5.7 Eulerizing Graphs

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5.7 Eulerizing Graphs
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• Euler circuit and Euler path do not always exist
  .
• There are many graphs (in real life) that have
  more than 2 odd vertices.
• Instead of finding a route that travels along the
  edges of a graph and passes through each and
  every edge of the graph at least once, we want to
  find a a route that re-cross the fewest number of
  edges.

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Eulerizing of a graph

• Eulerizing is the process of changing all odd
  vertices to even vertices by duplicating
  appropriate edges

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Eulerizing Graphs
OPTIMAL ROUTE duplicate the fewest number of
edges
NOT an optimal route

• First step is to identify the odd vertices.
• Second step is to add duplicate copies of
  edges to create all even vertices

illegal route
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Eulerizing the following graphs

• 1)
• 2)

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Semi-eulerizing of a graph

• Semi-eulerizing is the process of leaving 2 odd
  vertices on the graph unchanged and changing
  other odd vertices to even vertices.

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Semi-eulerizing Graphs
OPTIMAL ROUTEduplicate the fewest number of edges
NOT an optimal route

• First step is to identify the odd vertices.
• Second step is leave out 2 odd vertices and
  add duplicate copies of edges to create even
  vertices

illegal route
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Semi eulerizing the following graphs

• 1)
• 2)