NonEuler Circuits

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Non-Euler Circuits
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Non-Euler Circuits

• Goal Keep reused edges to a minimum
• Chinese Postman Problem minimize the length of
  a circuit by choosing which edges to retrace
• Ex.
• Parking control, mail routes

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How do we know if our graph has an Euler or a
non-Euler circuit?

• Eulers Theorem

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How to Solve Non-Euler circuits

• Assume all edges are same length
• Count reused edges
• Find a circuit that covers each edge with minimal
  reuse.

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Drawing the Graph
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Steps to solving

• Add edges to given graph by duplicating existing
  ones until you have a graph that is connected and
  even- valent. (Eulerizing a graph)
• Note Only add new edges that duplicate existing
  ones.
• 2) Find an Euler circuit on this new graph.

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Steps to solving

• 3) Squeeze new graph onto original by reusing an
  edge each time the Eulerized graph uses an added
  edge.
• 4) Check to see that the number of reused edges
  should equal the number of added edges.

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Steps to Solving

• Find a best Eulerization.
• Note We try to find a best Eulerization because
  there may be more than one.

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Better Eulerizations

• Rectangular network network composed of a
  series of rectangular blocks forming a rectangle
  a certain number of blocks high by a certain
  number of blocks wide
• Ex. 3x3, 3x4, 4x4
• Part of downtown Flagstaff could be rectangular

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Edge Walker Technique

• Pick a corner of the network.
• Begin walking an edge until you hit an odd-valent
  vertex.
• Start an edge and connect to next vertex.
• If that vertex is now even, repeat step 2.
• If that vertex is now odd, begin a new edge.
• Continue until you have walked around the entire
  graph.

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Homework

• Section 1.3
• Pg. 25
• Exercises 27, 28, 30, 32a,c , 33, 34, 35