Deck 7: Graph Theory

Question 1

Suppose that a graph containing four vertices A, B, C, and D has an Euler circuit starting from A. If the degree of the vertex B is 4, then how many times will the Euler circuit pass the vertex B?

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

Answer 1

B

Question 2

A circuit or cycle in a graph is a path that begins and ends at the same vertex.

Answer 2

True

Question 3

What is the degree of each vertex in the figure below?

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

Answer 3

C

Question 4

When graphs are represented pictorially using dots and segments, the dots are called ___________.

A)points
B)vertices
C)stops
D)edges

Question 5

How many Euler circuits starting from the vertex B are possible?

A)4
B)6
C)8
D)12

Question 6

Find an Euler circuit for the figure below.

A)ABDCA
B)ABDACDA
C)ADBACD
D)No Euler circuit exists.

Question 7

According to Euler's theorem, the figure below has an Euler circuit.

Question 8

The _______________ of a vertex is the number of edges that touch that vertex.

A)magnitude
B)amplitude
C)period
D)degree

Question 9

If a circuit traverses each edge of the graph exactly once, it is called a(n) _____________ circuit.

A)traveling
B)single
C)Hamilton
D)Euler

Question 10

Suppose the edges of a certain graph represent phone lines that must be maintained, and the vertices represent junctions. A worker maintaining the lines would like to find an Euler circuit for this graph to increase ___________.

A)payment
B)mileage
C)efficiency
D)workload

Question 11

The computers in an office area are connected as follows:

A)The computer in Office A is connected to all computers.
B)The computer in Office B is connected to Offices A and D.
C)The computer in Office C is connected to Offices A and D.
D)The computer in Office D is connected to Offices A, B, and C.

Question 12

What is the degree of vertex F?

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

Question 13

Find an Euler circuit for the figure below.

A)ABCDEFA
B)ABCDFEA
C)AFDECBA
D)No Euler circuit exists.

Question 14

Find an Euler circuit for the figure below.

A)AEFDCBA
B)EABCDFE
C)DFEABCD
D)No Euler circuit exists.

Question 15

What is the degree of each vertex in the figure below?

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

Question 16

Which of the following vertices have degree 2?

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

Question 17

According to Euler's theorem, a connected graph has an Euler circuit precisely when every vertex has a(n) ___________ degree.

A)nonzero
B)positive
C)even
D)odd

Question 18

Which of the following vertices have degree 2?

A)A and B
B)C and D
C)B and E
D)D and F

Question 19

According to Euler's theorem, the figure below has an Euler circuit.

Question 20

Find an Euler circuit for the figure below.

A)ABDCA
B)ABDACDA
C)ADBACA
D)No Euler circuit exists.

Question 21

The edges in a certain graph represent roads, and the vertices represent intersections. We want to time the traffic signals at each intersection and not visit any intersection twice. Should we look for an Euler circuit or a Hamilton circuit?

A)Euler
B)Hamilton
C)either
D)neither

Question 22

Find a Hamilton circuit for the figure below.

A)ABCDECFA
B)ABCEDEFA
C)AFECBDBA
D)No Hamilton circuit exists.

Question 23

If each vertex of a graph is connected to every other vertex by an edge, then it is called a(n) _______________ graph.

A)Hamilton
B)Euler
C)complete
D)direct

Question 24

The edges in a certain graph represent roads, and the vertices represent intersections. We want to clean all of the roads and not require the street sweeper to move along streets that have already been cleaned. Should we look for an Euler circuit or a Hamilton circuit?

A)Euler
B)Hamilton
C)either
D)neither

Question 25

The following table shows the distance (in miles) between cities A, B, C, and D. When a traveling salesman visits all four cities, how many miles does he travel if he uses the cheapest link algorithm?

A)3885
B)3950
C)4370
D)5135

Question 26

Finding a Hamilton circuit with the shortest distance for a given complete graph is called _______________.

A)the Hamilton process
B)the traveling salesman problem
C)the Fleury's algorithm
D)the optimal marketing problem

Question 27

A traveling salesman must visit all four cities indicated in the figure below. When he starts in A, what is the second city to visit according to the nearest-neighbor algorithm?

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

Question 28

If a graph has a vertex of degree _____, then each edge meeting that vertex must be part of any Hamilton circuit.

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

Question 29

When applying the nearest-neighbor algorithm, if there are two or more vertices equally nearby, any one of them may be selected.

Question 30

When applying the cheapest-link algorithm, do not choose an edge that would result in a vertex of degree ___.

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

Question 31

The nearest-neighbor algorithm constructs a(n) _______________ circuit in a complete graph by starting at a vertex.

A)Fleury
B)Euler
C)Vector
D)Hamilton

Question 32

Find a Hamilton circuit for the figure below.

A)ABCDEFA
B)AFDABCE
C)AFCDEBA
D)No Hamilton circuit exists.

Question 33

The following table shows the distance (in miles) between cities A, B, C, and D. When a traveling salesman visits all four cities, how many miles does he travel if he starts in city A and uses the nearest-neighbor algorithm?

A)3885
B)3950
C)4370
D)5135

Question 34

How many smaller circuits can be a part of a Hamilton circuit?

A)0
B)1
C)2
D)It varies according to the number of vertices of the Hamilton circuit.

Question 35

If a circuit of a graph visits each vertex exactly once, then it is called a(n) _______________.

A)Euler circuit
B)Hamilton circuit
C)Vector circuit
D)Fleury circuit

Question 36

A traveling salesman must visit all four cities indicated in the figure below. Which is the shortest route he can take if he wants to begin and end in city A?

A)ABCDA
B)ACDBA
C)ABDCA
D)ADBCA

Question 37

A traveling salesman must visit all four cities indicated in the figure below. How many miles does he travel if he starts in A and then visits B, C, and D before returning to A?

