Question 1

Every connected graph has an Euler circuit.

Accepted Answer

A) True 
 B)False

Question 2

Draw a graph with five vertices where the valence of each vertex is 4.

Accepted Answer

Answers wi

Question 3

If a graph had eight vertices of odd valence, what is the absolute minimum number of edges that would need to be duplicated to eulerize the graph?

Accepted Answer

A)  2 
B)  4 
C)  6 
D)  8 
A)  2 
B)  4 
C)  6 
D)  8

Question 4

Which of the graphs below have Euler circuits?

Accepted Answer

A)  I only 
B)  II only 
C)  Both I and II 
D)  Neither I nor II 
A)  I only 
B)  II only 
C)  Both I and II 
D)  Neither I nor II

Question 5

Use the following to answer Questions:
Add wiggly edges to find an efficient eulerization of the following graphs.
-

Accepted Answer

Answers ma

Question 6

What is the valence of vertex A in the graph below?

Accepted Answer

A)  3 
B)  4 
C)  5 
D)  6 
A)  3 
B)  4 
C)  5 
D)  6

Question 7

An Euler circuit of a graph passes through each vertex exactly once.

Accepted Answer

A) True 
 B)False

Question 8

Every graph with an Euler circuit has an even number of edges.

Accepted Answer

A) True 
 B)False

Question 9

Is it possible to have a connected graph with seven vertices and seven edges?

Accepted Answer

A) True 
 B)False

Question 10

How many vertices in the graph below have even valences?

Accepted Answer

A)  3 
B)  5 
C)  7 
D)  11 
A)  3 
B)  5 
C)  7 
D)  11

Question 11

What is the smallest number of edges that would need to be removed to disconnect the graph below?

Accepted Answer

A)  1 
B)  2 
C)  3 
D)  4 
A)  1 
B)  2 
C)  3 
D)  4

Question 12

Every graph that has an Euler circuit is connected.

Accepted Answer

A) True 
 B)False

Question 13

Consider the path represented by the sequence of numbered edges on the graph below. Why does the path NOT represent an Euler circuit?

Accepted Answer

A)  The path does not start and stop at the same vertex. 
B)  The path does not cover every edge of the graph. 
C)  The path uses some edges more than one time. 
D)  The path does not touch each vertex of the graph. 
A)  The path does not start and stop at the same vertex. 
B)  The path does not cover every edge of the graph. 
C)  The path uses some edges more than one time. 
D)  The path does not touch each vertex of the graph.

Question 14

Which of the graphs shown below gives the best eulerization of the given graph? (In the graphs below, added edges are denoted with zigzag lines.)

Accepted Answer

A)   
B)   
C)   
D)   
A)   
B)   
C)   
D)

Question 15

How many vertices in the graph below have odd valences?

Accepted Answer

A)  0 
B)  1 
C)  2 
D)  4 
A)  0 
B)  1 
C)  2 
D)  4

Question 16

Find an eulerization with nine added edges for a 3 F1F1F1S1F1F1F10 6-block rectangular street network.

Accepted Answer

The answer of Find an eulerization with nine added edges...

Question 17

Draw a graph with six vertices where the valences are 1, 2, 2, 3, 4, and 4.

Accepted Answer

Answers wi

Question 18

Use the following to answer Questions:
Add wiggly edges to find an efficient eulerization of the following graphs.
-

Accepted Answer

Answers ma

Question 19

If a graph of six vertices is connected, then there are at least five edges.

Accepted Answer

A) True 
 B)False

Question 20

Draw a connected graph with seven vertices where each vertex has valence 4.

Accepted Answer

Answers wi

Exam 1: Urban Services

Every connected graph has an Euler circuit.

Draw a graph with five vertices where the valence of each vertex is 4.

If a graph had eight vertices of odd valence, what is the absolute minimum number of edges that would need to be duplicated to eulerize the graph?

Which of the graphs below have Euler circuits?

Use the following to answer Questions: Add wiggly edges to find an efficient eulerization of the following graphs. -

What is the valence of vertex A in the graph below?

An Euler circuit of a graph passes through each vertex exactly once.

Every graph with an Euler circuit has an even number of edges.

Is it possible to have a connected graph with seven vertices and seven edges?

How many vertices in the graph below have even valences?

What is the smallest number of edges that would need to be removed to disconnect the graph below?

Every graph that has an Euler circuit is connected.

Consider the path represented by the sequence of numbered edges on the graph below. Why does the path NOT represent an Euler circuit?

Which of the graphs shown below gives the best eulerization of the given graph? (In the graphs below, added edges are denoted with zigzag lines.)

How many vertices in the graph below have odd valences?

Find an eulerization with nine added edges for a 3  6-block rectangular street network.

Draw a graph with six vertices where the valences are 1, 2, 2, 3, 4, and 4.

Use the following to answer Questions: Add wiggly edges to find an efficient eulerization of the following graphs. -

If a graph of six vertices is connected, then there are at least five edges.

Draw a connected graph with seven vertices where each vertex has valence 4.

Business Efficiency

Planning and Scheduling

Linear Programming

Exploring Data: Distributions

Exploring Data: Relationships

Data for Decisions

Probability: the Mathematics of Chance

Social Choice: the Impossible Dream

The Manipulability of Voting Systems

Weighted Voting Systems

Electing the President

Fair Division

Apportionment

Game Theory: the Mathematics of Competition

Identification Numbers

Information Science

Growth and Form

Symmetry and Patterns

Tilings

Savings Models

Borrowing Models

The Economics of Resources

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