Chapter 1 Euler Circuits

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Chapter 1Euler Circuits

• Graph a finite set of dots called Vertices and
  connecting links called Edges which connect two
  different vertices. Graphs can represent a city
  map, communications network, system of air routes.

Graph
Not a graph
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Forming a Graph from a street network
a)
d)
c)
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Chapter 1Euler Circuits
Path connected sequence of edges showing a route
on the graph that starts at a vertex and ends at
a vertex. Notation Name the vertices visited
AFEB
Circuit a path that starts and ends at the same
vertex AFEDCBA
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• Euler Circuit Circuit that covers every edge
  only once
• AFEBEDCBA

Valence of a vertex in a graph is the number of
edges meeting at the vertex. Vertices A,F,C,D
each have valence 2 while vertices B and E each
have valence 4.
Connected Graph a graph is connected if for
every pair of its vertices, there is at least
one path connecting the two vertices.
Connected
Not Connected
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Eulers Theorem a graph has an Euler circuit if
and only if it has even valences is connected.
Example Finding an Euler Circuit
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Chinese Postman Problem we know that there is no
Euler circuit for a graph with odd valences.
Therefore, we want to minimize the number of
edges that must be reused.
How? Eulerize the graph by adding edges to the
graph to make all valences even! Then squeeze
this Eulerized circuit onto the original graph by
reusing an edge of the original graph each time
the circuit on the eulerized graph uses an added
edge.
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Diagraph a graph in which each edge has an arrow
indicating the direction of the edge. Used for
one way streets, etc.
Street network
the digraph model
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Credits

• COMAP, For All Practical Purposes, 5th ed.