Discrete Mathematics and Its Applications

Kenneth H. Rosen

Chapter 9

Graphs - all with Video Answers

Educators

Section 1

Graphs and Graph Models

Problem 1

Draw graph models, stating the type of graph (from Table 1) used, to represent airline routes where every day there are four flights from Boston to Newark, two flights from Newark to Boston, three flights from Newark to Miami, two flights from Miami to Newark, one flight from Newark to Detroit, two flights from Detroit to Newark, three flights from Newark to Washington, two flights from Washington to Newark, and one flight from Washington to Miami, with
a) an edge between vertices representing cities that have a flight between them
b) an edge between vertices representing cities for each flight that operates between them
c) an edge between vertices representing cities for each flight that operates between them (in either direction), plus a loop for a special sightseeing trip that takes off and lands in Miami.
d) an edge from a vertex representing a city where a flight starts to the vertex representing the city where it ends.
e) an edge for each flight from a vertex representing a city where the flight begins to the vertex representing the city where the flight ends.

Amia Palmer

Amia Palmer

Numerade Educator

Problem 2

What kind of graph (from Table 1) can be used to model a highway system between major cities where
a) there is an edge between the vertices representing cities if there is an interstate highway between them?
b) there is an edge between the vertices representing cities for each interstate highway between them?
c) there is an edge between the vertices representing cities for each interstate highway between them, and there is a loop at the vertex representing a city if there is an interstate highway that circles this city?

Prathan Jarupoonphol

Prathan Jarupoonphol

Numerade Educator

Problem 3

For Exercises 3-9, Determine whether the graph shown has directed or undirected edges, whether it has multiple edges, and whether it has one or more loops. Use your answers to determine the type of graph in Table 1 this graph is.
(FIGURE CAN'T COPY)

Chris Trentman

Numerade Educator

Problem 4

Determine whether the graph shown has directed or undirected edges, whether it has multiple edges, and whether it has one or more loops. Use your answers to determine the type of graph in Table 1 this graph is.
(FIGURE CAN'T COPY)

WM

William Mead

Numerade Educator

Problem 5

Determine whether the graph shown has directed or undirected edges, whether it has multiple edges, and whether it has one or more loops. Use your answers to determine the type of graph in Table 1 this graph is.
(FIGURE CAN'T COPY)

WM

William Mead

Numerade Educator

Problem 6

Determine whether the graph shown has directed or undirected edges, whether it has multiple edges, and whether it has one or more loops. Use your answers to determine the type of graph in Table 1 this graph is.
(FIGURE CAN'T COPY)

WM

William Mead

Numerade Educator

Problem 7

Determine whether the graph shown has directed or undirected edges, whether it has multiple edges, and whether it has one or more loops. Use your answers to determine the type of graph in Table 1 this graph is.
(FIGURE CAN'T COPY)

WM

William Mead

Numerade Educator

Problem 8

Determine whether the graph shown has directed or undirected edges, whether it has multiple edges, and whether it has one or more loops. Use your answers to determine the type of graph in Table 1 this graph is.
(FIGURE CAN'T COPY)

WM

William Mead

Numerade Educator

Problem 9

Determine whether the graph shown has directed or undirected edges, whether it has multiple edges, and whether it has one or more loops. Use your answers to determine the type of graph in Table 1 this graph is.
(FIGURE CAN'T COPY)

WM

William Mead

Numerade Educator

Problem 10

For each undirected graph in Exercises 3-9 that is not simple, find a set of edges to remove to make it simple.

Prathan Jarupoonphol

Prathan Jarupoonphol

Numerade Educator

Problem 11

Let $G$ be a simple graph. Show that the relation $R$ on the set of vertices of $G$ such that $u R v$ if and only if there is an edge associated to $\{u, v\}$ is a symmetric, irreflexive relation on $G$.

Chris Trentman

Numerade Educator

Problem 12

Let $G$ be an undirected graph with a loop at every vertex. Show that the relation $R$ on the set of vertices of $G$ such that $u R v$ if and only if there is an edge associated to $\{u, v\}$ is a symmetric, reflexive relation on $G$.

Prathan Jarupoonphol

Prathan Jarupoonphol

Numerade Educator

Problem 13

The intersection graph of a collection of sets $A_1$, $A_2, \ldots, A_n$ is the graph that has a vertex for each of these sets and has an edge connecting the vertices representing two sets if these sets have a nonempty intersection. Construct the intersection graph of these collections of sets.
a)
$$
\begin{aligned}
& A_1=\{0,2,4,6,8\}, A_2=\{0,1,2,3,4\}, \\
& A_3=\{1,3,5,7,9\}, A_4=\{5,6,7,8,9\}, \\
& A_5=\{0,1,8,9\}
\end{aligned}
$$
b)
$$
\begin{aligned}
& A_1=\{\ldots,-4,-3,-2,-1,0\}, \\
& A_2=\{\ldots,-2,-1,0,1,2, \ldots\}, \\
& A_3=\{\ldots,-6,-4,-2,0,2,4,6, \ldots\}, \\
& A_4=\{\ldots,-5,-3,-1,1,3,5, \ldots\}, \\
& A_5=\{\ldots,-6,-3,0,3,6, \ldots\}
\end{aligned}
$$
c)
$$
\begin{aligned}
& A_1=\{x \mid x<0\}, \\
& A_2=\{x \mid-1<x<0\}, \\
& A_3=\{x \mid 0<x<1\}, \\
& A_4=\{x \mid-1<x<1\}, \\
& A_5=\{x \mid x>-1\}, \\
& A_6=\mathbf{R}
\end{aligned}
$$

Chris Trentman

Numerade Educator

Problem 14

Use the niche overlap graph in Figure 6 to determine the species that compete with hawks.

