This Maple 10 worksheet demonstrates some of the functionality and design of the new GraphTheory package being developed by people at SFU. It was created by Mahdad Khatarinejad and presented by Mahdad at the Maple company premises on Wednesday July 20th, 2005. The content were subsequently adjusted somewhat by Mike so that the reader can infer what the purpose of some of the examples are.
This version of the worksheet has been updated by Michael Monagan in June 2006 to work with GTv24 - version 1.24 of the GraphTheory package.
This version of the worksheet has been updated by Michael Monagan in August 2006 to work with GTv26 - version 1.26 of the GraphTheory package
GraphTheory vs networks
The networks package
over 10 years old
designed primarily with applications in networks in mind
data structure is too heavy and cumbersome for using in elementary graph theory problems
The GraphTheory package (main goals)
design primarily for solving problems related to graphs rather than networks
keep data structure simple, yet flexible, able to treat graphs of 1000 vertices
visualization of graphs is important
Some timing comparisons
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 : Random graph on 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 vertices and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEhRicvJSdpdGFsaWNHUSV0cnVlRicvJStmb3JlZ3JvdW5kR1EqWzI1NSwwLDBdRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiVGKy1GIzYkLUkjbW5HRiQ2JVEiM0YnRjIvRjZRJ25vcm1hbEYnLUYsNiZRIm5GJ0YvRjJGNUYrRis= edges
n
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
250
0.009
0.05
500
0.009
0.18
1000
0.069
0.201
2000
0.06
0.479
4000
0.139
1.211
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 : Random graph on 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 vertices and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEhRicvJSdpdGFsaWNHUSV0cnVlRicvJStmb3JlZ3JvdW5kR1EqWzI1NSwwLDBdRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiVGKy1GIzYkLUkjbW5HRiQ2JVEiM0YnRjIvRjZRJ25vcm1hbEYnLUYsNiZRIm5GJ0YvRjJGNUYrRis= edges
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 : Half of the vertices of 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
n
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
250
0.0
0.139
500
0.009
0.521
1000
0.01
0.1072
2000
0.019
3.163
4000
0.05
9.274
n
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
20
0.0
0.46
40
0.0
12.22
60
0.0
104.12
1000
0.048
?
2000
0.088
?
4000
0.219
?
8000
0.503
?
Commands and Features
Constructing graphs ....
C1 := CompleteGraph([1,2,3,4]);
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C2 := CompleteGraph([5,6,7,8]);
LUkoR1JBUEhMTkclKnByb3RlY3RlZEc2KEkrdW5kaXJlY3RlZEdGJEkrdW53ZWlnaHRlZEdGJDcmIiImIiInIiIoIiIpLUkmQXJyYXlHRiQ2Iy9JJCVpZEc2IiIpOyMzMCJJMUdSQVBITE4vdGFibGUvMTBHRjIiIiE=
C := DisjointUnion(C1,C2);
LUkoR1JBUEhMTkclKnByb3RlY3RlZEc2KEkrdW5kaXJlY3RlZEdGJEkrdW53ZWlnaHRlZEdGJDcqIiIiIiIjIiIkIiIlIiImIiInIiIoIiIpLUkmQXJyYXlHRiQ2Iy9JJCVpZEc2IiIpVyJcOyJJMUdSQVBITE4vdGFibGUvMTJHRjYiIiE=
AddEdge(C, {4,5});
LUkoR1JBUEhMTkclKnByb3RlY3RlZEc2KEkrdW5kaXJlY3RlZEdGJEkrdW53ZWlnaHRlZEdGJDcqIiIiIiIjIiIkIiIlIiImIiInIiIoIiIpLUkmQXJyYXlHRiQ2Iy9JJCVpZEc2IiIpVyJcOyJJMUdSQVBITE4vdGFibGUvMTJHRjYiIiE=
DrawGraph(C);
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
IsBiconnected(C);
SSZmYWxzZUclKnByb3RlY3RlZEc=
BiconnectedComponents(C);
NyU3JCIiJSIiJjcmIiIiIiIjIiIkRiQ3JkYlIiInIiIoIiIp
Problem: Find a Spanning Tree and the Spanning Polynomial of a given connected graph
G := RandomGraph(12,.3):
while ChromaticNumber(G)=1 or not(IsConnected(G))
do G := RandomGraph(12,.3): od:
DrawGraph(SpanningTree(G));
