Deck 13: Graph Theory

Question 1

How many edges does the graph have?

A)2
B)7
C)6
D)5

Answer 1

C

Question 2

Find a Hamiltonian path.

A)F, D, C, B, A, E
B)F, D, A, B, C, E
C)E, A, D, C, B, A, D, F
D)D, E, A, B, C, D, F

Answer 2

A

Question 3

List the vertices of the graph.

A)A, D, E, F
B)A, B, C, D
C)B, C
D)A, B, C, D, E, F

Answer 3

D

Question 4

Suppose you're trying to make a final exam schedule for six classes: anthropology 107 (A),
Economics 212 (E), information systems 194 (I), statistics 261 (S), history 201 (H), and geology 101
(G). The graph below shows which classes students have in common: these are connected by edges.
Use graph coloring to find the chromatic number.

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

Question 5

Find the number of Hamilton circuits if a complete graph has seven vertices.

A)720
B)40,320
C)5,040
D)128

Question 6

Decide whether the connected graph has an Euler path, an Euler circuit, an Euler circuit butnot an
Euler path, or neither an Euler circuit nor an Euler path.
The graph has 4 odd vertices and 2 even vertices.

A)Euler path
B)Euler circuit
C)neither an Euler circuit nor an Euler path
D)Euler circuit but not an Euler path

Question 7

State whether the graph has an Euler path, an Euler circuit, or neither. If it has an Euler path or an
Euler circuit, find one.

A)neither
B)Euler path: H, C, A, B, C, D, E, F, D, G, H, G
C)Euler circuit: A, B, C, D, E, F, D, G, H, C, A
D)Euler path: H, C, A, B, C, D, E, F, D, G

Question 8

Represent the figure using a graph. Use vertices for islands and edges for bridges.

A)
B)
C)
D)

Question 9

Draw a graph that represents the floor plan. Use vertices to represent the rooms and outside area
And edges to represent the connecting doors. Determine if the graph has an Euler path, and Euler
Circuit, or neither. If the graph has an Euler path or an Euler circuit, find one.

A)
B)
C)
D)

Question 10

Find a circuit that includes vertex D.

A)D, F, D
B)D, A, B, C
C)D, A, B, C, D
D)D, A, E, D

Question 11

Decide whether the connected graph has an Euler path, an Euler circuit, an Euler circuit butnot an
Euler path, or neither an Euler circuit nor an Euler path.
The graph has 2 odd vertices and 3 even vertices.

A)neither an Euler circuit nor an Euler path
B)Euler circuit but not an Euler path
C)Euler path
D)Euler circuit

Question 12

Determine if an Euler path or an Euler circuit exists so that a person who plows the roads does not
Have to pass over any street twice. If an Euler path or an Euler circuit exists, find one. The
Intersections of the streets have been labeled for you.

A)Euler path: B-A-H-G-B-C-D-E-F-I-G-F-C
B)neither exists
C)Euler circuit: A-B-C-D-E-F-I-G-H-A
D)Euler path: C-B-A-H-G-I-F-C-D-E-F

Question 13

Using graph coloring, find the smallest number of colors needed to color the map so that no regions
Sharing a common border are the same color

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

Question 14

Are the two graphs equivalent?

A)Yes
B)No

Question 15

List the edges that are bridges.

A)DF
B)BC, DF
C)F
D)D

Question 16

Find a Hamiltonian circuit.

A)F, D, E, A, B, C, F
B)A, E, B, C, F, D, E, A
C)F, D, C, B, A, E, D, F
D)E, B, A, D, F, C, E

Question 17

State whether the graph has an Euler path, an Euler circuit, or neither. If it has an Euler path or an
Euler circuit, find one.

A)neither
B)Euler path: A, B, C, D, E, F, G, H, C, A, H
C)Euler path: A, B, C, D, E, F, G, H, A
D)Euler circuit: A, B, C, D, E, F, G, H, A

Question 18

Draw a graph that represents the floor plan. Use vertices to represent the rooms and outside area and edges to
Represent the connecting doors.

A)
B)
C)
D)

Question 19

List the even vertices of the graph.

A)A, F
B)A, B, C, D, E, F
C)F
D)B, C, D, E

Question 20

Draw a graph for the figure using vertices for the islands and edges for the bridges. Determine if the
Graph has an Euler path, an Euler circuit, or neither. If it has an Euler path or Euler circuit, find one.

A)
B)
C)
D)

Question 21

Determine whether or not the graph is a tree. If it is not, explain why.

A)The graph is not a tree because it is disconnected.
B)The graph is not a tree because it contains a circuit.
C)The graph is a tree.
D)The graph is not a tree because two edges overlap.

Question 22

Determine whether or not the graph is a tree. If it is not, explain why.

A)The graph is not a tree because two edges overlap.
B)The graph is a tree.
C)The graph is not a tree because it is disconnected.
D)The graph is not a tree because it contains a circuit.

Question 23

A pizza delivery person has five prearranged deliveries. The driving times in seconds between those locations
And the pizza place are shown in the table below. Draw a complete weighted graph for the information in the
Table.

A)
B)
C)
D)

Question 24

Use the nearest neighbor method to approximate the optimal solution. Start at vertex A.

A)299
B)232
C)205
D)294

Question 25

Based on the information in the table, use the nearest neighbor method to find an approximation to the shortest
Route from Angleton to all the other cities and back to Angleton.

A)249 miles
B)251 miles
C)252 miles
D)225 miles

Question 26

Determine whether or not the graph is a tree. If it is not, explain why.

A)The graph is a tree.
B)The graph is not a tree because two edges overlap.
C)The graph is not a tree because it is disconnected.
D)The graph is not a tree because it contains a circuit.

Question 27

Based on the information in the table, use the brute force method to find the shortest route
From Angleton to all the other cities and back to Angleton.

A)219 miles
B)250 miles
C)255 miles
D)216 miles

Question 28

Use Kruskal's algorithm to find a minimum spanning tree for the graph below. Give the total weight
For the graph.

A)
B) w
C)
D)

Question 29

Based on the airfare information in the table below, draw a complete weighted graph for the table.

A)
B)
C)
D)

Question 30

At a concert, the production manager needs to run electricity to several lights. The distances in feet
Are shown. Use a minimum spanning tree to determine the shortest amount of wire that will be
Needed.

A)193 feet
B)173 feet
C)178 feet
D)182 feet

Question 31

Edges may be removed from the graph that follows so that is becomes a spanning tree. Identify a set
Of edges that can be removed to accomplish this.

A)GH and FG
B)GH and HI
C)EF and EH
D)HI and FI

Question 32

Use the cheapest link algorithm to approximate the optimal solution.

A)A, E, C, B, D, A: 294
B)A, D, C, B, E, A: 230
C)A, E, B, C, D, A: 297
D)A, B, D, C, E, A: 276

Question 33

Use the brute force method to find the optimal solution for the weighted graph.

A)258
B)222
C)254
D)216