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PPT – Aim: What is an Euler Path and Circuit? PowerPoint presentation | free to view - id: 670615-ZjAwO

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Aim What is an Euler Path and Circuit?

Do Now

Represent the following with a graph

Euler Path

Euler Path a path that travels through every

edge of a graph once and only once. Each edge

must be traveled and no edge can be retraced.

A,

B,

E,

D,

B,

C,

E,

D,

G

F,

Euler Path

Euler Circuit a circuit that travels through

every edge of a graph once and only once, and

must begin and end at the same vertex.

A,

B,

E,

D,

F,

B,

C,

E,

D,

G,

A

Every Euler circuit is an Euler path

Not every Euler path is an Euler circuit

Some graphs have no Euler paths

Other graphs have several Euler paths

Some graphs with Euler paths have no Euler

circuits

Euler Theorem

- Eulers Theorem
- The following statements are true for connected

graphs - If a graph has exactly two odd vertices, then it

has at least one Euler path, but no Euler

circuit. Each Euler path must start at one of

the odd vertices and end at the other. - If a graph has no odd vertices (all even

vertices), it has at least one Euler circuit. An

Euler circuit can start and end at any vertex. - If a graph has more than two odd vertices, then

it has no Euler paths and no Euler circuits.

Model Problem

Explain why the graph has at least one Euler path.

number of edges at each vertex

degree 2

degree 4

degree 4

2

A

D

3

B

E

3

4

C

4

Two odd vertices

degree 3

degree 3

- If a graph has exactly two odd vertices, then it

has at least one Euler path, but no Euler

circuit. Each Euler path must start at one of

the odd vertices and end at the other.

Model Problem

Find a Euler path.

5

6

2

3

4

7

1

8

D,

C,

B,

C,

A,

B,

D,

E

E,

Model Problem

Explain why the graph has a least one Euler

circuit.

degree 4

degree 4

degree 2

degree 2

degree 4

degree 4

no odd vertices

2. If a graph has no odd vertices (all even

vertices), it has at least one Euler circuit. An

Euler circuit can start and end at any vertex.

Model Problem

Find an Euler circuit.

11

10

12

9

13

3

8

4

2

14

1

7

5

6

H,

G,

E,

G,

I,

J,

H,

D,

C,

C,

A,

B,

D,

F,

H

Euler Circuit a circuit that travels through

every edge of a graph once and only once, and

must begin and end at the same vertex.

Graph Theorys Beginnings

In the early 18th century, the Pregel River in a

city called Konigsberg, surround an island before

splitting into two. Seven bridges crossed the

river and connected four different land area.

Many citizens wished to take a stroll that would

lead them across each bridge and return them to

the starting point without traversing the same

bridge twice. Possible?

They couldnt do it.

Euler proved that it was not possible.

The Theorem at Work

3. If a graph has more than two odd vertices,

then it has no Euler paths and not Euler circuits.

Model Problem

A, B, C, and D represent rooms. The outside of

the house is labeled E. The openings represent

doors. a) Is it possible to find a path that

uses each door exactly once?

degree 6

look for a Euler path or circuit

exactly 2 odd vertices

- If a graph has exactly two odd vertices, then it

has at least one Euler path, but no Euler

circuit. Each Euler path must start at one of

the odd vertices and end at the other.

Model Problem

A, B, C, and D represent rooms. The outside of

the house is labeled E. The openings represent

doors. b) If possible, find such a path.

10

9

7

2

1

3

8

6

4

5

start at one of the odd vertices

B,

E,

A,

D,

C,

A,

E,

C,

B,

E,

D

Fluerys Algorithm

- If Eulers Theorem indicates the existence of an

Euler path or Euler circuit, one can be found

using the following procedure. - If the graph has exactly two odd vertices, chose

one of the two odd vertices as the starting

point. If the graph has no odd vertices, choose

any vertex as the starting point. - Number edges as you trace through the graph

according to the following rules - After you traveled over an edge, change it

to a dashed line. - When faced with a choice of edges to trace,

choose an edge that is not a bridge (an edge,

which, if removed from a connected graph would

leave behind a disconnected graph). Travel over

an edge that is a bridge only if there is no

alternative.

Model Problem

The graph has at least one Euler circuit. Find

it using Fleurys Algorithm.

E

F

D

C

A

B

no odd vertices, begin at any vertex.