Question 1

Draw a graph with vertices A, B, and C in which A has valence 4 and B and C have valence 5.

Accepted Answer

The answer of Draw a graph with vertices A, B,...

Question 2

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 3

Consider the path represented by the numbered sequence of edges on the graph below. Which statement is true?

Accepted Answer

A)  The path is not a circuit. 
B)  The path is an Euler circuit. 
C)  The path is a circuit, but not an Euler circuit. 
D)  None of the above. 
A)  The path is not a circuit. 
B)  The path is an Euler circuit. 
C)  The path is a circuit, but not an Euler circuit. 
D)  None of the above.

Question 4

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

Question 5

Which of the following statements about a connected graph is always true?

Accepted Answer

A)  Every pair of vertices is joined by a single edge. 
B)  A path of edges exists between any two vertices of the graph. 
C)  There is an even number of vertices on the graph. 
D)  There is an even number of edges on the graph. 
A)  Every pair of vertices is joined by a single edge. 
B)  A path of edges exists between any two vertices of the graph. 
C)  There is an even number of vertices on the graph. 
D)  There is an even number of edges on the graph.

Question 6

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 7

The map below gives the territory of a parking control officer. The dots represent meters that must be checked. Draw the graph that would be useful for finding an efficient route.

Accepted Answer

The answer of The map below gives the territory of...

Question 8

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 9

Explain why the graph shown below does NOT have an Euler circuit.

Accepted Answer

The graph does not h

Question 10

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

Accepted Answer

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

Question 11

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

Question 12

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

Accepted Answer

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

Question 13

For which of the two situations below is it desirable to find an Euler circuit or an efficient eulerization of a graph? I. After a storm, a health department worker inspects all the houses of a small village to check for damage.
II) A veteran planning a visit to all the war memorials in Washington, D.C., plots a route to follow.

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 14

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

Accepted Answer

A)  4 
B)  6 
C)  10 
D)  12 
A)  4 
B)  6 
C)  10 
D)  12

Question 15

It is possible for a graph with all vertices of even valence to NOT have an Euler circuit.

Accepted Answer

A) True 
 B)False

Question 16

It is possible to construct a graph such that the sum of all valences is odd.

Accepted Answer

A) True 
 B)False

Question 17

Draw a graph with vertices A, B, C, and D in which the valence of vertices A and D is 3 and the valence of vertices B and C is 2.

Accepted Answer

Answers ma

Question 18

Explain why the graph shown below does NOT have an Euler circuit.

Accepted Answer

The graph does not h

Question 19

A graph that is not connected must have at least one vertex with valence 0.

Accepted Answer

A) True 
 B)False

Question 20

Use the following to answer Questions:
Identify an Euler circuit on the following graphs by numbering the sequence of edges in the order traveled.
-

Accepted Answer

Answers ma

Exam 1: Urban Services

Draw a graph with vertices A, B, and C in which A has valence 4 and B and C have valence 5.

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

Consider the path represented by the numbered sequence of edges on the graph below. Which statement is true?

Which of the graphs below are connected?

Which of the following statements about a connected graph is always true?

Which of the graphs below have Euler circuits?

The map below gives the territory of a parking control officer. The dots represent meters that must be checked. Draw the graph that would be useful for finding an efficient route.

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.)

Explain why the graph shown below does NOT have an Euler circuit.

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

Which of the graphs below are connected?

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

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

It is possible for a graph with all vertices of even valence to NOT have an Euler circuit.

It is possible to construct a graph such that the sum of all valences is odd.

Draw a graph with vertices A, B, C, and D in which the valence of vertices A and D is 3 and the valence of vertices B and C is 2.

Explain why the graph shown below does NOT have an Euler circuit.

A graph that is not connected must have at least one vertex with valence 0.

Use the following to answer Questions: Identify an Euler circuit on the following graphs by numbering the sequence of edges in the order traveled. -

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