Prathan Jarupoonphol

Prathan Jarupoonphol

Numerade Educator

Problem 15

Construct a niche overlap graph for six species of birds, where the hermit thrush competes with the robin and with the blue jay, the robin also competes with the mockingbird, the mockingbird also competes with the blue jay, and the nuthatch competes with the hairy woodpecker.

Chris Trentman

Numerade Educator

Problem 16

Draw the acquaintanceship graph that represents that Tom and Patricia, Tom and Hope, Tom and Sandy, Tom and Amy, Tom and Marika, Jeff and Patricia, Jeff and Mary, Patricia and Hope, Amy and Hope, and Amy and Marika know each other, but none of the other pairs of people listed know each other.

Prathan Jarupoonphol

Prathan Jarupoonphol

Numerade Educator

Problem 17

We can use a graph to represent whether two people were alive at the same time. Draw such a graph to represent whether each pair of the mathematicians and computer scientists with biographies in the first four chapters of this book who died before 1900 were contemporaneous. (Assume two people lived at the same time if they were alive during the same year.)

JS

Jamie Shive

Numerade Educator

Problem 18

Who can influence Fred and whom can Fred influence in the influence graph in Example 3?

Prathan Jarupoonphol

Prathan Jarupoonphol

Numerade Educator

Problem 19

Construct an influence graph for the board members of a company if the President can influence the Director of Research and Development, the Director of Marketing, and the Director of Operations; the Director of Research and Development can influence the Director of Operations; the Director of Marketing can influence the Director of Operations; and no one can influence, or be influenced by, the Chief Financial Officer.

Chris Trentman

Numerade Educator

Problem 20

Which other teams did Team 4 beat and which teams beat Team 4 in the round-robin tournament represented by the graph in Figure 9?

Prathan Jarupoonphol

Prathan Jarupoonphol

Numerade Educator

Problem 21

In a round-robin tournament the Tigers beat the Blue Jays, the Tigers beat the Cardinals, the Tigers beat the Orioles, the Blue Jays beat the Cardinals, the Blue Jays beat the Orioles, and the Cardinals beat the Orioles. Model this outcome with a directed graph.

Chris Trentman

Numerade Educator

Problem 22

Draw the call graph for the telephone numbers 555-0011, 555-1221,555-1333,555-8888,555-2222,555-0091, and 555-1200 if there were three calls from 555-0011 to 5558888 and two calls from $555-8888$ to 555-0011, two calls from 555-2222 to 555-0091, two calls from 555-1221 to each of the other numbers, and one call from 555-1333 to each of 555-0011, 555-1221, and 555-1200.

Prathan Jarupoonphol

Prathan Jarupoonphol

Numerade Educator

Problem 23

Explain how the two telephone call graphs for calls made during the month of January and calls made during the month of February can be used to determine the new telephone numbers of people who have changed their telephone numbers.

Chris Trentman

Numerade Educator

Problem 24

a) Explain how graphs can be used to model electronic mail messages in a network. Should the edges be directed or undirected? Should multiple edges be allowed? Should loops be allowed?
b) Describe a graph that models the electronic mail sent in a network in a particular week.

Chris Trentman

Numerade Educator

Problem 25

How can a graph that models e-mail messages sent in a network be used to find people who have recently changed their primary e-mail address?

Prashant Bana

Numerade Educator

Problem 26

How can a graph that models e-mail messages sent in a network be used to find electronic mail mailing lists used to send the same message to many different e-mail addresses?

Nick Johnson

Numerade Educator

Problem 27

Describe a graph model that represents whether each person at a party knows the name of each other person at the party. Should the edges be directed or undirected? Should multiple edges be allowed? Should loops be allowed?

Chris Trentman

Numerade Educator

Problem 28

Describe a graph model that represents a subway system in a large city. Should edges be directed or undirected? Should multiple edges be allowed? Should loops be allowed?

Prathan Jarupoonphol

Prathan Jarupoonphol

Numerade Educator

Problem 29

Describe a graph model that represents traditional marriages between men and women. Does this graph have any special properties?

Chris Trentman

Numerade Educator

Problem 30

Which statements must be executed before $S_6$ is executed in the program in Example 9? (Use the precedence graph in Figure 11.)

Chris Trentman

Numerade Educator

Problem 31

Construct a precedence graph for the following program:
$$
\begin{aligned}
& S_1: x:=0 \\
& S_2: x:=x+1 \\
& S_3: y:=2 \\
& S_4: z:=y \\
& S_5: x:=x+2 \\
& S_6: y:=x+z \\
& S_7: z:=4
\end{aligned}
$$

Chris Trentman

Numerade Educator

Problem 32

Describe a discrete structure based on a graph that can be used to model airline routes and their flight times.

Prathan Jarupoonphol

Prathan Jarupoonphol

Numerade Educator

Problem 33

Describe a discrete structure based on a graph that can be used to model relationships between pairs of individuals in a group, where each individual may either like, dislike, or be neutral about another individual, and the reverse relationship may be different.

Chris Trentman

Numerade Educator

Problem 34

Describe a graph model that can be used to represent all forms of electronic communication between two people in a single graph. What kind of graph is needed?

Prathan Jarupoonphol

Prathan Jarupoonphol

Numerade Educator