Deck 1: Urban Services

Question 1

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.

Answer 1

Answers may vary. One solution is:

Question 2

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

Answer 2

Answers may vary. One solution is:

Question 3

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

Answer 3

Question 4

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

Question 5

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

Question 6

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

Question 7

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

Question 8

Describe the two conditions for a graph to have an Euler circuit.

Question 9

Consider the path represented by the sequence of numbered edges on the graph below. Explain why the path is NOT an Euler circuit.

Question 10

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

Question 11

Consider the path represented by the sequence of numbered edges on the graph below. Explain why the path is NOT an Euler circuit.

Question 12

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

Question 13

Draw a graph with eight vertices, with the valence of each vertex being even, that does NOT have an Euler circuit.

Question 14

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

Question 15

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

Question 16

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?

Question 17

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

Question 18

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

Question 19

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

Question 20

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

Question 21

Give three real-world applications in which a worker would want to find an Euler circuit on a street network.

Question 22

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

Question 23

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

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

Question 24

For the street network shown below, draw a graph that would be useful for routing a garbage truck. Assume all streets are two-way and that passing once down the street is sufficient for collecting trash from both sides.

Question 25

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

Question 26

Draw a graph where every vertex has a valence of at least 2 but removing a single edge disconnects the graph.

Question 27

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.

Question 28

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.

Question 29

How many vertices in the graph below have odd valences?

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

Question 30

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

Question 31

How many vertices in the graph below have even valences?

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

Question 32

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

Question 33

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

Question 34

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.

Question 35

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

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

Question 36

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.

Question 37

Why would a city street department want its snowplow operator's path to follow an Euler circuit if possible?

Question 38

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

Question 39

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.

Question 40

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

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

Question 41

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

Question 42

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

Question 43

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

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 44

Every graph with an Euler circuit has only vertices with even valences.

Question 45

Which of the graphs below are connected?

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

Question 46

Which of the following defines the valence of vertex A of a graph?

A) The total number of vertices of the graph
B) The number of edges meeting at vertex A
C) The total number of edges of the graph
D) None of the above

Question 47

Which of the graphs below are connected?

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

Question 48

Which of the graphs below are connected?

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

Question 49

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

Question 50

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

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

Question 51

Which of the graphs below have Euler circuits?

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

Question 52

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

Question 53

What is the sum of all valences of the graph shown below?

A) 6
B) 8
C) 10
D) 12

Question 54

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

Question 55

Which of the graphs below have Euler circuits?

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

Question 56

Every graph that has an Euler circuit is connected.

Question 57

Which of the graphs below have Euler circuits?

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

Question 58

Which of the graphs below have Euler circuits?

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

Question 59

Which of the graphs below have Euler circuits?

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

Question 60

Every connected graph has an Euler circuit.

Question 61

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

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

Question 62

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

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 63

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

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 64

After a major natural disaster, such as a flood, hurricane, or tornado, many tasks need to be completed as efficiently as possible. For which situation below would finding an Euler circuit or an efficient eulerization of a graph be the appropriate mathematical technique to apply?

A) Relief food supplies must be delivered to eight emergency shelters located at different sites in a large city.
B) The Department of Public Works must inspect traffic lights at intersections in the city to determine which are still working.
C) An insurance claims adjuster must visit 10 homes in various neighborhoods to write reports.
D) The Department of Public Works must inspect all streets in the city to remove dangerous debris.

Question 65

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

Question 66

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

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 67

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

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

Question 68

For which of the two situations below is it desirable to find an Euler circuit or an efficient eulerization of a graph? I. Plowing the streets of a small village after a snow.
II) Painting the lines down the center of the roads in a town with only two-way roads.

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

Question 69

After a major natural disaster, such as a flood, hurricane, or tornado, many tasks need to be completed as efficiently as possible. For which situation below would finding an Euler circuit or an efficient eulerization of a graph be the appropriate mathematical technique to apply?

A) The electric company must check several substations for malfunctions.
B) The gas company must check along all gas lines for possible leaks.
C) The phone company must respond to customers' needs in several parts of town.
D) The water company must spot-check the integrity of eight water towers located throughout the city.

Question 70

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

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

Question 71

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

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

Question 72

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

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

Question 73

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

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

Question 74

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

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 75

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.

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

Question 76

For which of the two situations below is it desirable to find an Euler circuit or an efficient eulerization of a graph? I. A street department employee must check the traffic signals at each intersection in a downtown area to be certain they are working.
II) An employee of a power company reads the electric meters outside each house along the streets in a residential area.

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

Question 77

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

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

Question 78

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

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 79

For which of the two situations below is it desirable to find an Euler circuit or an efficient eulerization of a graph? I. A pizza delivery person takes pizzas to 10 houses in a neighborhood and then returns to pick up the next set to be delivered.
II) A postal carrier picks up mail from six collection boxes around a city.

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

Question 80

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

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