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Ch. 15: Graph Theory Some practical uses

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 Theory Some practical uses

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  1. 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

  2. Collaboration graphs

  3. See book and written handouts on Graph Coloring, mailroute, and Konigsberg bridge

  4. 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:

  5. Konigsberg bridge Konigsberg bridge problem

  6. A B C D

  7. 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

  8. … A B C A B A B C C D C D D

  9. 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 _____.

  10. Does this graph have an Euler circuit or Euler path? (look at degrees)

  11. 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:

  12. 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

  13. Hamilton paths and circuits A A B C A B B C D D E C D

  14. Hamilton paths and circuits A B A B A B C C D C D D E F G E

  15. Hamilton paths and circuits

  16. Hamilton paths and circuits

  17. Traveling salesman- p. 845

  18. use Brute force or nearest neighbor approximation

  19. hw

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