 +2901 0.207333
98.16 +2967 0.235
98.17 +3169 0.313333
98.18 +3279 1.13867
98.19 +3320 0.268333
98.20 +3510 0.271333
98.21 +4337 0.383
98.22 +4535 0.784333
98.23 +4841 0.376
98.24 +5020 11.99
98.25 +5111 0.631333
98.26 +5163 0.456
98.27 +5359 0.532
98.28 +5542 0.409
98.29 +5547 0.390667
98.30 +5562 1.90533
98.31 +5631 1.079
98.32 +5877 0.505333
98.33 +6101 0.489667
98.34 +6295 0.470333
98.35 +6584 2.44433
98.36 +6594 19.3737
98.37 +6788 2.752
98.38 +6849 0.527333
98.39 +7173 1.077
98.40 +7279 2.22867
98.41 +7361 0.739667
98.42 +7384 2.746
98.43 +7461 127.963
98.44 +7467 1.13867
98.45 +8091 1.29667
98.46 +8451 0.656667
98.47 +8827 66.1093
98.48 +9031 0.911667
98.49 +9134 20.6043
98.50 +9427 0.798
98.51 +9639 1.06033
98.52 +9720 5.394
98.53 +9943 1.997
99.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
99.2 +++ b/randGraph/ind/vf2ppInd10_0.1.txt Tue Nov 22 08:15:16 2016 +0100
99.3 @@ -0,0 +1,48 @@
99.4 +70 0.008
99.5 +377 0.0476667
99.6 +390 0.0596667
99.7 +788 0.116667
99.8 +1086 0.154333
99.9 +1922 0.260333
99.10 +2057 0.265667
99.11 +2071 0.306333
99.12 +2169 0.358333
99.13 +2406 0.573667
99.14 +2958 0.496333
99.15 +3323 1.10033
99.16 +3631 0.498333
99.17 +4224 2.97633
99.18 +4380 1.394
99.19 +4427 1.05067
99.20 +4663 1.098
99.21 +4942 1.568
99.22 +5118 0.946667
99.23 +5183 0.800667
99.24 +5204 1.32433
99.25 +5368 1.427
99.26 +5704 1.175
99.27 +5826 1.62467
99.28 +5859 1.14
99.29 +5925 5.22067
99.30 +5948 5.151
99.31 +5961 2.336
99.32 +6052 7.86367
99.33 +6472 4.58433
99.34 +6871 5.176
99.35 +7015 2.91567
99.36 +7039 1.56433
99.37 +7221 5.88167
99.38 +7238 1.15867
99.39 +7776 3.652
99.40 +7806 11.2547
99.41 +8478 14.108
99.42 +8679 9.199
99.43 +8966 8.07367
99.44 +9251 36.0683
99.45 +9336 3.62633
99.46 +9499 40.213
99.47 +9583 11.5847
99.48 +9585 5.372
99.49 +9649 499.196
99.50 +9816 19.6343
99.51 +9932 3.18467
100.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
100.2 +++ b/randGraph/ind/vf2ppInd10_0.3.txt Tue Nov 22 08:15:16 2016 +0100
100.3 @@ -0,0 +1,50 @@
100.4 +326 0.122667
100.5 +399 0.158
100.6 +476 0.19
100.7 +1069 0.440333
100.8 +1389 0.578
100.9 +1415 0.588
100.10 +1654 0.699333
100.11 +1909 0.830667
100.12 +1961 0.811333
100.13 +2638 1.361
100.14 +3005 1.39367
100.15 +3045 1.407
100.16 +3144 1.42767
100.17 +3229 1.69733
100.18 +3275 1.49733
100.19 +3803 1.963
100.20 +4459 2.233
100.21 +4816 2.28267
100.22 +4988 2.47933
100.23 +5097 2.67933
100.24 +5424 2.91633
100.25 +5465 2.99933
100.26 +5634 3.07333
100.27 +5645 3.224
100.28 +6022 2.97333
100.29 +6094 3.03967
100.30 +6159 3.767
100.31 +6291 2.952
100.32 +6361 2.99367
100.33 +6638 4.44067
100.34 +6677 3.07867
100.35 +6838 4.14167
100.36 +7043 4.32567
100.37 +7531 3.77567
100.38 +7566 3.97833
100.39 +7623 4.11233
100.40 +7647 4.552
100.41 +7822 4.82467
100.42 +7994 4.35267
100.43 +8003 4.042
100.44 +8475 4.786
100.45 +8996 4.78967
100.46 +9118 5.09367
100.47 +9256 5.06733
100.48 +9702 6.66567
100.49 +9750 5.32933
100.50 +9753 6.09333
100.51 +9787 5.00067
100.52 +9848 6.21267
100.53 +9849 5.135
101.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
101.2 +++ b/randGraph/ind/vf2ppInd10_0.6.txt Tue Nov 22 08:15:16 2016 +0100
101.3 @@ -0,0 +1,50 @@
101.4 +379 0.363667
101.5 +431 0.392333
101.6 +665 0.625
101.7 +718 0.686333
101.8 +826 0.797333
101.9 +1251 1.23167
101.10 +1279 1.29267
101.11 +1443 1.50733
101.12 +1500 1.52067
101.13 +2193 2.234
101.14 +2231 2.262
101.15 +2280 2.35133
101.16 +2632 2.76867
101.17 +2687 2.781
101.18 +2907 3.16533
101.19 +3053 3.29767
101.20 +3091 3.36433
101.21 +3437 3.69
101.22 +3587 3.81933
101.23 +3976 4.31133
101.24 +4702 5.154
101.25 +4751 5.19467
101.26 +5349 5.89933
101.27 +5350 5.955
101.28 +5353 5.91467
101.29 +5722 6.85567
101.30 +5764 6.48367
101.31 +5844 6.50733
101.32 +5946 6.64233
101.33 +5969 6.803
101.34 +5993 6.76567
101.35 +6013 6.766
101.36 +6323 7.207
101.37 +6883 8.03367
101.38 +6917 7.98533
101.39 +7218 8.47467
101.40 +7383 8.57
101.41 +7663 8.92767
101.42 +8038 9.49367
101.43 +8119 9.51667
101.44 +8133 10.128
101.45 +8165 9.59467
101.46 +8204 9.61967
101.47 +8694 10.1283
101.48 +8805 10.378
101.49 +9262 10.8383
101.50 +9334 10.9447
101.51 +9569 11.28
101.52 +9811 11.663
101.53 +9876 11.675
102.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
102.2 +++ b/randGraph/ind/vf2ppInd10_0.8.txt Tue Nov 22 08:15:16 2016 +0100
102.3 @@ -0,0 +1,50 @@
102.4 +22 0.0266667
102.5 +155 0.21
102.6 +176 0.236667
102.7 +911 1.297
102.8 +1065 1.57533
102.9 +1107 1.61733
102.10 +1526 2.259
102.11 +1623 2.48567
102.12 +1782 2.81767
102.13 +1863 2.838
102.14 +1911 2.89667
102.15 +2165 3.31867
102.16 +2192 3.348
102.17 +2217 3.445
102.18 +2247 3.59367
102.19 +3050 4.935
102.20 +3167 5.151
102.21 +3398 5.51833
102.22 +3627 5.90267
102.23 +4092 6.87
102.24 +4240 7.22467
102.25 +4346 7.32867
102.26 +4615 7.695
102.27 +4673 7.90567
102.28 +4708 7.988
102.29 +4739 8.083
102.30 +4818 8.18833
102.31 +5058 8.73
102.32 +5222 9.14267
102.33 +5414 9.27467
102.34 +5868 10.5013
102.35 +5946 10.2697
102.36 +6061 10.8493
102.37 +6263 11.3743
102.38 +6611 11.4547
102.39 +6699 11.75
102.40 +6799 12.2597
102.41 +6915 12.1893
102.42 +7100 12.796
102.43 +7136 12.5183
102.44 +7218 12.9163
102.45 +7369 13.036
102.46 +7736 13.905
102.47 +8219 14.67
102.48 +8377 15.2837
102.49 +8482 15.0247
102.50 +8632 15.2837
102.51 +8886 15.868
