Ch. 15: Graph Theory Some practical uses

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Ch. 15: Graph Theory Some practical uses. Degree of separation- Hollywood, acquaintance, collaboration Travel between cities Konigsberg bridge Shortest path Least cost Schedule exams, assign channels, rooms Number of colors on a map Highway inspecting, snow removal, street sweeping

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Ch. 15: Graph TheorySome practical uses
• Degree of separation- Hollywood, acquaintance, collaboration
• Travel between cities
• Konigsberg bridge
• Shortest path
• Least cost
• Schedule exams, assign channels, rooms
• Number of colors on a map
• Highway inspecting, snow removal, street sweeping
• Mail delivery
• Niche overlap- ecology
• Influence graphs
• Round-robin tournaments
• Precedence graphs
Euler paths and circuits- definitions
• Euler circuit – a simple circuit containing every edge of G

Note: circuits start and end at the same point

• Euler path – a simple path containing every edge of G

Practical applications of Euler circuits:

Konigsberg bridge

Konigsberg bridge problem

A

B C

D

Are there Euler paths or circuits for these graphs?

A A

A B A B C CC

B D

C D D E F B

E

A B

C D

A B C A B A

B C

C D C D D

Q—When is there an Euler circuit or path?

A connected multigraph has an Euler circuit iff each of its vertices has _______.

A connected multigraph has an Euler path but not an Euler circuit iff it has exactly _____.

Hamilton circuits and paths
• Just touch every vertex once and only once
• We are not concerned with traveling along each edge
• Practical applications of Hamilton paths and circuits:
Do these graphs have Hamilton paths or circuits?

A

A B A B C A C

C B D

C D D E F B

E

A B

C D

Hamilton paths and circuits

A

A B C A B

B C

D

D E C D

Hamilton paths and circuits

A B A B A B C

C D C D D E F G

E