Question 1

Draw a graph representing four cities, A, B, C, and D, with a road that connects each pair of cities given:
AB, AC, BC, BD, CD

Accepted Answer

The answer of Draw a graph representing four cities, A,...

Question 2

A graph that represents six cities and the roads among them is connected. What does this mean in real-world terms?

Accepted Answer

Any pair of two citi

Question 3

In order to eulerize the graph below, give the fewest number of edges that need to be added or duplicated.

Accepted Answer

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

Question 4

Consider the paths represented by the numbered sequence of edges on the graphs below. Which path represents an Euler circuit?

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

If a graph of four vertices has a vertex with valence 0, then the graph is not connected.

Accepted Answer

A) True 
 B)False

Question 6

Draw a graph that has four vertices, with valences 1, 2, 3, and 4.

Accepted Answer

The answer of Draw a graph that has four vertices,...

Question 7

The cost of a route is computed by summing the numbers along the edges used. Consider the weighted graph below.   The cheapest route starting and ending at A and traversing each edge at least once has what cost? (You are allowed to visit A between the start and end of the route.)

Accepted Answer

A)  20 
B)  28 
C)  37 
D)  42 
A)  20 
B)  28 
C)  37 
D)  42

Question 8

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 9

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 10

Find an Euler circuit on the graph on the left and use it to find a circuit on the graph on the right that reuses three edges.

Accepted Answer

Answers ma

Question 11

Find an Euler circuit on the graph on the left and use it to find a circuit on the graph on the right that reuses three edges.

Accepted Answer

Answers ma

Question 12

Find the minimum number of edges that must be duplicated to create the best possible eulerization of this graph.

Accepted Answer

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

Question 13

Find the minimum number of edges that must be duplicated to create the best possible eulerization of this graph.

Accepted Answer

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

Question 14

In order to eulerize the graph below, give the fewest number of edges that need to be added or duplicated.

Accepted Answer

A)  2 
B)  3 
C)  4 
D)  5 
A)  2 
B)  3 
C)  4 
D)  5

Question 15

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 16

Find an Euler circuit on the graph on the left and use it to find a circuit on the graph on the right that reuses four edges.

Accepted Answer

Answers ma

Question 17

On a graph that represents six cities and the roads between them, the valence of vertex A is 4. What does this mean in real-world terms?

Accepted Answer

If the valence of ve

Question 18

Find an Euler circuit on the graph on the left and use it to find a circuit on the graph on the right that reuses one edge.

Accepted Answer

Answers ma

Question 19

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

Accepted Answer

A)  0 
B)  1 
C)  2 
D)  3 
A)  0 
B)  1 
C)  2 
D)  3

Question 20

Which of the graphs below are connected?

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

Exam 1: Urban Services

Draw a graph representing four cities, A, B, C, and D, with a road that connects each pair of cities given: AB, AC, BC, BD, CD

A graph that represents six cities and the roads among them is connected. What does this mean in real-world terms?

In order to eulerize the graph below, give the fewest number of edges that need to be added or duplicated.

Consider the paths represented by the numbered sequence of edges on the graphs below. Which path represents an Euler circuit?

If a graph of four vertices has a vertex with valence 0, then the graph is not connected.

Draw a graph that has four vertices, with valences 1, 2, 3, and 4.

The cost of a route is computed by summing the numbers along the edges used. Consider the weighted graph below. The cheapest route starting and ending at A and traversing each edge at least once has what cost? (You are allowed to visit A between the start and end of the route.)

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

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

Find an Euler circuit on the graph on the left and use it to find a circuit on the graph on the right that reuses three edges.

Find an Euler circuit on the graph on the left and use it to find a circuit on the graph on the right that reuses three edges.

Find the minimum number of edges that must be duplicated to create the best possible eulerization of this graph.

Find the minimum number of edges that must be duplicated to create the best possible eulerization of this graph.

In order to eulerize the graph below, give the fewest number of edges that need to be added or duplicated.

Which of the graphs below have Euler circuits?

Find an Euler circuit on the graph on the left and use it to find a circuit on the graph on the right that reuses four edges.

On a graph that represents six cities and the roads between them, the valence of vertex A is 4. What does this mean in real-world terms?

Find an Euler circuit on the graph on the left and use it to find a circuit on the graph on the right that reuses one edge.

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

Which of the graphs below are connected?

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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