102.52 +9351 16.7927
102.53 +9771 17.4753
103.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
103.2 +++ b/randGraph/ind/vf2ppInd10_0.95.txt Tue Nov 22 08:15:16 2016 +0100
103.3 @@ -0,0 +1,39 @@
103.4 +550 0.977333
103.5 +628 1.12967
103.6 +660 1.16167
103.7 +1192 2.21433
103.8 +1663 3.18033
103.9 +1749 3.43967
103.10 +2082 4.02067
103.11 +2355 4.75233
103.12 +2375 4.75433
103.13 +2946 6.37567
103.14 +3499 7.39967
103.15 +3567 7.56767
103.16 +3774 8.10367
103.17 +4189 9.01667
103.18 +4205 9.05533
103.19 +4214 9.09067
103.20 +4414 9.57467
103.21 +4430 10.0363
103.22 +4481 9.95667
103.23 +4500 9.74033
103.24 +4861 10.8177
103.25 +5162 11.4517
103.26 +5650 12.598
103.27 +5724 13.0937
103.28 +5795 13.2703
103.29 +5944 13.2853
103.30 +6006 13.8167
103.31 +6057 13.4997
103.32 +6462 14.4263
103.33 +6536 14.7693
103.34 +6668 15.2183
103.35 +6996 15.9877
103.36 +7594 17.3567
103.37 +7936 18.2223
103.38 +8716 20.4137
103.39 +9142 21.4613
103.40 +9164 21.742
103.41 +9435 21.7353
103.42 +9465 22.3537
104.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
104.2 +++ b/randGraph/ind/vf2ppInd15_0.05.txt Tue Nov 22 08:15:16 2016 +0100
104.3 @@ -0,0 +1,30 @@
104.4 +216 0.0146667
104.5 +407 0.0306667
104.6 +538 0.0413333
104.7 +927 0.0796667
104.8 +1009 0.0803333
104.9 +1277 0.0966667
104.10 +1668 0.133667
104.11 +2391 0.220667
104.12 +3229 0.483667
104.13 +3754 0.379
104.14 +3953 0.362333
104.15 +4254 0.448333
104.16 +4525 0.44
104.17 +4780 0.854
104.18 +5281 1.765
104.19 +5861 1.72167
104.20 +6317 28.055
104.21 +6434 8.12967
104.22 +6766 0.83
104.23 +6881 4.948
104.24 +7047 1.42567
104.25 +7316 20.8757
104.26 +7396 4.272
104.27 +8696 45.728
104.28 +8897 3.91867
104.29 +9051 22.0687
104.30 +9075 47.6293
104.31 +9377 33.6793
104.32 +9573 5.967
104.33 +9687 56.9007
105.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
105.2 +++ b/randGraph/ind/vf2ppInd15_0.1.txt Tue Nov 22 08:15:16 2016 +0100
105.3 @@ -0,0 +1,30 @@
105.4 +344 0.0553333
105.5 +1434 0.264
105.6 +2070 0.424333
105.7 +2175 0.442333
105.8 +2480 0.467333
105.9 +2685 0.545
105.10 +3150 0.735333
105.11 +3611 0.881
105.12 +3814 1.27267
105.13 +4085 0.890333
105.14 +4089 0.840333
105.15 +4565 0.805667
105.16 +4742 11.2947
105.17 +4778 14.677
105.18 +5395 2.54333
105.19 +5587 3.559
105.20 +5598 1.28433
105.21 +5741 17.7933
105.22 +6033 1.878
105.23 +6357 3.084
105.24 +6401 4.80367
105.25 +6581 3.23567
105.26 +6593 2.65867
105.27 +6869 3.31867
105.28 +7079 3.54667
105.29 +7280 2.801
105.30 +7882 9.438
105.31 +8847 18.4517
105.32 +9039 13.3743
105.33 +9728 7.01267
106.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
106.2 +++ b/randGraph/ind/vf2ppInd15_0.2.txt Tue Nov 22 08:15:16 2016 +0100
106.3 @@ -0,0 +1,30 @@
106.4 +32 0.012
106.5 +471 0.158667
106.6 +1247 0.422667
106.7 +2258 0.848
106.8 +2333 0.870667
106.9 +2339 0.860667
106.10 +2344 0.877333
106.11 +2642 1.05267
106.12 +2756 0.966
106.13 +3207 1.196
106.14 +3593 1.27933
106.15 +4212 1.534
106.16 +5038 1.90733
106.17 +5085 2.155
106.18 +5543 2.518
106.19 +5801 2.947
106.20 +5861 3.33867
106.21 +6042 2.218
106.22 +6385 2.66867
106.23 +6959 2.95133
106.24 +7205 2.58767
106.25 +8042 4.22967
106.26 +8173 3.53733
106.27 +9063 5.05867
106.28 +9074 5.36933
106.29 +9736 4.149
106.30 +9770 6.145
106.31 +9805 4.74167
106.32 +9833 4.91367
106.33 +9914 5.165
107.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
107.2 +++ b/randGraph/ind/vf2ppInd15_0.3.txt Tue Nov 22 08:15:16 2016 +0100
107.3 @@ -0,0 +1,30 @@
107.4 +262 0.135333
107.5 +598 0.309333
107.6 +694 0.366667
107.7 +722 0.370333
107.8 +752 0.379333
107.9 +1116 0.585667
107.10 +1224 0.645667
107.11 +1293 0.698333
107.12 +1314 0.681333
107.13 +1836 0.994333
107.14 +3127 1.80733
107.15 +3213 1.82233
107.16 +3390 2.109
107.17 +3613 2.00933
107.18 +4832 2.88533
107.19 +5319 3.144
107.20 +5330 3.16367
107.21 +5379 3.43933
107.22 +5394 3.18233
107.23 +5456 3.38367
107.24 +6354 4.65767
107.25 +6579 3.926
107.26 +6889 4.30833
107.27 +7096 4.39633
107.28 +8039 5.09067
107.29 +8436 5.38967
107.30 +8686 5.465
107.31 +9377 6.039
107.32 +9455 6.445
107.33 +9608 6.16967
108.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
108.2 +++ b/randGraph/ind/vf2ppInd15_0.4.txt Tue Nov 22 08:15:16 2016 +0100
108.3 @@ -0,0 +1,30 @@
108.4 +30 0.0193333
108.5 +315 0.226333
108.6 +439 0.312333
108.7 +717 0.533667
108.8 +1550 1.21567
108.9 +1859 1.512
108.10 +1867 1.473
108.11 +1868 1.52367
108.12 +2687 2.25567
108.13 +2734 2.37
108.14 +3357 2.74433
108.15 +3709 3.01867
108.16 +5091 4.33667
108.17 +5510 4.62033
108.18 +5514 4.83333
108.19 +5654 5.03467
108.20 +5732 5.07067
108.21 +6082 5.78367
108.22 +6549 5.85133
108.23 +6816 6.15333
108.24 +6920 6.08967
108.25 +7859 7.083
108.26 +8363 7.626
108.27 +8757 7.96433
108.28 +8854 7.919
108.29 +8868 8.09233
108.30 +9036 8.47033
108.31 +9515 8.78633
108.32 +9686 8.83533
108.33 +9782 9.45933
109.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
109.2 +++ b/randGraph/ind/vf2ppInd15_0.5.txt Tue Nov 22 08:15:16 2016 +0100
109.3 @@ -0,0 +1,30 @@
109.4 +313 0.319667
109.5 +542 0.548667
109.6 +623 0.64
109.7 +1531 1.55567
109.8 +1753 1.84767
109.9 +1856 1.99567
109.10 +1903 1.96733
109.11 +2022 2.211
109.12 +2064 2.209
109.13 +3364 3.75433
109.14 +3982 4.50333
109.15 +4041 4.603
109.16 +5037 5.99967
109.17 +6089 7.01633