NjwtSSlQT0xZR09OU0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYwNyY3JCQiKysrK19lISM1JCIrKytTRTUhIio3JCQiKysrK19lISM1JCIrKysrTygqISM1NyQkIisrKytbXiEjNSQiKysrK08oKiEjNTckJCIrKysrW14hIzUkIisrK1NFNSEiKjcmNyQkIisrKytfOCEjNSQiKysrK2t4ISM1NyQkIisrKytfOCEjNSQiKysrK09zISM1NyQkIissKyshWychIzYkIisrKytPcyEjNTckJCIrLCsrIVsnISM2JCIrKysra3ghIzU3JjckJCIrKysrX0IhIzUkIisrKytrRiEjNTckJCIrKysrX0IhIzUkIisrKytPQSEjNTckJCIrKysrWzshIzUkIisrKytPQSEjNTckJCIrKysrWzshIzUkIisrKytrRiEjNTcmNyQkIisrKytfQiEjNSQiKysrK1NFISM2NyQkIisrKytfQiEjNSQhKysrK1NFISM2NyQkIisrKytbOyEjNSQhKysrK1NFISM2NyQkIisrKytbOyEjNSQiKysrK1NFISM2NyY3JCQiKysrK18kKSEjNSQiKysrK2tGISM1NyQkIisrKytfJCkhIzUkIisrKytPQSEjNTckJCIrKysrW3chIzUkIisrKytPQSEjNTckJCIrKysrW3chIzUkIisrKytrRiEjNTcmNyQkIisrKytfViEjNSQiKysrK2t4ISM1NyQkIisrKytfViEjNSQiKysrK09zISM1NyQkIisrKytbTyEjNSQiKysrK09zISM1NyQkIisrKytbTyEjNSQiKysrK2t4ISM1NyY3JCQiKysrK19qISM1JCIrKysra18hIzU3JCQiKysrK19qISM1JCIrKysrT1ohIzU3JCQiKysrK1tjISM1JCIrKysrT1ohIzU3JCQiKysrK1tjISM1JCIrKysra18hIzU3JjckJCIrKioqKioqPk4hIzYkIisrKytrXyEjNTckJCIrKioqKioqPk4hIzYkIisrKytPWiEjNTckJCErKioqKioqPk4hIzYkIisrKytPWiEjNTckJCErKioqKioqPk4hIzYkIisrKytrXyEjNTcmNyQkIisrKytfQiEjNSQiKysrK2tfISM1NyQkIisrKytfQiEjNSQiKysrK09aISM1NyQkIisrKytbOyEjNSQiKysrK09aISM1NyQkIisrKytbOyEjNSQiKysrK2tfISM1NyY3JCQiKysrK18kKSEjNSQiKysrK2tfISM1NyQkIisrKytfJCkhIzUkIisrKytPWiEjNTckJCIrKysrW3chIzUkIisrKytPWiEjNTckJCIrKysrW3chIzUkIisrKytrXyEjNTcmNyQkIisrKytfdCEjNSQiKysrK2t4ISM1NyQkIisrKytfdCEjNSQiKysrK09zISM1NyQkIisrKytbbSEjNSQiKysrK09zISM1NyQkIisrKytbbSEjNSQiKysrK2t4ISM1NyY3JCQiKysrP041ISIqJCIrKysra3ghIzU3JCQiKysrP041ISIqJCIrKysrT3MhIzU3JCQiKysrK1snKiEjNSQiKysrK09zISM1NyQkIisrKytbJyohIzUkIisrKytreCEjNS1JJkNPTE9SRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNkdJJFJHQkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IiIiIiIiIiQiIiMhIiIiIiIiIiIkIiIjISIiIiIiIiIiJCIiIyEiIiIiIiIiIiQiIiMhIiIiIiIiIiIkIiIjISIiIiIiIiIiJCIiIyEiIiIiIiIiIiQiIiMhIiIiIiIiIiIkIiIjISIiIiIiIiIiJCIiIyEiIiIiIiIiIiQiIiMhIiIiIiIiIiIkIiIjISIiIiIiIiIiJCIiIyEiIi1JJlNUWUxFRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJLFBBVENITk9HUklERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiLUklVEVYVEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYlNyQkIisrKysrYiEjNSQiIiIiIiFRIjE2Ii1JJUZPTlRHNiI2JUkqSEVMVkVUSUNBRzYiSSVCT0xERzYiIiM2LUklVEVYVEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYlNyQkIisrKysrNSEjNSQiKysrKyt2ISM1USIyNiItSSVGT05URzYiNiVJKkhFTFZFVElDQUc2IkklQk9MREc2IiIjNi1JJVRFWFRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JTckJCIrKysrKz8hIzUkIisrKysrRCEjNVEiMzYiLUklRk9OVEc2IjYlSSpIRUxWRVRJQ0FHNiJJJUJPTERHNiIiIzYtSSVURVhURzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiU3JCQiKysrKys/ISM1JCIiISIiIVEiNDYiLUklRk9OVEc2IjYlSSpIRUxWRVRJQ0FHNiJJJUJPTERHNiIiIzYtSSVURVhURzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiU3JCQiKysrKyshKSEjNSQiKysrKytEISM1USI1NiItSSVGT05URzYiNiVJKkhFTFZFVElDQUc2IkklQk9MREc2IiIjNi1JJVRFWFRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JTckJCIrKysrK1MhIzUkIisrKysrdiEjNVEiNjYiLUklRk9OVEc2IjYlSSpIRUxWRVRJQ0FHNiJJJUJPTERHNiIiIzYtSSVURVhURzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiU3JCQiKysrKytnISM1JCIrKysrK10hIzVRIjc2Ii1JJUZPTlRHNiI2JUkqSEVMVkVUSUNBRzYiSSVCT0xERzYiIiM2LUklVEVYVEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYlNyQkIiIhIiIhJCIrKysrK10hIzVRIjg2Ii1JJUZPTlRHNiI2JUkqSEVMVkVUSUNBRzYiSSVCT0xERzYiIiM2LUklVEVYVEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYlNyQkIisrKysrPyEjNSQiKysrKytdISM1USI5NiItSSVGT05URzYiNiVJKkhFTFZFVElDQUc2IkklQk9MREc2IiIjNi1JJVRFWFRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JTckJCIrKysrKyEpISM1JCIrKysrK10hIzVRIzEwNiItSSVGT05URzYiNiVJKkhFTFZFVElDQUc2IkklQk9MREc2IiIjNi1JJVRFWFRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JTckJCIrKysrK3EhIzUkIisrKysrdiEjNVEjMTE2Ii1JJUZPTlRHNiI2JUkqSEVMVkVUSUNBRzYiSSVCT0xERzYiIiM2LUklVEVYVEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYlNyQkIiIiIiIhJCIrKysrK3YhIzVRIzEyNiItSSVGT05URzYiNiVJKkhFTFZFVElDQUc2IkklQk9MREc2IiIjNi1JKVBPTFlHT05TRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiY3JDckJCIrKysrKzUhIzUkIisrKysrdiEjNTckJCIrKysrK2IhIzUkIiIiIiIhLUkmQ09MT1JHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JkkkUkdCRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiIiIhIiIhIiIiLUkqVEhJQ0tORVNTRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMiIiMtSSZTVFlMRUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSVMSU5FRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiLUkpUE9MWUdPTlNHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JjckNyQkIisrKysrPyEjNSQiIiEiIiE3JCQiKysrKys/ISM1JCIrKysrK0QhIzUtSSZDT0xPUkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYmSSRSR0JHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiIiIiEiIiEiIiItSSpUSElDS05FU1NHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyIiIy1JJlNUWUxFRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJJUxJTkVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiItSSlQT0xZR09OU0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYmNyQ3JCQiKysrKytTISM1JCIrKysrK3YhIzU3JCQiKysrKytiISM1JCIiIiIiIS1JJkNPTE9SRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiZJJFJHQkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IiIiISIiISIiIi1JKlRISUNLTkVTU0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjIiIjLUkmU1RZTEVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0klTElORUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2Ii1JKVBPTFlHT05TRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiY3JDckJCIiISIiISQiKysrKytdISM1NyQkIisrKysrNSEjNSQiKysrKyt2ISM1LUkmQ09MT1JHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JkkkUkdCRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiIiIhIiIhIiIiLUkqVEhJQ0tORVNTRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMiIiMtSSZTVFlMRUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSVMSU5FRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiLUkpUE9MWUdPTlNHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JjckNyQkIisrKysrPyEjNSQiKysrKytdISM1NyQkIisrKysrNSEjNSQiKysrKyt2ISM1LUkmQ09MT1JHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JkkkUkdCRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiIiIhIiIhIiIiLUkqVEhJQ0tORVNTRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMiIiMtSSZTVFlMRUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSVMSU5FRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiLUkpUE9MWUdPTlNHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JjckNyQkIisrKysrPyEjNSQiKysrKytdISM1NyQkIisrKysrPyEjNSQiKysrKytEISM1LUkmQ09MT1JHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JkkkUkdCRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiIiIhIiIhIiIiLUkqVEhJQ0tORVNTRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMiIiMtSSZTVFlMRUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSVMSU5FRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiLUkpUE9MWUdPTlNHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JjckNyQkIisrKysrISkhIzUkIisrKysrXSEjNTckJCIrKysrKyEpISM1JCIrKysrK0QhIzUtSSZDT0xPUkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYmSSRSR0JHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiIiIiEiIiEiIiItSSpUSElDS05FU1NHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyIiIy1JJlNUWUxFRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJJUxJTkVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiItSSlQT0xZR09OU0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYmNyQ3JCQiKysrKytxISM1JCIrKysrK3YhIzU3JCQiKysrKytiISM1JCIiIiIiIS1JJkNPTE9SRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiZJJFJHQkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IiIiISIiISIiIi1JKlRISUNLTkVTU0