Chapter 5: Euler Circuits
1. Vertex A is adjacent to
A) vertex D only.
C) vertex C and vertex D only.
E) None of the above
B) every other vertex.
D) vertex B and vertex E only.
2. The degree of vertex D is
A) 3
B) 4
C) 5
D) 4.5
E) None of the above
3. The degree of vertex C is
A) 5
B)4
C) 3
D) 2
E) None of the above
4. Which of the following is not a path from vertex D to vertex A?
A)
D, E, C, D, A
B)
D, E, D, C, C, A
D)
D, C, C, A
E, D, C, C, E
D, C, A
E) All of the above are paths from D to A.
5. Which of the following is not a circuit in the graph?
A)
A, D, C, D, E, D, A
B)
A, D, E, C, A
D)
C)
C)
C, A, D, C
E) All of the above are circuits in the graph.
6. Which of the following is a bridge of the graph?
A) EC
B) BD
C) AD
7. Which of the graphs has an Euler circuit?
A) Graph 1 only
B)Graph 3 only
E) None of the above
D) CC
C) Graph 2 only
E) None of the above
D)Graphs 1 and 3
8. Which of the graphs has an Euler path but no Euler circuit?
A) Graph 1 only
E) None of the above
B) Graph 3 only
C)Graph 2 only
D)Graphs 1 and 2
Questions 9 to 11 refer 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 described by the street grid shown at
the right. The walk starts and ends at A. The officer
wants to minimize the total number of blocks he has to
walk each night.
9. How many vertices of odd degree are there in the graph representing this problem?
A) 24
B) 18
C) 22
D) 20
E) None of the above
10 . An optimal eulerization of the graph representing this problem can be obtained by adding
A) 10 edges.
B) 12 edges.
C) 9 edges.
D) 11 edges. E) None of the above
11. Suppose that it takes the officer 5 minutes to walk a block. In an optimal trip, the officer will
cover the entire neighborhood in
A) 4 hours and 45 minutes.
B) 5 hours and 10 minutes.
C)4 hours and 50 minutes.
D) 5 hours.
E) None of the above
12 . A graph has an Euler circuit if
A) it is connected and has an even number of vertices.
B) it is connected and has an even number of edges.
C) it is connected and every vertex has even degree.
D) every vertex has even degree.
E) None of the above
13 . A graph with 11 vertices has an Euler path but no Euler circuit. The graph must have
A) 2 vertices of odd degree and 9 vertices of even degree.
B) 11 vertices of even degree.
C) 11 vertices of odd degree.
D) 2 vertices of even degree and 9 vertices of odd degree
E) None of the above
14 . A graph has six vertices—two vertices of degree 4, two vertices of degree 3, and two vertices of
degree 2. The number of edges in the graph is
A) 8
B) 9
C) 6
D) 5
E) None of the above
15 . The basic rule in Fleury’s algorithm is
A)only travel across a bridge of the untraveled part of the graph if there is no other alternative.
B) only travel across a bridge of the original graph if there is no other alternative.
C) never travel across a bridge of the untraveled part of the graph.
D)never travel across a bridge of the original graph.
E) None of the above
16. Vertex E is adjacent to
A) vertex B, C and D only.
B) vertex C and vertex D only.
C) vertex B and vertex D only. D) every other vertex.
D) None of the above
17. The degree of vertex D is
A) 3
B) 4
C) 2
D)1
E)None of the above
18. The degree of vertex C is
A) 3
B) 4
C) 2
D)1
E)None of the above
19. Which of the following is not a path from vertex D to vertex A?
A)
D)
D, E, B, D, C, B, A
D, A
B)
D, C, C, B, A
C)
D, C, B, A
E) All of the above are paths from D to A.
20. Which of the following is not a circuit in the graph?
A)
D)
B, C, D, B
B, D, E, B
B)
B, D, E, B, A, D, B
C)
B, C, C, D, B
E) All of the above are circuits in the graph.
21. Which of the following is a bridge of the graph?
A) AD
B) DE
C) BD
D) AB
E) There are no bridges in this graph.
Questions 22 and 23 refer to a graph with vertices A, B, C, D, E and edges BC, CE, AB, AC, and BD.
Draw IT:
22. The degree of vertex B is
A) 1.
B) 0.
C) 3.
D) 2.
E) None of the above
D) BC
E) None of the above
23. Which of the following is a bridge of the graph?
A)
AB
B)
BD
C) AC
24.
25 .
Which graph has an Euler circuit?
A) Graph 4
B) Graph 2
C) Graph 3
E) None of the above
Which graph(s) are disconnected?
A) Graph 3 only
D) Graph 2 only
26.
D) Graph 1
B) Graph 1 and Graph 4
E) None of the above
C) Graph 2 and Graph 3
Which graph has an Euler path but no Euler circuit?
A) Graph 3
B) Graph 1
C) Graph 2
D) Graph 4
E) None of the above
27. Which of the drawings has an Euler Path?
A) Figure 1 only
B)Figure 3 only
C) Figure 2 only
D) Figure 2 and Figure 3 only
E)None of the above
28. Which of the drawings has an Euler Circuit?
A) Figure 2 and Figure 3 only
B) Figure 3 only
D) Figure 2 only
E) None of the above
29
C) Figure 1 only
A graph has seven vertices–two vertices of degree 6, four vertices of degree 5, and one vertex
of degree 2. The number of edges in the graph is
A) 17.
B) 7.
C) 13.
D) 14.
E) None of the above
30. In a certain city there is a river running through the middle
of the city. There are four islands and nine bridges as shown. A
graph model might have:
A) 4 vertices and 9 edges.
B) 9 vertices and 6 edges.
C) 9 vertices and 4 edges.
D) 6 vertices and 9 edges.
E) None of the above