109.18 +6443 7.47167
109.19 +6638 7.72567
109.20 +6892 8.05567
109.21 +7005 8.47933
109.22 +7133 8.693
109.23 +7764 9.36267
109.24 +7826 9.21667
109.25 +7839 9.55367
109.26 +7916 9.45433
109.27 +8055 9.772
109.28 +8075 9.816
109.29 +8403 9.97067
109.30 +8627 10.559
109.31 +8701 10.5947
109.32 +9207 11.37
109.33 +9907 12.3923
110.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
110.2 +++ b/randGraph/ind/vf2ppInd15_0.6.txt Tue Nov 22 08:15:16 2016 +0100
110.3 @@ -0,0 +1,30 @@
110.4 +136 0.165667
110.5 +1037 1.349
110.6 +1104 1.43567
110.7 +1198 1.58233
110.8 +1204 1.641
110.9 +1400 1.924
110.10 +1533 2.02433
110.11 +1669 2.29033
110.12 +1723 2.36067
110.13 +2230 3.12033
110.14 +3701 5.424
110.15 +4063 6.157
110.16 +4258 6.25533
110.17 +4476 6.64567
110.18 +5205 7.83367
110.19 +5304 8.06333
110.20 +5718 8.58367
110.21 +6123 9.443
110.22 +6141 9.535
110.23 +6192 9.48433
110.24 +6711 10.6077
110.25 +6973 11.155
110.26 +7075 11.1143
110.27 +7294 11.6163
110.28 +7653 12.2893
110.29 +7869 12.6847
110.30 +7878 12.678
110.31 +8668 13.8913
110.32 +8726 14.3277
110.33 +9523 15.6823
111.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
111.2 +++ b/randGraph/ind/vf2ppInd15_0.7.txt Tue Nov 22 08:15:16 2016 +0100
111.3 @@ -0,0 +1,49 @@
111.4 +115 0.172667
111.5 +197 0.288333
111.6 +364 0.552
111.7 +392 0.621667
111.8 +914 1.50733
111.9 +950 1.57567
111.10 +1556 2.613
111.11 +1561 2.69567
111.12 +1702 2.94833
111.13 +1740 2.94633
111.14 +1834 3.12
111.15 +1982 3.45367
111.16 +1998 3.46433
111.17 +2023 3.472
111.18 +2092 3.586
111.19 +2187 3.82633
111.20 +2297 4.02933
111.21 +2314 4.08033
111.22 +2399 4.28333
111.23 +2450 4.43367
111.24 +2534 4.54633
111.25 +2570 4.59367
111.26 +2666 4.739
111.27 +2712 4.90267
111.28 +2725 4.784
111.29 +2729 4.82333
111.30 +2730 4.883
111.31 +2895 5.25267
111.32 +2920 5.23633
111.33 +2929 5.23967
111.34 +2997 5.37767
111.35 +3117 5.81833
111.36 +3149 5.81033
111.37 +3182 5.79967
111.38 +3451 6.29767
111.39 +3464 6.463
111.40 +3537 6.45167
111.41 +3564 6.591
111.42 +3594 6.57067
111.43 +3825 7.19667
111.44 +3892 7.172
111.45 +3972 7.33367
111.46 +4198 8.09167
111.47 +4243 7.99033
111.48 +4281 8.07633
111.49 +4327 8.302
111.50 +4776 9.09833
111.51 +4882 9.21733
111.52 +4938 9.47633
112.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
112.2 +++ b/randGraph/ind/vf2ppInd15_0.8.txt Tue Nov 22 08:15:16 2016 +0100
112.3 @@ -0,0 +1,30 @@
112.4 +57 0.099
112.5 +864 1.69167
112.6 +1355 2.68633
112.7 +1402 2.852
112.8 +1580 3.21867
112.9 +2161 4.597
112.10 +2295 4.885
112.11 +2368 5.01667
112.12 +2585 5.51133
112.13 +2720 5.82433
112.14 +3126 6.803
112.15 +3151 6.917
112.16 +4218 9.72633
112.17 +4886 11.0813
112.18 +5022 11.658
112.19 +5239 12.1167
112.20 +5344 12.2433
112.21 +5361 12.1497
112.22 +5375 12.3153
112.23 +5819 13.4813
112.24 +5930 13.69
112.25 +6273 14.7603
112.26 +6477 15.2887
112.27 +6947 16.3877
112.28 +7598 18.3013
112.29 +8269 19.8827
112.30 +8736 21.7027
112.31 +8868 21.4237
112.32 +8972 21.9797
112.33 +9725 23.7263
113.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
113.2 +++ b/randGraph/ind/vf2ppInd15_0.95.txt Tue Nov 22 08:15:16 2016 +0100
113.3 @@ -0,0 +1,30 @@
113.4 +28 0.058
113.5 +322 0.729
113.6 +449 1.04533
113.7 +561 1.30933
113.8 +848 2.12467
113.9 +1519 4.12433
113.10 +2125 5.89733
113.11 +2211 6.30633
113.12 +2218 6.33867
113.13 +2232 6.37933
113.14 +2533 7.33867
113.15 +2622 7.69733
113.16 +2892 8.50067
113.17 +3128 9.12
113.18 +3156 9.349
113.19 +3379 9.994
113.20 +3513 10.518
113.21 +3807 11.3437
113.22 +4061 12.1577
113.23 +4750 14.4103
113.24 +4767 14.4057
113.25 +4768 14.575
113.26 +5893 17.9827
113.27 +6203 19.2467
113.28 +8331 26.5803
113.29 +8433 27.073
113.30 +8547 27.4483
113.31 +8935 28.6703
113.32 +9613 31.179
113.33 +9627 31.0557
114.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
114.2 +++ b/randGraph/ind/vf2ppInd32_0.05.txt Tue Nov 22 08:15:16 2016 +0100
114.3 @@ -0,0 +1,30 @@
114.4 +216 0.0256667
114.5 +407 0.054
114.6 +538 0.0753333
114.7 +927 0.141667
114.8 +1009 0.13
114.9 +1277 0.182
114.10 +1668 0.246333
114.11 +2391 0.79
114.12 +3229 0.588667
114.13 +3754 1.11433
114.14 +3953 1.711
114.15 +4254 2.74767
114.16 +4525 0.831
114.17 +4780 2.55267
114.18 +5281 3.972
114.19 +5861 1.98633
114.20 +6317 2.189
114.21 +6434 2.517
114.22 +6766 4.74233
114.23 +6881 1.70067
114.24 +7047 1.75867
114.25 +7316 6.84733
114.26 +7396 2.67267
114.27 +8696 7.369
114.28 +8897 64.127
114.29 +9051 4.37267
114.30 +9075 19.596
114.31 +9377 7.534
114.32 +9573 26.88
114.33 +9687 30.3903
115.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
115.2 +++ b/randGraph/ind/vf2ppInd32_0.1.txt Tue Nov 22 08:15:16 2016 +0100
115.3 @@ -0,0 +1,30 @@
115.4 +344 0.0883333
115.5 +1434 0.382333
115.6 +2070 0.731333
115.7 +2175 0.609667
115.8 +2480 0.679
115.9 +2685 0.820667
115.10 +3150 1.18033
115.11 +3611 1.29167
115.12 +3814 1.39767
115.13 +4085 1.19267
115.14 +4089 1.387
115.15 +4565 1.44533
115.16 +4742 1.53067
115.17 +4778 1.50533
115.18 +5395 2.04567
115.19 +5587 1.93833
115.20 +5598 2.05633
115.21 +5741 1.82233
115.22 +6033 2.15367
115.23 +6357 1.87467
115.24 +6401 3.375
115.25 +6581 2.27433
115.26 +6593 2.21667
115.27 +6869 2.19067
115.28 +7079 2.73167
115.29 +7280 3.381
115.30 +7882 3.30433
115.31 +8847 4.94833
115.32 +9039 4.95233
115.33 +9728 4.448
