Graph Theory:Traveling Salesman Problem (TSP)

E3 Teacher Summer Research Program 2005

Texas A & M UniversityJune 29,

2005

TAKS Objectives9th & 10th Grade

Objective 1 – Ab1A, B, C, D & E

Objective 2 – Ab2A & D, Ab3A & B

Objective 3 – Ac1A & C

Objective 6 – 8.7D

Objective 10 - 8.14A, B & C, 8.15A, 8.16A & B

Material

GEO Board

Yarn

Rubber bands

Colored pencils

Grid paper

Graphing calculator

Vocabulary

Nodes (vertices)Edges (arcs)DegreeAdjacentPath

Vocabulary

LengthCircuitSimple GraphComplete Graph

Nodes (vertices)

Nodes (vertices)

Land Phone numbersPeopleJunction points (electric circuits)AtomChess playersCompanies or industries

Edges (arcs)

Edges (arcs)

Bridges connecting landCalls made from one number to another connecting phone numbersRelationships or acquaintances connecting by peopleWires connecting junction points (electric circuits)Bonds between atoms connecting atomsMatches connecting chess players (chess tournament)Transactions connecting companies or industries

Degree

A

F

H

G

B

E

D

C

Degree

Node A is of two degrees

Node B is of two degrees

Node C is of six degrees

Node D is of three degrees

How many degrees are nodes E, F, G and H?

Adjacent

A

F

H

G

B

E

D

C

Adjacent

Node A is adjacent to Nodes C and HNode B is adjacent to Nodes F and HNode C is adjacent to Nodes A, D, E, F, G and HNode D is adjacent to Nodes C, E and HWhat nodes are adjacent to nodes E, F, G and H?

Path

A

BC

D

E

F

Path

A

BC

D

E

F

Path

A

BC

D

E

F

No Path

A

BC

D

E

F

Draw two different paths.

A

BC

D

E

F

LengthA

BC

D

E

F

LengthA

BC

D

E

F

LengthA

BC

D

E

F

LengthA

BC

D

E

F

CircuitPoint A is the starting point

C

B D

A E

CircuitPoint A is the starting point

C

B D

A E

CircuitPoint A is the starting point

C

B D

A E

CircuitPoint A is the starting point

C

B

D

A

E

Simple Graphs

Not, Simple Graphs

Complete Graphs

Complete Graphs

Draw a complete graph with six nodes.

Draw a complete graph with seven nodes

Petroleum Delivers

745

325

490

825

520

565

370

570

565380

S

T

N

M

C

T TexasS South CarolinaN New YorkM MinnesotaC Colorado

1 Node

2 Nodes

2 Nodes

3 Nodes

3 Nodes

3 Nodes

3 Nodes

3 Nodes

3 Nodes

3 Nodes

3 Nodes

2 Routes

4 Nodes

4 Nodes

4 Nodes

4 Nodes

4 Nodes

4 Nodes

4 Nodes

4 Nodes

4 Nodes

4 Nodes

4 Nodes

6 Routes

2 Node

Node(s)

Process Column Route(s)

2 1

3 Nodes

Nodes

Process Column Route(s)

3 2

2 1

4 Nodes

Nodes

Process Column Route(s)

4 3

3 2

2 1

2 Nodes

Node(s)

Process Column Route(s)

2 2-1 1

3 Nodes

Nodes

Process Column Route(s)

3 3-1 2

2 1

3 Nodes

Nodes

Process Column Route(s)

3 3-1 2

2 2-1 1

4 Nodes

Nodes

Process Column Route(s)

4 4-1 3

3 2

2 1

n r1

4 Nodes

Nodes

Process Column Route(s)

4 4-1 3

3 3-1 2

2 1

n r1

4 Nodes

Nodes

Process Column Route(s)

4 4-1 3

3 3-1 2

2 2-1 1

n r1

4 Nodes

Nodes

Process Column Route(s)

4 4-1 3

3 3-1 2

2 2-1 1

n n-1 r1

Function Ruler1 equals number of routes from each remaining nodes

r1 = n - 1

4 Nodes3

2

1

Total Original Routes4 Nodes

3 Routes

1

2

3

2 Routes

21

21

21

1 Route11

1

1

11

Total Original Routes

3 x 2 x 1 = 6

or

(4-1)!=6

Function Ruler2 equals number of unique routes

2

2)!1(

rn

Petroleum Delivers

745

325

490

825

520

565

370

570

565380

S

T

N

M

C

T TexasS South CarolinaN New YorkM MinnesotaC Colorado