A)892 miles
B)1127 miles
C)1239 miles
D)1273 miles

Question 38

The graph below has two odd-degree vertices, so no Euler circuit exists. Suppose we would like to find a route that backtracks as little as possible. What duplicate edge could we create to help us find this route?

A)between A and B
B)between B and C
C)between D and E
D)between E and F

Question 39

Which of the following is not a practical situation where finding an Euler circuit may be important?

A)Delivery of groceries.
B)Garbage pickup.
C)Street sweepers.
D)Driving kids to school.

Question 40

If a complete graph has five vertices, then how many edges are in the graph?

A)10
B)14
C)18
D)20

Question 41

In a tree, if a parent has the level of 2, then what is the level of the grandchildren?

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

Question 42

Use the nearest-neighbor algorithm starting at vertex A of the figure below to find an approximate solution to the traveling salesman problem.

A)ADECBA
B)ADEBCA
C)ACBEDA
D)ABECDA

Question 43

The largest level of the tree is called the _____________.

A)length
B)width
C)height
D)depth

Question 44

In a complete binary tree of height H, the number of parents equals _________.

A)
B)
C)
D)

Question 45

Every vertex of a tree is either a parent or a _______________ but not both.

A)branch
B)child
C)leaf
D)stem

Question 46

Use the cheapest link algorithm to find an approximate solution to the traveling salesman problem for the figure below.

A)ABECDA
B)ACBEDA
C)ADEBCA
D)ADECBA

Question 47

In a complete binary tree of height H, the number of vertices equals _________.

A)
B)
C)
D)

Question 48

A binary tree is _______________ if all the leaves are at the highest level.

A)tall
B)complete
C)complex
D)fully grown

Question 49

A tree has the following characteristic. Find the total number of vertices.

A)13
B)18
C)20
D)26

Question 50

You want to use a dictionary so that you have to check any word at most six times. How big a dictionary can you accommodate?

A)64 words
B)63 words
C)12 words
D)120 words

Question 51

On a complete graph with n vertices, there are n!/2 Hamilton circuits starting at a given vertex, not counting reverse routes. If a traveling salesman wanted to make deliveries to nine cities in a region, how many possible routes would he have to check to find the shortest one?

A)2520
B)20,160
C)181,440
D)1,814,400

Question 52

In a complete binary tree, how many vertices have the level L or

A)
B)
C)
D)

Question 53

In a complete binary tree of height H, the number of leaves equals _________.

A)
B)
C)
D)

Question 54

Estimate the largest number of checks needed if you have a 60,000-word dictionary.

A)14
B)15
C)16
D)17

Question 55

Use the cheapest link algorithm to find an approximate solution to the traveling salesman problem for the figure below.

A)ABECDA
B)ACBEDA
C)ADEBCA
D)ADECBA

Question 56

If each parent of a tree has exactly two children, then the tree is called ___________.

A)alternating
B)binary
C)equivalent
D)complete

Question 57

A _______________ is a graph that contains no circuits.

A)tree
B)branch
C)leaf
D)stem

Question 58

Use the nearest-neighbor algorithm starting at vertex A of the figure below to find an approximate solution to the traveling salesman problem.

A)ADECBA
B)ADEBCA
C)ACBEDA
D)ABECDA

Question 59

A spell-checker is using a binary tree. If the size of the dictionary doubles, what will be the increase in the number of required checks?

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

Question 60

What is the level of the root of a tree?

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

Question 61

What size dictionary can be accommodated by a spell-checker that uses 30 checks?

A)1.1 million words
B)1.1 billion words
C)1.1 trillion words
D)1.1 quadrillion words

Question 62

Suppose you tell a story to two of your friends. Some people repeat the story to two others, and in turn some of these pass the story on. Each time the story is told, it is told to two new people and no one hears the story twice. Sometime later, it turns out that 384 people have heard the story, but not related it to others. How many people have heard the story from others?

A)766
B)767
C)768
D)769

Question 63

One night you put a quarter into a slot machine and get back two quarters. The next night, you put these two quarters back in the machine one at a time, and each time two quarters came out. This pattern continued through the sixth night, after which you took your winnings and went home. How much money did you take home?

A)64 quarters
B)48 quarters
C)32 quarters
D)16 quarters

Question 64

There are 65 members of an organization. The plan for disseminating news requires one member to contact two others with the information. Each member who receives the news contacts two members who have not yet gotten the news. This continues until all members have the information. How many members contact others?

A)64
B)48
C)32
D)16

Question 65

Suppose you tell a story to two of your friends. Some people repeat the story to two others, and in turn some pass the story on. Each time the story is told, it is told to two new people and no one hears the story twice. Sometime later, 271 people know the story. How many people have related the story to others?

A)271
B)270
C)169
D)135

Question 66

One night you put a quarter into a slot machine and get back two quarters. The next night, you put these two quarters back in the machine one at a time, and each time two quarters came out. This pattern continued through the sixth night, after which you took your winnings and went home. How many quarters were fed into the machine?

A)64
B)63
C)47
D)31

Question 67

One night you put a quarter into a slot machine and get back two quarters. The next night, you put these two quarters back in the machine one at a time, and each time two quarters came out. If you want to go home taking your winnings of $500, then what nights should this pattern be continued through?

A)10^th night
B)11^th night
C)12^th night
D)13^th night

Question 68

We have a dictionary consisting of 500 words. If we use a binary tree as a spell-checker, how many checks might be required to test whether "tat" is a valid word?

A)7
B)8
C)9
D)10

Question 69

Which of the following is NOT a child in the tree below?

A)A
B)D
C)G
D)H

Question 70

Which of the following is NOT a leaf in the tree below?

A)F
B)G
C)H
D)J