116.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
116.2 +++ b/randGraph/ind/vf2ppInd32_0.3.txt Tue Nov 22 08:15:16 2016 +0100
116.3 @@ -0,0 +1,30 @@
116.4 +262 0.222
116.5 +598 0.523667
116.6 +694 0.625333
116.7 +722 0.641667
116.8 +752 0.686667
116.9 +1116 1.033
116.10 +1224 1.14567
116.11 +1293 1.21533
116.12 +1314 1.231
116.13 +1836 1.76667
116.14 +3127 3.10133
116.15 +3213 3.18767
116.16 +3390 3.39567
116.17 +3613 3.69567
116.18 +4832 4.86967
116.19 +5319 5.41767
116.20 +5330 5.293
116.21 +5379 5.45167
116.22 +5394 5.40433
116.23 +5456 5.504
116.24 +6354 6.41133
116.25 +6579 6.688
116.26 +6889 7.098
116.27 +7096 7.31133
116.28 +8039 8.348
116.29 +8436 8.902
116.30 +8686 9.26267
116.31 +9377 9.98467
116.32 +9455 10.0533
116.33 +9608 10.3483
117.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
117.2 +++ b/randGraph/ind/vf2ppInd32_0.6.txt Tue Nov 22 08:15:16 2016 +0100
117.3 @@ -0,0 +1,30 @@
117.4 +136 0.3
117.5 +1037 2.623
117.6 +1104 2.71433
117.7 +1198 2.94267
117.8 +1204 3.004
117.9 +1400 3.534
117.10 +1533 3.877
117.11 +1669 4.27267
117.12 +1723 4.46333
117.13 +2230 5.88833
117.14 +3701 10.3217
117.15 +4063 12.1177
117.16 +4258 11.9453
117.17 +4476 12.628
117.18 +5205 14.783
117.19 +5304 15.1423
117.20 +5718 16.596
117.21 +6123 17.8053
117.22 +6141 17.9073
117.23 +6192 18.3197
117.24 +6711 19.5373
117.25 +6973 20.4103
117.26 +7075 20.6033
117.27 +7294 21.3557
117.28 +7653 22.3623
117.29 +7869 23.215
117.30 +7878 23.1713
117.31 +8668 25.939
117.32 +8726 26.0967
117.33 +9523 28.605
118.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
118.2 +++ b/randGraph/ind/vf2ppInd32_0.8.txt Tue Nov 22 08:15:16 2016 +0100
118.3 @@ -0,0 +1,30 @@
118.4 +248 0.872333
118.5 +439 1.55767
118.6 +443 1.63633
118.7 +1328 5.37267
118.8 +1471 6.00033
118.9 +2396 10.006
118.10 +2454 10.3023
118.11 +2968 12.4067
118.12 +2997 12.942
118.13 +3149 13.509
118.14 +3328 14.336
118.15 +3472 14.9353
118.16 +3850 16.56
118.17 +4055 17.5867
118.18 +5013 22.144
118.19 +5059 23.2843
118.20 +5663 25.5683
118.21 +5728 26.1993
118.22 +6077 28.2707
118.23 +6448 29.18
118.24 +6510 28.466
118.25 +6532 29.3593
118.26 +7140 31.576
118.27 +7297 32.294
118.28 +7470 33.6157
118.29 +7628 33.767
118.30 +8250 38.49
118.31 +8333 39.233
118.32 +8834 42.1883
118.33 +9980 48.5733
119.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
119.2 +++ b/randGraph/ind/vf2ppInd32_0.95.txt Tue Nov 22 08:15:16 2016 +0100
119.3 @@ -0,0 +1,30 @@
119.4 +28 0.0763333
119.5 +322 1.45067
119.6 +449 2.122
119.7 +561 2.68233
119.8 +848 4.34533
119.9 +1519 8.448
119.10 +2125 11.5473
119.11 +2211 12.187
119.12 +2218 12.574
119.13 +2232 12.2737
119.14 +2533 14.052
119.15 +2622 14.1437
119.16 +2892 16.2433
119.17 +3128 18.7247
119.18 +3156 17.7963
119.19 +3379 18.994
119.20 +3513 20.174
119.21 +3807 20.4843
119.22 +4061 22.896
119.23 +4750 25.737
119.24 +4767 25.796
119.25 +4768 26.983
119.26 +5893 32.2643
119.27 +6203 34.7197
119.28 +8331 51.787
119.29 +8433 53.5687
119.30 +8547 54.17
119.31 +8935 55.877
119.32 +9613 59.2407
119.33 +9627 59.24
120.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
120.2 +++ b/randGraph/ind/vf2ppInd35_0.05.txt Tue Nov 22 08:15:16 2016 +0100
120.3 @@ -0,0 +1,47 @@
120.4 +33 0.002
120.5 +684 0.103333
120.6 +1057 0.156333
120.7 +1125 0.226333
120.8 +1578 0.288
120.9 +1860 0.441667
120.10 +1867 0.375
120.11 +1990 0.387
120.12 +2036 0.582667
120.13 +2595 0.420333
120.14 +2607 0.611
120.15 +2756 0.716667
120.16 +3185 0.749333
120.17 +3275 0.71
120.18 +3362 1.171
120.19 +3466 0.787667
120.20 +3492 0.979667
120.21 +3564 0.615667
120.22 +3656 1.07733
120.23 +3834 1.38467
120.24 +4345 2.50567
120.25 +4605 1.11367
120.26 +4740 1.585
120.27 +4921 1.08533
120.28 +4941 1.28733
120.29 +4993 1.187
120.30 +5513 2.28133
120.31 +5767 3.42267
120.32 +6134 4.55433
120.33 +6410 6.12933
120.34 +6534 4.74933
120.35 +6904 2.03367
120.36 +7010 11.2137
120.37 +7067 3.02833
120.38 +7231 2.602
120.39 +7314 5.80833
120.40 +7525 4.636
120.41 +7538 5.51833
120.42 +7660 3.01067
120.43 +8422 8.947
120.44 +8696 19.85
120.45 +8885 13.444
120.46 +8916 8.696
120.47 +8919 4.75467
120.48 +9337 8.49733
120.49 +9878 10.3503
120.50 +9999 12.4133
121.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
121.2 +++ b/randGraph/ind/vf2ppInd35_0.1.txt Tue Nov 22 08:15:16 2016 +0100
121.3 @@ -0,0 +1,50 @@
121.4 +302 0.0766667
121.5 +450 0.119
121.6 +697 0.204
121.7 +744 0.202333
121.8 +805 0.222333
121.9 +938 0.259667
121.10 +963 0.287667
121.11 +1043 0.302
121.12 +1240 0.402667
121.13 +1797 0.548
121.14 +2018 0.637667
121.15 +2124 0.641
121.16 +2435 0.864333
121.17 +2810 0.999333
121.18 +3368 1.06533
121.19 +3401 1.177
121.20 +3531 1.17
121.21 +3674 1.36033
121.22 +3734 1.14833
121.23 +4359 1.41933
121.24 +4403 1.459
121.25 +4406 1.82333
121.26 +4466 1.49833
121.27 +4579 1.647
121.28 +4685 1.49167
121.29 +4940 1.68467
121.30 +5007 1.70733
121.31 +5312 1.733
121.32 +5471 1.767
121.33 +5564 2.71533
121.34 +5962 2.27367
121.35 +6114 2.51267
121.36 +6677 2.802
121.37 +7019 2.60933
121.38 +7128 2.97933
121.39 +7468 2.61133
121.40 +7488 2.498
121.41 +7636 3.679
121.42 +7668 4.28667
121.43 +7823 2.94333
121.44 +8143 4.04233
121.45 +8156 3.09033
121.46 +8333 3.767
121.47 +8572 3.66367
121.48 +8961 5.04967
121.49 +9009 3.94867
121.50 +9443 4.829
121.51 +9784 4.55167
121.52 +9868 3.76567