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjIiIjLUkmU1RZTEVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0klTElORUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2Ii1JKVBPTFlHT05TRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiY3JDckJCIrKysrK3EhIzUkIisrKysrdiEjNTckJCIrKysrK2chIzUkIisrKysrXSEjNS1JJkNPTE9SRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiZJJFJHQkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IiIiISIiISIiIi1JKlRISUNLTkVTU0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjIiIjLUkmU1RZTEVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0klTElORUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2Ii1JKVBPTFlHT05TRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiY3JDckJCIrKysrK3EhIzUkIisrKysrdiEjNTckJCIrKysrKyEpISM1JCIrKysrK10hIzUtSSZDT0xPUkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYmSSRSR0JHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiIiIiEiIiEiIiItSSpUSElDS05FU1NHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyIiIy1JJlNUWUxFRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJJUxJTkVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiItSSlQT0xZR09OU0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYmNyQ3JCQiIiIiIiEkIisrKysrdiEjNTckJCIrKysrK2IhIzUkIiIiIiIhLUkmQ09MT1JHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JkkkUkdCRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiIiIhIiIhIiIiLUkqVEhJQ0tORVNTRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMiIiMtSSZTVFlMRUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSVMSU5FRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiLUkoU0NBTElOR0c2IjYjSSxDT05TVFJBSU5FREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2Ii1JKkFYRVNTVFlMRUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSVOT05FRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYi
f := SpanningPolynomial(G, p);
LDAqJEkicEc2IiIjNiIlLDoqJEYkIiM3ISUmNCgqJEYkIiM4IiZsVCIqJEYkIiM5ISZsXyIqJEYkIiM6IiVgJCoqJEYkIiM7ISUnMyQqJEYkIiM8IiRHJQ==
eval(f, p=.5);
JCIrLTBXd1IhIzY=
Problem: Find the maximum clique (maximum independent set) in a graph.
Algorithm: Backtracking (Branch & Bound)
MaximumIndependentSet(G);
NygiIiMiIiQiIiYiIiciIigiIzc=
MaximumClique(G);
NyUiIiMiIikiIio=
ChromaticNumber(G,bound);
OyIiJCIiJQ==
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiMtRiw2J1EibkYnLyUlYm9sZEdGMUYvL0YzUSxib2xkLWl0YWxpY0YnLyUrZm9udHdlaWdodEdRJWJvbGRGJy1GLDYnRi5GOkYvRjxGPg== : Number of vertices of G
\316\261 : Independence number of G (maximum size of an independent set)
\317\207 : Chromatic number of G
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiMtRiw2J1EiZ0YnLyUlYm9sZEdGMUYvL0YzUSxib2xkLWl0YWxpY0YnLyUrZm9udHdlaWdodEdRJWJvbGRGJy1GLDYnRi5GOkYvRjxGPg== : Number of colors in a greedy coloring of G
Then LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiVGKy1JJm1mcmFjR0YkNigtRiM2Iy1GLDYlUSJuRidGL0YyLUYjNiRGKy1GLDYlUSgmYWxwaGE7RicvRjBRJmZhbHNlRicvRjNRJ25vcm1hbEYnLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZNLyUpYmV2ZWxsZWRHRkVGK0Yr \342\251\275 \317\207 \342\251\275 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== .
Problem: Proper (circular) coloring of vertices or edges of a graph
Algorithm: Backtracking
ChromaticNumber(G);
IiIk
Pet := PetersenGraph():
DrawGraph(Pet);
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EdgeChromaticNumber(Pet, 'ColorClasses');
IiIl
ColorClasses;
NyY8JjwkIiIiIiIjPCQiIioiIzU8JCIiKCIiKTwkIiIkIiIlPCY8JEYvRik8JEYlIiImPCRGJkYoPCQiIidGKzwmPCRGJUY2PCRGLkYrPCRGL0YzPCRGLEYoPCU8JEYzRiw8JEY2Rik8JEYmRi4=
HighlightEdges(Pet,ColorClasses[1],red);
HighlightEdges(Pet,ColorClasses[2],green);
HighlightEdges(Pet,ColorClasses[3],magenta);
DrawGraph(Pet);
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Problem: Find the max flow of a given network.
