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Euler Circuits

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[ In case (1) this can be any vertex; in case (2) it must be one of the two odd vertices. ... This graph shows the many possible Euler circuits,with the edges ... – PowerPoint PPT presentation

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Title: Euler Circuits


1
Euler Circuits
  • 5.6 Fleurys Algorithm

2
Euler Circuits
  • Fleurys Algorithm for Finding an Euler Circuit
    (Path)
  • Preliminaries. Make sure that the graph is
    connected and either (1) has no odd vertices
    (circuit), or (2) has two odd vertices (path).

3
Euler Circuits
  • Fleurys Algorithm for Finding an Euler Circuit
    (Path)
  • Start. Choose a starting vertex. In case (1)
    this can be any vertex in case (2) it must be
    one of the two odd vertices.

4
Euler Circuits
  • Fleurys Algorithm for Finding an Euler Circuit
    (Path)
  • Intermediate steps. At each step, if you have a
    choice, dont choose a bridge of the
    yet-to-be-traveled part of the graph. However,
    if you have only one choice, take it.

5
Euler Circuits
  • Fleurys Algorithm for Finding an Euler Circuit
    (Path)
  • Intermediate steps. At each step, if you have a
    choice, dont choose a bridge of the
    yet-to-be-traveled part of the graph. However,
    if you have only one choice, take it.

6
Euler Circuits
  • Fleurys Algorithm for Finding an Euler Circuit
    (Path)
  • Intermediate steps. At each step, if you have a
    choice, dont choose a bridge of the
    yet-to-be-traveled part of the graph. However,
    if you have only one choice, take it.

7
Euler Circuits
  • Fleurys Algorithm for Finding an Euler Circuit
    (Path)
  • Intermediate steps. At each step, if you have a
    choice, dont choose a bridge of the
    yet-to-be-traveled part of the graph. However,
    if you have only one choice, take it.

8
Euler Circuits
  • Fleurys Algorithm for Finding an Euler Circuit
    (Path)
  • Intermediate steps. At each step, if you have a
    choice, dont choose a bridge of the
    yet-to-be-traveled part of the graph. However,
    if you have only one choice, take it.

9
Euler Circuits
  • Fleurys Algorithm for Finding an Euler Circuit
    (Path)
  • End. When you cant travel any more, the circuit
    (path) is complete. In case (1) you will be
    back at the starting vertex in case (2) you will
    end at the other odd vertex.

10
Euler Circuits
  • 5.7 Eulerizing Graphs

11
Euler Circuits
  • Eulerizing Graphs
  • Our first step is to identify the odd vertices.
    This graph has eight odd vertices
    (B,C,E,F,H,I,K,and L), shown in red.

12
Euler Circuits
  • Eulerizing Graphs
  • When we add a duplicate copy of edges BC,EF,HI,
    and KL, we get this graph. This is the eulerized
    version of the original graph.

13
Euler Circuits
  • Eulerizing Graphs
  • This graph shows the many possible Euler
    circuits,with the edges numbered in the order
    they are traveled..

14
Euler Circuits
  • Eulerizing Graphs
  • With the four duplicate edges (BC,EF,HI,and KL)
    indicating the deadhead blocks where a second
    pass is required. The total length of this route
    is 28 blocks (24 blocks in the grid plus 4
    deadhead blocks).

15
Euler Circuits
  • Eulerizing Graphs
  • In some situations we need to find an exhaustive
    route, but there is no requirement that it be
    closedthe route may start and end at different
    points.

16
Euler Circuits
  • Eulerizing Graphs
  • In these cases, we want to leave two odd
    vertices on the graph unchanged, and change the
    other odd vertices into even vertices.

17
Euler Circuits
  • Eulerizing Graphs
  • This process id called a semi-eulerization of
    the graph. In this case the route will start at
    one of the two odd vertices and end at the other
    one.

18
Euler Circuits Conclusion
  • Concept of a graph
  • This idea can be traced back to Euler some 270
    years ago.
  • Concept of a graph model.
  • We used graphs and mathematical theory of graphs
    to solve certain types of routing problems.
  • Concept of an algorithm
  • A set of procedural rules that helps us find
    Euler circuits or Euler path in a graph
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