121.53 +9959 3.98567
122.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
122.2 +++ b/randGraph/ind/vf2ppInd35_0.3.txt Tue Nov 22 08:15:16 2016 +0100
122.3 @@ -0,0 +1,50 @@
122.4 +326 0.305333
122.5 +399 0.372333
122.6 +476 0.450333
122.7 +1069 1.09433
122.8 +1389 1.456
122.9 +1415 1.48867
122.10 +1654 1.74733
122.11 +1909 1.978
122.12 +1961 2.06433
122.13 +2638 2.839
122.14 +3005 3.26433
122.15 +3045 3.39367
122.16 +3144 3.45867
122.17 +3229 3.54833
122.18 +3275 3.603
122.19 +3803 4.04133
122.20 +4459 4.956
122.21 +4816 5.32667
122.22 +4988 5.50933
122.23 +5097 5.647
122.24 +5424 6.02467
122.25 +5465 6.09067
122.26 +5634 6.33367
122.27 +5645 6.24233
122.28 +6022 6.81033
122.29 +6094 6.83167
122.30 +6159 6.803
122.31 +6291 7.09933
122.32 +6361 7.144
122.33 +6638 7.70633
122.34 +6677 7.56567
122.35 +6838 7.53833
122.36 +7043 8.01667
122.37 +7531 8.29867
122.38 +7566 8.32933
122.39 +7623 8.47833
122.40 +7647 8.54567
122.41 +7822 8.67933
122.42 +7994 8.86367
122.43 +8003 8.98133
122.44 +8475 10.0393
122.45 +8996 10.6107
122.46 +9118 10.5393
122.47 +9256 11.244
122.48 +9702 11.1997
122.49 +9750 11.3573
122.50 +9753 11.3957
122.51 +9787 11.4
122.52 +9848 11.467
122.53 +9849 12.043
123.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
123.2 +++ b/randGraph/ind/vf2ppInd35_0.6.txt Tue Nov 22 08:15:16 2016 +0100
123.3 @@ -0,0 +1,50 @@
123.4 +379 0.943333
123.5 +431 1.11667
123.6 +665 1.804
123.7 +718 1.83133
123.8 +826 2.151
123.9 +1251 3.39333
123.10 +1279 3.49767
123.11 +1443 4.16333
123.12 +1500 4.27933
123.13 +2193 6.26067
123.14 +2231 6.265
123.15 +2280 6.45067
123.16 +2632 7.407
123.17 +2687 7.696
123.18 +2907 8.28067
123.19 +3053 8.77467
123.20 +3091 8.99
123.21 +3437 10.057
123.22 +3587 10.6527
123.23 +3976 11.5893
123.24 +4702 14.2217
123.25 +4751 14.1807
123.26 +5349 16.8533
123.27 +5350 16.7427
123.28 +5353 16.4263
123.29 +5722 17.5457
123.30 +5764 17.609
123.31 +5844 17.6013
123.32 +5946 18.1777
123.33 +5969 18.356
123.34 +5993 18.331
123.35 +6013 18.586
123.36 +6323 19.724
123.37 +6883 21.5973
123.38 +6917 21.8227
123.39 +7218 22.7117
123.40 +7383 23.3807
123.41 +7663 24.1823
123.42 +8038 26.0197
123.43 +8119 25.9057
123.44 +8133 25.7447
123.45 +8165 25.8963
123.46 +8204 26.8807
123.47 +8694 27.3007
123.48 +8805 29.443
123.49 +9262 29.2483
123.50 +9334 30.2993
123.51 +9569 30.4333
123.52 +9811 31.097
123.53 +9876 31.6503
124.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
124.2 +++ b/randGraph/ind/vf2ppInd35_0.8.txt Tue Nov 22 08:15:16 2016 +0100
124.3 @@ -0,0 +1,50 @@
124.4 +22 0.0393333
124.5 +155 0.571667
124.6 +176 0.630333
124.7 +911 3.962
124.8 +1065 4.58933
124.9 +1107 4.76133
124.10 +1526 7.139
124.11 +1623 7.289
124.12 +1782 7.93
124.13 +1863 8.31633
124.14 +1911 8.60933
124.15 +2165 10.078
124.16 +2192 10.0547
124.17 +2217 10.0457
124.18 +2247 9.949
124.19 +3050 14.1343
124.20 +3167 14.7493
124.21 +3398 16.012
124.22 +3627 16.2417
124.23 +4092 19.0117
124.24 +4240 19.2673
124.25 +4346 20.274
124.26 +4615 21.8037
124.27 +4673 23.27
124.28 +4708 22.9527
124.29 +4739 21.8833
124.30 +4818 23.3217
124.31 +5058 23.948
124.32 +5222 24.4
124.33 +5414 25.2407
124.34 +5868 27.4257
124.35 +5946 30.0323
124.36 +6061 28.5753
124.37 +6263 30.288
124.38 +6611 32.98
124.39 +6699 32.8473
124.40 +6799 33.324
124.41 +6915 33.991
124.42 +7100 34.5173
124.43 +7136 35.7403
124.44 +7218 36.1373
124.45 +7369 36.3943
124.46 +7736 39.464
124.47 +8219 43.719
124.48 +8377 43.019
124.49 +8482 43.3823
124.50 +8632 43.822
124.51 +8886 47.0287
124.52 +9351 49.2603
124.53 +9771 52.7797
125.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
125.2 +++ b/randGraph/ind/vf2ppInd35_0.95.txt Tue Nov 22 08:15:16 2016 +0100
125.3 @@ -0,0 +1,49 @@
125.4 +20 0.0413333
125.5 +550 2.86733
125.6 +628 3.361
125.7 +660 3.513
125.8 +1192 6.79433
125.9 +1663 9.78133
125.10 +1749 10.8153
125.11 +1768 10.407
125.12 +2082 12.0513
125.13 +2355 14.6867
125.14 +2375 13.79
125.15 +2617 15.323
125.16 +2946 17.9003
125.17 +3499 21.046
125.18 +3567 21.276
125.19 +3774 25.3113
125.20 +3923 23.951
125.21 +4002 24.817
125.22 +4189 25.452
125.23 +4205 26.0877
125.24 +4214 26.499
125.25 +4414 26.8313
125.26 +4430 26.8737
125.27 +4481 27.4707
125.28 +4500 28.364
125.29 +4706 28.8917
125.30 +4861 29.77
125.31 +5162 31.4247
125.32 +5650 35.2547
125.33 +5724 35.5233
125.34 +5795 35.4867
125.35 +5938 37.4263
125.36 +5944 37.0227
125.37 +6006 38.4617
125.38 +6057 42.5057
125.39 +6462 41.0057
125.40 +6536 41.8393
125.41 +6668 43.6183
125.42 +6996 46.7797
125.43 +7594 49.0137
125.44 +7936 52.2893
125.45 +8716 56.329
125.46 +8871 55.0203
125.47 +9032 58.0967
125.48 +9142 58.6687
125.49 +9164 66.3373
125.50 +9435 58.8493
125.51 +9465 67.1867
125.52 +9792 69.378
126.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
126.2 +++ b/randGraph/ind/vf2ppInd5_0.05.txt Tue Nov 22 08:15:16 2016 +0100
126.3 @@ -0,0 +1,49 @@
126.4 +354 0.0166667
126.5 +446 0.022
126.6 +631 0.033
126.7 +668 0.033
126.8 +842 0.0453333
126.9 +1296 0.065
126.10 +1327 0.0686667
126.11 +1495 0.073
126.12 +1713 0.089
126.13 +2027 0.103
126.14 +2106 0.111667
126.15 +2901 0.144
126.16 +2967 0.152333
126.17 +3169 0.180333
126.18 +3279 0.189
126.19 +3320 0.176667
126.20 +3510 0.183
126.21 +4337 0.236333
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126.23 +4841 0.256333
126.24 +5020 0.272
