70
Math 115 – 61 & 71 Quiz #4
Name:
Do all of your work on a separate sheet of paper and clearly write your answers on this sheet.
Questions 1 - 7 refer to the graph below.
B
A
E
C
D
1.
What is the degree of vertex D? (3 points)
2.
What is the degree of vertex C? (3 points)
3.
Does the graph have any bridges? If so, which edge(s)? (3 points)
4.
Is the following a path from A to D? If not explain why not. (3 points)
ACCDED
5.
Is the following a path from D to B? If not explain why not. (3 points)
DEDCEDB
6.
Does the graph have an Euler Circuit? If so, find it. If not, explain why not. (5 points)
7.
Does the graph have an Euler Path? If so, find it. If not, explain why not. (5 points)
Question 8 refers to the following situation: An undercover police officer is assigned the job of walking once a
night each of the 48 blocks of a certain section of town shown in the street grid below. The walk should start and
end at his car at point A. The officer wants to minimize the total number of blocks he has to walk each night.
A
PARK
8.
How many blocks must the officer repeat in order to complete his walk in the most efficient manner? (10 points)
Questions 9 - 11 refer to the following situation: In a certain town there is a river running through the middle of the
city. There are three islands and nine bridges as shown in the figure below.
North Bank
A
B
C
South Bank
9.
Draw a graph that models this situation. (5 points)
10. Is it possible to take a walk in this town, starting on the South Bank, crossing each bridge exactly once (and
only once) and ending on island A? If so, how? If not, explain why not? (10 points)
11. Is it possible to take a walk in this town, starting on the South Bank, crossing each bridge exactly once (and
only once) and ending on island C? If so, how? If not, explain why not? (10 points)
12. Define the following graph theory terms: (5 points each)
(a) Euler Circuit
(b) bridge
Quiz 4 – Answers
1.
5
2.
5
3.
Yes – BD
4.
Yes
5.
No because edge ED is used three times, but there are only 2 such edges in the graph.
6.
No because there are some vertices with odd degree.
7.
No because there are more than 2 vertices with odd degree.
8.
He must repeat 12 blocks.
9.
N
A
C
B
S
10. No. This describes an Euler path (which is supposed to start and end at an odd degree vertex) and vertex A has
even degree.
11. Yes, for example – SABSCBNBNC
12. An Euler circuit is a path in which we start and end at the same vertex and use every edge exactly once.
A bridge is an edge such that if removed causes the graph to become disconnected.