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~ a Graph Theory Problem - PowerPoint PPT Presentation

Konigsberg bridge. ~ a Graph Theory Problem. C. 6. 1. 2. Edges – (Bridges) A link joining two vertices in a graph. . A. 5. B. Vertex (vertices) - A point on a graph where one or more edges end.

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~ a Graph Theory Problem

C

6

1

2

Edges– (Bridges) A link joining two vertices in a graph.

A

5

B

Vertex(vertices) - A point on a graph where one or more edges end.

Graph– A finite set of dots (vertices) and connecting links (edges). Each edge must connect two different vertices.

3

4

D

7

A graph of the bridge problem:

C

6

2

1

5

B

A

Can you draw this graph without lifting your pencil and without redrawing an edge?

4

3

7

D

Example:

The graph represents cities and nonstop airline routes between them.

a) How many vertices? How many edges?

5 vertices ~ the cities

7 edges ~ the nonstop airline routes between them

• Path ~ a connected sequence of edges showing a route on the graph that starts at a vertex and ends at a vertex.

b) Describe a path a person may travel from New York to Berlin.

NLB

NMRB

NMRLB

A ONE-WAY ticket to destination

Example:

The graph represents cities and nonstop airline routes between them.

• Circuit ~ a path that starts and ends at the same vertex

c) Describe a possible circuit that starts at Miami.

MRLM

A ROUND TRIP ticket returning to original city (vertex)

Example:

The graph represents cities and nonstop airline routes between them.

• Connected ~ a graph is CONNECTED if for every pair of its vertices there is at least one PATH connecting the two vertices

d) Is this graph connected?

yes

• Complete ~ a graph is COMPLETE of there is an edge for every pair of vertices

e) Is this graph complete?

no

Valence– the number of edges meeting at a given vertex

A á

2

E á

2

B á

4

F á

4

C á

3

3

G á

Is this graph connected?

D á

2

2

H á

Is this graph complete?