126.25 +5111 0.274333
126.26 +5163 0.357333
126.27 +5359 0.331333
126.28 +5542 0.288333
126.29 +5547 0.279333
126.30 +5562 0.296667
126.31 +5631 0.298
126.32 +5877 0.299333
126.33 +6101 0.361333
126.34 +6295 0.339667
126.35 +6584 0.346667
126.36 +6594 0.573333
126.37 +6788 0.375333
126.38 +6849 0.406
126.39 +7173 0.384667
126.40 +7279 0.385333
126.41 +7361 0.722
126.42 +7384 0.412667
126.43 +7461 0.425
126.44 +7467 0.461
126.45 +8091 0.476667
126.46 +8451 0.506
126.47 +8827 0.451333
126.48 +9031 0.574
126.49 +9134 1.27267
126.50 +9639 0.583
126.51 +9720 0.682
126.52 +9943 0.648333
127.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
127.2 +++ b/randGraph/ind/vf2ppInd5_0.1.txt Tue Nov 22 08:15:16 2016 +0100
127.3 @@ -0,0 +1,48 @@
127.4 +70 0.006
127.5 +377 0.035
127.6 +390 0.038
127.7 +788 0.0723333
127.8 +1086 0.102333
127.9 +1922 0.177
127.10 +2057 0.193
127.11 +2071 0.196
127.12 +2169 0.227333
127.13 +2406 0.238667
127.14 +2958 0.366333
127.15 +3323 0.347333
127.16 +3631 0.4
127.17 +4224 0.624333
127.18 +4380 0.49
127.19 +4427 0.642
127.20 +4663 0.521667
127.21 +4942 0.742333
127.22 +5118 0.578
127.23 +5183 0.657667
127.24 +5204 0.573
127.25 +5368 0.611667
127.26 +5704 0.642
127.27 +5826 0.613667
127.28 +5859 0.588
127.29 +5925 0.634667
127.30 +5948 0.676667
127.31 +5961 0.786
127.32 +6052 0.672333
127.33 +6472 0.674667
127.34 +6871 0.838667
127.35 +7015 0.833
127.36 +7039 1.065
127.37 +7221 1.371
127.38 +7238 15.657
127.39 +7776 1.20033
127.40 +7806 0.818
127.41 +8478 2.17133
127.42 +8679 3.082
127.43 +8966 7.78167
127.44 +9251 1.44833
127.45 +9336 15.544
127.46 +9499 1.076
127.47 +9583 1.40233
127.48 +9585 58.2783
127.49 +9649 1.74167
127.50 +9816 4.50733
127.51 +9932 2.14333
128.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
128.2 +++ b/randGraph/ind/vf2ppInd5_0.3.txt Tue Nov 22 08:15:16 2016 +0100
128.3 @@ -0,0 +1,48 @@
128.4 +326 0.0933333
128.5 +399 0.118333
128.6 +476 0.136
128.7 +1069 0.345667
128.8 +1389 0.458667
128.9 +1415 0.462333
128.10 +1654 0.489
128.11 +1909 0.566333
128.12 +1961 0.579
128.13 +2638 0.995333
128.14 +3005 0.898667
128.15 +3045 1.058
128.16 +3144 1.033
128.17 +3229 1.057
128.18 +3275 1.29667
128.19 +3803 1.158
128.20 +4459 1.47133
128.21 +4816 1.68333
128.22 +4988 2.487
128.23 +5097 2.03067
128.24 +5424 2.01667
128.25 +5465 2.03633
128.26 +5634 1.9
128.27 +5645 2.21233
128.28 +6022 2.241
128.29 +6094 3.47567
128.30 +6159 2.819
128.31 +6291 2.505
128.32 +6361 6.54433
128.33 +6638 11.287
128.34 +6677 3.48567
128.35 +6838 3.951
128.36 +7043 3.713
128.37 +7531 2.717
128.38 +7566 2.86233
128.39 +7623 4.40867
128.40 +7647 4.656
128.41 +7822 4.36233
128.42 +7994 3.64
128.43 +8003 5.27433
128.44 +8475 6.34667
128.45 +8996 6.25967
128.46 +9118 4.21533
128.47 +9750 23.6907
128.48 +9753 5.349
128.49 +9787 5.91533
128.50 +9848 6.192
128.51 +9849 10.102
129.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
129.2 +++ b/randGraph/ind/vf2ppInd5_0.6.txt Tue Nov 22 08:15:16 2016 +0100
129.3 @@ -0,0 +1,50 @@
129.4 +379 0.226
129.5 +431 0.26
129.6 +665 0.398
129.7 +718 0.424667
129.8 +826 0.503333
129.9 +1251 0.784333
129.10 +1279 0.785667
129.11 +1443 0.904
129.12 +1500 0.946667
129.13 +2193 1.41733
129.14 +2231 1.445
129.15 +2280 1.49333
129.16 +2632 1.776
129.17 +2687 1.836
129.18 +2907 1.90867
129.19 +3053 2.04767
129.20 +3091 2.11767
129.21 +3437 2.47367
129.22 +3587 2.62867
129.23 +3976 2.693
129.24 +4702 3.98433
129.25 +4751 3.34633
129.26 +5349 4.24767
129.27 +5350 4.414
129.28 +5353 3.749
129.29 +5722 4.31667
129.30 +5764 4.34967
129.31 +5844 4.79067
129.32 +5946 4.254
129.33 +5969 4.63667
129.34 +5993 4.403
129.35 +6013 4.73267
129.36 +6323 4.629
129.37 +6883 5.247
129.38 +6917 6.433
129.39 +7218 5.329
129.40 +7383 5.45233
129.41 +7663 5.998
129.42 +8038 6.07867
129.43 +8119 5.91267
129.44 +8133 6.55067
129.45 +8165 6.58833
129.46 +8204 6.14433
129.47 +8694 6.402
129.48 +8805 7.31933
129.49 +9262 7.337
129.50 +9334 7.96067
129.51 +9569 8.64567
129.52 +9811 7.92667
129.53 +9876 9.43567
130.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
130.2 +++ b/randGraph/ind/vf2ppInd5_0.8.txt Tue Nov 22 08:15:16 2016 +0100
130.3 @@ -0,0 +1,50 @@
130.4 +22 0.016
130.5 +155 0.129667
130.6 +176 0.139667
130.7 +911 0.797333
130.8 +1065 0.919
130.9 +1107 0.971333
130.10 +1526 1.38433
130.11 +1623 1.472
130.12 +1782 1.65067
130.13 +1863 1.70633
130.14 +1911 1.76667
130.15 +2165 2.38267
130.16 +2192 2.04267
130.17 +2217 2.12933
130.18 +2247 2.09467
130.19 +3050 2.965
130.20 +3167 3.10467
130.21 +3398 3.517
130.22 +3627 3.529
130.23 +4092 4.08667
130.24 +4240 4.17667
130.25 +4346 4.381
130.26 +4615 4.61533
130.27 +4673 4.68533
130.28 +4708 4.75833
130.29 +4739 4.811
130.30 +4818 4.84
130.31 +5058 5.27267
130.32 +5222 5.36067
130.33 +5414 5.62467
130.34 +5868 6.06767
130.35 +5946 6.277
130.36 +6061 6.354
130.37 +6263 6.612
130.38 +6611 7.035
130.39 +6699 7.41967
130.40 +6799 7.291
130.41 +6915 7.52767
130.42 +7100 8.37267
130.43 +7136 7.69
130.44 +7218 7.85467
130.45 +7369 8.376
130.46 +7736 8.47367
130.47 +8219 8.76167
130.48 +8377 8.928
130.49 +8482 9.275
130.50 +8632 9.286
130.51 +8886 9.56033
130.52 +9351 10.1563
130.53 +9771 10.635
131.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
131.2 +++ b/randGraph/ind/vf2ppInd5_0.95.txt Tue Nov 22 08:15:16 2016 +0100
