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Thinking Mathematically, 10.7 & 15.1–15.3 Excursions in Mathematics, 5.1–5.7

(2)

ORIGINS OF GRAPH THEORY

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ORIGINS OF

GRAPH THEORY

Königsberg, Germany

Seven Bridges

• Salesman’s Bridge

• Green Bridge

• Slaughter Bridge

• Blacksmith’s Bridge

• Timber Bridge

• High Bridge

• Honey Bridge

Four City Areas

• Alstadt & Löbnicht

• Kneiphof Island

• Vorstadt

• Lomse Island

A

K

V

L

(4)

Five Bridges Today

• Estacada Bridge

• Timber (October) Bridge

• High Bridge

• Honey Bridge

• NEW: Emperor Bridge

Kaliningrad, Russia

(5)

Origins of Graph Theory

Königsberg, Germany in early 1700s had seven bridges Residents of city took walks on Sunday, developed a game

“Can a person cross all bridges only once and return to start?” 1735 Leonhard Euler was given the

problem and determined impossible

Leonhard Euler

Swiss mathematician

St. Petersburg Academy of Sciences Started “geometris situs” which became graph theory

(6)

DEFINITIONS

vertex – point (NOTE: plural is vertices)

edge – line segment or curve that starts at one vertex

and ends at another vertex

odd vertex – vertex with an odd number of attached edges

even vertex – vertex with an even number of attached edges

traversable graph – a graph for which all edges can be

traced once and only once

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EXAMPLE

Consider the graph. Is it traversable?

That is, can each line of the graph be traced once and

only once without lifting your pencil?

(8)

RULES

Euler’s Rules of Traversability

1. A graph with all even vertices is traversable.

One can start at any vertex and end at same vertex.

2. A graph with two odd vertices is traversable.

One must start at one odd vertex and

end at the other odd vertex.

(9)

BACK TO EXAMPLE

Consider the graph. Is it traversable?

That is, can each line of the graph be traced once and

only once without lifting your pencil?

Yes, the graph is traversable!

Odd

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Odd

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Even

(2)

There are two, and only two, odd

vertices so it was traversable.

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EXAMPLE

Consider the graph. Is it traversable?

Yes, the graph is traversable!

Even

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Even

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Even

(2)

Even

(2)

Even

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Recall Euler’s Rules of Traversability:

1. A graph with all even vertices is traversable.

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Let’s model the Seven Bridges of Königsberg as a graph.

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EXAMPLE

Consider the graph. Is it traversable?

No, the graph is not traversable!

Odd

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Odd

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Odd

(3)

Odd

(5)

Recall Euler’s Rules of Traversability:

3. A graph with more than two odd vertices is NOT traversable.

The Bridges of Königsberg problem was not a

(13)

EXAMPLE

Consider the graph. Is it traversable? If so, find the path.

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EXAMPLE

Consider the graph. Is it traversable? If so, find the path.

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EXAMPLE

Consider the graph. Is it traversable? If so, find the path.

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GRAPHS,

PATHS, & CIRCUITS

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DEFINITIONS

vertex – point (NOTE: plural is vertices)

edge – line segment or curve that starts at one vertex

and ends at another vertex

loop – curve that starts and ends at the same vertex

point must be marked with a dot; denoted with letter

denoted with two vertex letters

(18)

EXAMPLE

Consider the graph below.

Identify the vertices, edges, and loops.

How many vertices?

4

How many edges?

4

This is NOT a vertex!

edge AD edge DB

edge BC edge CA

How many loops?

1

loop CC

Not every point where two edges cross is a vertex!

If the point is not marked with a dot, it is not a vertex (think of it as one line is above the other).

What is C’s degree?

4

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DEFINITION

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EXAMPLE

Which are equivalent graphs?

4

1

2

3

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EXAMPLE

(22)

EXAMPLE

Consider the map of the Rocky Mountain states.

Model the states that share a common border.

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EXAMPLE

Can you cross the border of each state once and only

once (that is, is the graph traversable)?

Odd(3)

Odd(3)

Even (2)

Even (4)

Even (2)

Recall Euler’s Rules of Traversability:

2. A graph with two odd vertices is traversable.

One must start at one odd vertex and end at the other odd vertex.

(24)

EXAMPLE

Find the path that traverses the graph.

Find the path on the map that crosses each border

exactly one time.

Odd(3)

Odd(3)

Even (2)

Even (4)

Even (2)

(25)

EXAMPLE

(26)

EXAMPLE

Can you cross the border of each state once and only

once (that is, is the graph traversable)?

Even (2)

Even (2)

Even (2)

Even (4)

Odd (3)

Odd (1)

Yes, it is traversable.

Recall Euler’s Rules of Traversability:

2. A graph with two odd vertices is traversable.

(27)

EXAMPLE

Find the path that traverses the graph.

Find the path on the map that crosses each border

exactly one time.

Even (2)

Even (2)

Even (2)

Even (4)

(28)

EXAMPLE

Consider the floor plan of a four-room house.

(Consider all the outdoors as one location.)

Model the ways to walk from one area to another.

(29)

EXAMPLE

Consider the floor plan of a four-room house.

(Consider all the outdoors as one location.)

(30)

EXAMPLE

Consider the bridges of Madison County.

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EXAMPLE

Consider this neighborhood. A mail carrier delivers the

mail by walking to each house. If houses are on both

sides of the street, he walks it twice. Model this.

Houses are purple squares.

Roads are pink lines.

(32)

DEFINITIONS

adjacent vertices – vertices that are connected directly and

thus share at least one edge

D is adjacent to C

C is adjacent to A & D

B is adjacent to A & E

A is adjacent to B, C, E, & F

E is adjacent to A & B

F is adjacent to A

F is NOT adjacent to E

(33)

DEFINITIONS

path – a sequence of adjacent vertices and the edges

connecting them, denoted by list of vertices in order

path: A,B,F,G,H,M

not a path:

A,B,F,G,H,M,H,D

NOTE: A path can use a vertex more than once but can use each edge only once. However, the entire graph

does not have to be touched to have a path.

(34)

DEFINITIONS

circuit – a path that begins and ends at the same vertex

circuit: A,B,F,G,L,K,J,E,A

NOTE: Every circuit is a path.

(35)

DEFINITIONS

connected graph – a graph in which there is at least one

path connecting any two vertices

disconnected graph – a graph in which there is no path

connecting any two vertices

disconnected graph

(36)

DEFINITIONS

bridge – an edge that, if removed, would make a connected

graph into a disconnected graph

edge BC is a bridge

(37)

EXAMPLE

Find the bridge(s) in the graph.

edge AB is a bridge

(38)

HOMEWORK

From the Cow book

References

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