Hamiltonian Circuits

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• Draw a graph where every vertex has valence of at
  least 3 but where removing a single edge
  disconnects the graph.
• What are the two conditions necessary for an
  Euler circuit to exist and what do these
  conditions mean?
• For graph number 1, what is the least number of
  edges that need to be dropped in order to have an
  Euler circuit? Draw the Euler circuit with the
  dropped edges.
• In graph 2, add one or more edges to produce an
  Euler circuit and draw the circuit.

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Hamiltonian Circuits
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Hamiltonian Circuits

• Hamiltonian circuits tour that starts at a
  vertex of a graph and ends at the same vertex and
  visits each vertex once and only once

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Using Hamiltonian Circuits to plan for vacation

• Apply weights weights incorporate factors
  the impact travel plans
• ex. Distances, cost, time
• Minimum-cost Hamiltonian Circuit Ham. Circuit
  with the lowest possible sum of weights of its
  edges

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Procedure

• Generate all possible Hamiltonian circuits.
• Add up total weights for each circuit.
• Choose circuit w/ minimum weight total.

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Fundamental Principle of Counting

• If there are a ways of selecting a 1st item,
• b ways of selecting a 2nd item and ...z ways of
  selecting the last item, then the total number of
  choices for combining ways is
• a x b x c x .x z

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Traveling Salesman Problem

• Goal Determine a trip of minimum cost that a
  salesperson can make by starting and ending in
  the same city.
• Ex. Route for meter readers, bus routes, limo
  pickups.
• Note cost can be in dollars, distance, time

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Solving Traveling Salesman Problems

• Nearest Neighbor Algorithm continue visiting
  the nearest city until no other choice is
  available then return to start

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Homework

• Pg. 61
• 1,2,3 a,c,e
• 5a,b
• 9,19,21