131.3 @@ -0,0 +1,49 @@
131.4 +20 0.017
131.5 +550 0.581333
131.6 +628 0.666667
131.7 +660 0.68
131.8 +1192 1.278
131.9 +1663 1.866
131.10 +1749 1.997
131.11 +1768 2.01533
131.12 +2082 2.39367
131.13 +2355 2.81433
131.14 +2375 2.87133
131.15 +2617 3.139
131.16 +2946 3.63033
131.17 +3499 4.28467
131.18 +3567 4.39733
131.19 +3774 4.69433
131.20 +3923 4.88467
131.21 +4002 4.96033
131.22 +4189 5.24467
131.23 +4205 5.207
131.24 +4214 5.22567
131.25 +4414 5.62767
131.26 +4430 5.663
131.27 +4481 5.681
131.28 +4500 5.72767
131.29 +4706 6.06867
131.30 +4861 6.368
131.31 +5162 6.70533
131.32 +5650 7.55067
131.33 +5724 7.42167
131.34 +5795 7.68333
131.35 +5938 7.843
131.36 +5944 7.83067
131.37 +6006 7.95533
131.38 +6057 8.11267
131.39 +6462 8.7
131.40 +6536 8.81
131.41 +6668 9.71067
131.42 +6996 9.32
131.43 +7594 10.5173
131.44 +7936 10.8687
131.45 +8716 11.93
131.46 +8871 12.0603
131.47 +9032 12.4133
131.48 +9142 12.7773
131.49 +9164 12.6763
131.50 +9435 12.9623
131.51 +9465 13.0667
131.52 +9792 13.6357
132.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
132.2 +++ b/randGraph/ind/vf2ppInd7_0.05.txt Tue Nov 22 08:15:16 2016 +0100
132.3 @@ -0,0 +1,30 @@
132.4 +78 0.00366667
132.5 +104 0.00533333
132.6 +132 0.00633333
132.7 +142 0.00866667
132.8 +173 0.00833333
132.9 +196 0.0103333
132.10 +218 0.0123333
132.11 +276 0.014
132.12 +293 0.0156667
132.13 +296 0.0143333
132.14 +366 0.0203333
132.15 +423 0.0233333
132.16 +458 0.0263333
132.17 +460 0.0263333
132.18 +486 0.0263333
132.19 +561 0.033
132.20 +575 0.032
132.21 +641 0.039
132.22 +792 0.0436667
132.23 +870 0.049
132.24 +922 0.0526667
132.25 +992 0.0726667
132.26 +1114 0.0606667
132.27 +1166 0.0656667
132.28 +1202 0.0696667
132.29 +1608 0.0873333
132.30 +1737 0.107333
132.31 +2022 0.123
132.32 +2837 0.154
132.33 +3271 0.196333
133.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
133.2 +++ b/randGraph/ind/vf2ppInd7_0.1.txt Tue Nov 22 08:15:16 2016 +0100
133.3 @@ -0,0 +1,30 @@
133.4 +47 0.00366667
133.5 +85 0.008
133.6 +150 0.0143333
133.7 +192 0.02
133.8 +200 0.02
133.9 +206 0.02
133.10 +273 0.0296667
133.11 +302 0.032
133.12 +387 0.0383333
133.13 +394 0.0386667
133.14 +536 0.0566667
133.15 +790 0.0783333
133.16 +816 0.0896667
133.17 +851 0.0913333
133.18 +925 0.0976667
133.19 +1085 0.140333
133.20 +1107 0.114
133.21 +1135 0.119333
133.22 +1460 0.164333
133.23 +1488 0.178333
133.24 +1954 0.235667
133.25 +2054 0.225
133.26 +2170 0.392
133.27 +2190 0.242333
133.28 +2253 1.291
133.29 +2257 0.244
133.30 +2311 0.34
133.31 +2390 0.256333
133.32 +2463 0.263
133.33 +3366 0.619
134.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
134.2 +++ b/randGraph/ind/vf2ppInd7_0.3.txt Tue Nov 22 08:15:16 2016 +0100
134.3 @@ -0,0 +1,28 @@
134.4 +20 0.00566667
134.5 +47 0.0143333
134.6 +65 0.02
134.7 +82 0.0273333
134.8 +85 0.0256667
134.9 +101 0.0316667
134.10 +141 0.0433333
134.11 +213 0.069
134.12 +228 0.071
134.13 +272 0.09
134.14 +277 0.0963333
134.15 +355 0.124
134.16 +491 0.160667
134.17 +587 0.187
134.18 +654 0.223
134.19 +698 0.231333
134.20 +716 0.241
134.21 +763 0.272
134.22 +851 0.289333
134.23 +1032 0.357
134.24 +1095 0.358333
134.25 +1603 0.537333
134.26 +1969 0.708667
134.27 +2176 0.757333
134.28 +2412 1.07333
134.29 +2587 1.00167
134.30 +2958 1.048
134.31 +4266 1.70767
135.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
135.2 +++ b/randGraph/ind/vf2ppInd7_0.6.txt Tue Nov 22 08:15:16 2016 +0100
135.3 @@ -0,0 +1,29 @@
135.4 +20 0.015
135.5 +25 0.0166667
135.6 +43 0.033
135.7 +56 0.0423333
135.8 +227 0.162667
135.9 +397 0.290333
135.10 +448 0.333333
135.11 +474 0.351
135.12 +487 0.353667
135.13 +509 0.369667
135.14 +630 0.456333
135.15 +727 0.545
135.16 +732 0.553333
135.17 +777 0.593333
135.18 +826 0.62
135.19 +899 0.666333
135.20 +1034 0.788
135.21 +1078 0.797667
135.22 +1107 0.845333
135.23 +1146 0.889333
135.24 +1282 1.01867
135.25 +1393 1.06933
135.26 +1410 1.108
135.27 +1484 1.154
135.28 +1600 1.25333
135.29 +2053 1.76267
135.30 +2076 1.62967
135.31 +2201 1.75733
135.32 +2813 2.30467
136.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
136.2 +++ b/randGraph/ind/vf2ppInd7_0.8.txt Tue Nov 22 08:15:16 2016 +0100
136.3 @@ -0,0 +1,30 @@
136.4 +37 0.0366667
136.5 +70 0.0686667
136.6 +157 0.164333
136.7 +199 0.208667
136.8 +271 0.276667
136.9 +376 0.403
136.10 +379 0.399667
136.11 +380 0.404
136.12 +522 0.564333
136.13 +554 0.597667
136.14 +595 0.658667
136.15 +634 0.67
136.16 +718 0.777
136.17 +721 0.789333
136.18 +730 0.787
136.19 +928 1.02833
136.20 +1046 1.168
136.21 +1088 1.19933
136.22 +1156 1.28133
136.23 +1357 1.53
136.24 +1443 1.62433
136.25 +1564 1.788
136.26 +1579 1.79267
136.27 +1780 2.067
136.28 +1860 2.16867
136.29 +1876 2.22433
136.30 +2079 2.429
136.31 +2866 3.45133
136.32 +3318 4.071
136.33 +4068 5.20433
137.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
137.2 +++ b/randGraph/ind/vf2ppInd7_0.95.txt Tue Nov 22 08:15:16 2016 +0100
137.3 @@ -0,0 +1,28 @@
137.4 +20 0.029
137.5 +59 0.0793333
137.6 +80 0.106
137.7 +108 0.137667
137.8 +166 0.235
137.9 +176 0.238667
137.10 +222 0.303333
137.11 +444 0.61
137.12 +624 0.854333
137.13 +631 0.857
137.14 +669 0.906333
137.15 +757 1.02633
137.16 +876 1.198
137.17 +891 1.20467
137.18 +1124 1.565
137.19 +1193 1.66233
137.20 +1217 1.75233
137.21 +1345 1.89033
137.22 +1449 2.063
137.23 +1603 2.35167