Algorithm: Goldberg Push-Preflow
Complexity: O( 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 )
V := [s,v1,v2,v3,v4,t];
NyhJInNHNiJJI3YxR0YkSSN2MkdGJEkjdjNHRiRJI3Y0R0YkSSJ0R0Yk
N := Digraph(V, {
[[s,v1],16],[[s,v2],13],
[[v1,v2],10],[[v2,v1],4],
[[v1,v3],12],[[v3,v2],9],
[[v2,v4],14],[[v4,v3],7],
[[v3,t],20],[[v4,t],4] } ):
Vertex_Positions := [[0,0.35],[0.35,0.7],[0.35,0],[0.7,0.7],[0.7,0],[1.05,0.35]]:
SetVertexPositions(N,Vertex_Positions);
DrawGraph(N);
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MaxFlow(N,s,t);
NiQiI0ItSSdNYXRyaXhHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2Iy9JJCVpZEdGKSIpJ3B2VyI=
Problem: Given a graph, test whether or not the graph is planar. If yes, return the plane embedding.
Algorithm: Demoucron, Malgrange, Pertuiset
IsPlanar(DeleteEdge(CompleteGraph(5),{1,2}));
SSV0cnVlRyUqcHJvdGVjdGVkRw==
IsPlanar(PetersenGraph());
SSZmYWxzZUclKnByb3RlY3RlZEc=
IsPlanar(DodecahedronGraph(), 'Faces');
SSV0cnVlRyUqcHJvdGVjdGVkRw==
Faces;
Ny43JyIjPCIjNyIiKSIjOCIjPTcnIiM1IiImIiIiIiInIiM6NydGJiIiJCIiJSIiKkYnNydGJ0YyIiM5IiM+Rig3JyIiKCIiI0YwRiZGJTcnIiM7IiM/Ri5GLSIjNjcnRitGMUYwRjhGLDcnRixGOEY3RjxGLTcnRjJGMUYrRipGNDcnRjxGN0YlRiRGOjcnRjVGNEYqRi5GOzcnRjtGOkYkRihGNQ==
Problem: Find the minimum cost trip of a salesman that must visit n cities and return home.
Algorithm: Backtracking (Branch & Bound)
M := LinearAlgebra[RandomMatrix](11,11,generator=rand(100)):
for i to 10 do M[i,i] := infinity od:
G11:=CompleteGraph(11):
s:=time(): TravelingSalesman(G11,M); time()-s;
NiQiJCU+Ny4iIiIiIioiIikiIiYiIzUiIigiIiciIiMiIiUiIiQiIzZGJQ==
JCIkTCUhIiQ=
Problem: Construct a random graph R, sort it's vertices into decreasing order by degree and plot the adjacency matrix to inspect the density of the graph.
R := RandomGraph(100,.5);
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
sigma := sort([op(Vertices(R))],proc(i,j) evalb(Degree(R,i) > Degree(R,j)) end proc);
N2BxIiNTIiMnKiIjeiIjcCIjbSIjKCoiIyYpIiNjIiNiIiNhIiNbIiNQIiImIiMmKiIjIioiIycpIiN0IiNeIiNSIiNJIiM7IiN1IiNfIiNWIiQrIiIjISkiI2wiI2YiI2UiI10iI1ciI0IiIzgiIigiIyEqIiMpKSIjdyIjcyIjaiIjZyIjWiIjVSIjSyIjSCIjPiIjOiIjJCoiIyQpIiMjKSIjIikiI3IiI3EiI2giI1kiI0wiIzYiIiUiI3YiI24iI1EiIiMiIyoqIiMqKSIjRCIjQyIjPyIiKSIiJyIiJCIiIiIjbyIjWCIjVCIjSiIjRSIjQSIiKiIjKCkiI3giI2kiI08iI04iI0YiIzwiIzkiIzUiI3kiI2AiI00iI0ciIyUpIiNrIiNkIiM9IiMlKiIjNyIjXCIjIyoiIykqIiNA
plots[matrixplot](AdjacencyMatrix(PermuteVertices(R, sigma)),
orientation=[0,0], shading=Z, style=PATCHNOGRID);
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The above plot converges to the following plot when n approaches infinity, where n is the number of vertices of the random graph in which every edge is present with probability .5:
plot3d(1-max(x,y), x=0..1, y=0..1,
orientation=[0,0], shading=Z, style=PATCHNOGRID);
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