APPENDIX
GRAPH THEORY
Exercise Set
1. A vertex is a designated point.
2. An edge (or an arc) is any line, either straight or curved, that begins and ends at a vertex.
3. To determine whether a vertex is odd or even, count the number of edges attached to the vertex.
If the number of edges is odd, the vertex is odd. If the number of edges is even, the vertex is even.
4. Answers will vary.
5.
7.
9.
11.
5 vertices, 7 edges
7 vertices, 11 edges
Each graph has the same number of edges from the
corresponding vertices.
Odd vertices: C, D
Even vertices: A, B
6.
8.
10.
12.
6 vertices, 8 edges
5 vertices, 6 edges
Each graph has the same number of edges from
the corresponding vertices.
Odd vertices: A, C, E, F
Even vertices: B, D
13. Yes. The figure has exactly two odd vertices, namely C and D. Therefore, the figure is traversable.
You may start at C and end at D, or start at D and end at C.
14. No. All four vertices are odd. There are more than two odd vertices. Therefore, the figure is not traversable.
15. Yes. The figure has no odd vertices. Therefore, the figure is traversable. You may start at any point
and end where you started.
16. Yes. The figure has no odd vertices. Therefore, the figure is traversable. You may start at any point
and end where you started.
17. No. The figure has four odd vertices, namely A, B, E, and F. There are more than two odd vertices.
Therefore, the figure is not traversable.
18. Yes. The figure has exactly two odd vertices, namely C and G. Therefore, the figure is traversable.
You may start at C and end at G, or start at G and end at C.
19. Yes. The figure has exactly two odd vertices, namely A and C. Therefore, the figure is traversable.
You may start at A and end at C, or start at C and end at A.
20. Yes. The figure has no odd vertices. Therefore, the figure is traversable. You may start at any point
and end where you started.
21. a) 0 rooms have an odd number of doors.
5 rooms have an even number of doors.
b) Yes because the figure would have no odd vertices.
c) Start in any room and end where you began. For example: A to D to B to C to E to A.
22. a) 4 rooms have an odd number of doors.
1 room has an even number of doors.
b) No because the figure would have more than two odd vertices.
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518 APPENDIX Graph Theory
23. a) 2 rooms have an odd number of doors.
4 rooms have an even number of doors.
b) Yes because the figure would have exactly two odd vertices.
c) Start at B and end at F, or start at F and end at B.
For example: B to C to F to E to D to A to B to E to F
24. a) 2 rooms have an odd number of doors.
4 rooms have an even number of doors.
b) Yes because the figure would have exactly two odd vertices.
c) Start at B and end at E, or start at E and end at B. For example: B to A to D to E to F to C to B to E
25. a) 4 rooms have an odd number of doors.
1 room has an even number of doors.
b) No because the figure would have more than two odd vertices.
26. a) 5 rooms have an odd number of doors.
1 room has an even number of doors.
b) No because the figure would have more than two odd vertices.
27. a) 3 rooms have an odd number of doors.
2 rooms have an even number of doors.
b) No because the figure would have more than two odd vertices.
28. a) 3 rooms have an odd number of doors.
4 rooms have an even number of doors.
b) No because the figure would have more than two odd vertices.
29. The door must be placed in room D. Adding a door to any other room would create two rooms with
an odd number of vertices. You would then be unable to enter the building through the door marked
"enter" and exit through the new door without going through a door at least twice.
30. The door must be placed in room D. Adding a door to any other room would create two rooms with
an odd number of vertices. You would then be unable to enter the building through the door marked
"enter" and exit through the new door without going through a door at least twice.
31. Yes because the figure would have exactly two odd vertices. Begin at either the island on the left or
on the right and end at the other island.
32. Yes because the figure would have exactly two odd vertices. Begin at the island on the right
and end on the land below the island, or vice versa.
33.
A
B
34.
B
A
D
C
C
E
D
APPENDIX
35. a) Kentucky, Virginia, North Carolina, Georgia, Alabama, Mississippi, Arkansas, Missouri
b) Illinois, Arkansas, Tennessee
36. a) French Guiana, Surinam, Guyana, Venezuela, Columbia, Peru, Bolivia, Paraguay, Argentina,
Uruguay
b) Peru, Chile, Argentina, Paraguay, Brazil
37. a) 4
b) 4
c) 11
38.
Dawn
Pam
Bill
Scott
Jessica
Ed
39.
France
Spain
Belgium
Netherlands
Germany
Switzerland
Italy
40.
41.
42.
Poland
Czech Republic
Austria
No, it is not possible, assuming that your starting and ending points are considered vertices.
a) Yes, the graph has exactly two odd vertices, namely C and G.
b) C, A, B, E, F, D, G, C
Number of Edges = Number of Vertices + Number of Regions - 2
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