137.24 +1655 2.424
137.25 +2079 3.209
137.26 +2129 3.17833
137.27 +2567 3.89
137.28 +2716 4.233
137.29 +3103 4.88267
137.30 +3657 5.85833
137.31 +4945 8.292
138.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
138.2 +++ b/randGraph/iso/vf2pIso.txt Tue Nov 22 08:15:16 2016 +0100
138.3 @@ -0,0 +1,149 @@
138.4 +127 0.3269
138.5 +162 0.432
138.6 +240 0.7383
138.7 +376 1.2215
138.8 +425 1.4418
138.9 +491 1.7892
138.10 +653 2.6752
138.11 +690 2.954
138.12 +896 4.3193
138.13 +985 4.9454
138.14 +1003 5.0853
138.15 +1030 5.3009
138.16 +1132 6.187
138.17 +1328 7.918
138.18 +1942 14.7931
138.19 +2226 19.1137
138.20 +2472 22.4367
138.21 +2804 28.1592
138.22 +2818 28.1566
138.23 +2907 30.4856
138.24 +2930 30.6408
138.25 +3076 34.0425
138.26 +3200 35.2525
138.27 +3364 39.1791
138.28 +3467 41.12
138.29 +3544 42.2586
138.30 +3613 44.5656
138.31 +3704 45.6906
138.32 +3946 51.1882
138.33 +4023 53.1333
138.34 +4076 55.1514
138.35 +4145 56.5863
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138.37 +4426 64.6284
138.38 +4644 71.1045
138.39 +4767 73.4162
138.40 +5112 81.686
138.41 +5204 84.8276
138.42 +5243 88.8607
138.43 +5263 87.1636
138.44 +5384 90.1785
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138.47 +5653 101.436
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138.51 +6014 110.53
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139.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
139.2 +++ b/randGraph/iso/vf2pIso100_1.txt Tue Nov 22 08:15:16 2016 +0100
139.3 @@ -0,0 +1,150 @@
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140.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
140.2 +++ b/randGraph/iso/vf2pIso10_1.txt Tue Nov 22 08:15:16 2016 +0100
140.3 @@ -0,0 +1,150 @@
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141.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
141.2 +++ b/randGraph/iso/vf2pIso15_1.txt Tue Nov 22 08:15:16 2016 +0100
141.3 @@ -0,0 +1,150 @@
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142.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
142.2 +++ b/randGraph/iso/vf2pIso30_1.txt Tue Nov 22 08:15:16 2016 +0100
142.3 @@ -0,0 +1,78 @@
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143.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
143.2 +++ b/randGraph/iso/vf2pIso35_1.txt Tue Nov 22 08:15:16 2016 +0100
143.3 @@ -0,0 +1,149 @@
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144.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
144.2 +++ b/randGraph/iso/vf2pIso45_1.txt Tue Nov 22 08:15:16 2016 +0100
144.3 @@ -0,0 +1,150 @@
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145.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
145.2 +++ b/randGraph/iso/vf2pIso5_1.txt Tue Nov 22 08:15:16 2016 +0100
145.3 @@ -0,0 +1,149 @@
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146.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
146.2 +++ b/randGraph/iso/vf2pIso7_1.txt Tue Nov 22 08:15:16 2016 +0100
146.3 @@ -0,0 +1,30 @@
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147.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
147.2 +++ b/randGraph/iso/vf2pIso80_1.txt Tue Nov 22 08:15:16 2016 +0100
147.3 @@ -0,0 +1,30 @@
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148.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
148.2 +++ b/randGraph/iso/vf2ppIso.txt Tue Nov 22 08:15:16 2016 +0100
148.3 @@ -0,0 +1,149 @@
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149.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
149.2 +++ b/randGraph/iso/vf2ppIso100_1.txt Tue Nov 22 08:15:16 2016 +0100
149.3 @@ -0,0 +1,150 @@
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150.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
150.2 +++ b/randGraph/iso/vf2ppIso10_1.txt Tue Nov 22 08:15:16 2016 +0100
150.3 @@ -0,0 +1,150 @@
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151.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
151.2 +++ b/randGraph/iso/vf2ppIso15_1.txt Tue Nov 22 08:15:16 2016 +0100
151.3 @@ -0,0 +1,150 @@
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152.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
152.2 +++ b/randGraph/iso/vf2ppIso30_1.txt Tue Nov 22 08:15:16 2016 +0100
152.3 @@ -0,0 +1,78 @@
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153.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
153.2 +++ b/randGraph/iso/vf2ppIso35_1.txt Tue Nov 22 08:15:16 2016 +0100
153.3 @@ -0,0 +1,149 @@
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154.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
154.2 +++ b/randGraph/iso/vf2ppIso45_1.txt Tue Nov 22 08:15:16 2016 +0100
154.3 @@ -0,0 +1,150 @@
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155.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
155.2 +++ b/randGraph/iso/vf2ppIso5_1.txt Tue Nov 22 08:15:16 2016 +0100
155.3 @@ -0,0 +1,149 @@
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156.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
156.2 +++ b/randGraph/iso/vf2ppIso7_1.txt Tue Nov 22 08:15:16 2016 +0100
156.3 @@ -0,0 +1,30 @@
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157.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
157.2 +++ b/randGraph/iso/vf2ppIso80_1.txt Tue Nov 22 08:15:16 2016 +0100
157.3 @@ -0,0 +1,30 @@
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