Graph Theory (Networks)
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Transcript of Graph Theory (Networks)
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Copyright © 2014, 2010, 2007 Pearson Education, Inc.Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 4.1, Slide 1
4 Graph Theory (Networks)
The Mathematics of Relationships
4
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Copyright © 2014, 2010, 2007 Pearson Education, Inc.Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 4.1, Slide 2
Graphs, Puzzles, and Map Coloring
4.1
• Understand graph terminology
• Apply Euler’s theorem to graph tracing
• Understand when to use graphs as models
(continued on next slide)
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Copyright © 2014, 2010, 2007 Pearson Education, Inc.Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 4.1, Slide 3
Graphs, Puzzles, and Map Coloring
4.1
• Use Fleury’s theorem to find Euler circuits
• Utilize graph coloring to simplify a problem
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Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 4.1, Slide 4
Graph Terminology
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Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 4.1, Slide 5
Examples of Graphs
http://oak.cats.ohiou.edu/~ridgely/Volvo_docs/
http://maps.google.com/?ie=UTF8&ll=57.703528,11.966429&spn=0.017449,0.036993&z=15
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Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 4.1, Slide 6
Graph Terminology
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Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 4.1, Slide 7
Graph Tracing
• The Koenigsberg bridge problem
Starting at some point, can you visit all parts of the city, crossing each bridge once and only once, and return to the starting point?
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Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 4.1, Slide 8
Graph Tracing
• We can model the problem with a graph model.
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Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 4.1, Slide 9
Graph Tracing
• The Koenigsberg problem, phased in graph theory language, is “Can the graph be traced?”
To trace a graph means to begin at some vertex and draw the entire graph without lifting the pencil and without going over any edge more than once.
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Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 4.1, Slide 10
Graph Tracing
*Connected graphs are also called networks.
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Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 4.1, Slide 11
Graph Tracing
• Example:
(solution on next slide)
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Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 4.1, Slide 12
Graph Tracing
• Example:
Odd: B and C
Even: A, D, E, and F
(Zero edges is considered “even”)
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Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 4.1, Slide 13
Euler’s Theorem
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Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 4.1, Slide 14
Euler’s Theorem
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Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 4.1, Slide 15
Euler’s Theorem
• Example: Which of the graphs can be traced?
(solution on next slide)
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Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 4.1, Slide 16
Euler’s Theorem
• Solution:
Not trace-able
Trace-able
Trace-ableTrace-able
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Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 4.1, Slide 17
Euler’s Theorem
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Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 4.1, Slide 18
Euler’s Theorem
• Example:
• Solution:Path ACEB has length 3.
Path ACEBDA is an Euler path of length 5. It is also an Euler circuit.
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Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 4.1, Slide 19
Fleury’s Algorithm
• We use Fleury’s algorithm to find Euler circuits.
(example on next slide)
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Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 4.1, Slide 20
Fleury’s Algorithm
• Example:
(solution on next 3 slides)
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Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 4.1, Slide 21
Fleury’s Algorithm
• Solution: (answers may vary)
(continued on next slide)
erase
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Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 4.1, Slide 22
Fleury’s Algorithm
• Solution: (answers may vary)
(continued on next slide)
erase
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Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 4.1, Slide 23
Fleury’s Algorithm
• Solution: (answers may vary)
erase
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Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 4.1, Slide 24
Eulerizing a Graph
• We can add edges to convert a non-Eulerian graph to an Eulerian graph.
• This technique is called Eulerizing a graph.
(example on next slide)
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Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 4.1, Slide 25
Eulerizing a Graph
• Example:
(solution on next slide)
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Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 4.1, Slide 26
Eulerizing a Graph
• Solution: (answers may vary)
Eulerize
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Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 4.1, Slide 27
Map Coloring
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Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 4.1, Slide 28
Map Coloring
• Example:
(continued on next slide)
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Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 4.1, Slide 29
Map Coloring
• Solution:
(continued on next slide)
We can rephrase the map-coloring question now as follows: Using four or fewer colors, can we color the vertices so that no two vertices of the same edge receive the same color?
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Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 4.1, Slide 30
Map Coloring
• Solution:
There is no particular method for solving the problem.
A trial-and-error is shown.
Other solutions are possible.
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Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 4.1, Slide 31
Map Coloring
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Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 4.1, Slide 32
Map Coloring
• Example:Each member of a city council usually serves on several committees in city government. Assume council members serve on the following committees: police, parks, sanitation, finance, development, streets, fire department, and public relations. Use the table below to determine a conflict-free schedule for the meetings. We do not duplicate information - that is, because police conflicts with fire department, we do not also list that fire department conflicts with police.
(solution on next slide)
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Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 4.1, Slide 33
Map Coloring
• Solution:
Model the information with a graph. Join conflicting committees with an edge.
(continued on next slide)
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Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 4.1, Slide 34
Map Coloring
• Solution:
Using trial and error and four colors or less, color committees that can meet at the same time with a common color. That is, similar-colored vertices should not share a common edge (same problem as the four-color problem).