1. Playing with Graphs
Alex Andreotti

2. Countries of Bug World

3. Countries of Bug World

4. Countries of Bug World Bugs are stupid: can’t measure length, curving.
Tunnels are dark, and bugs can’t see much anyway.

5. Countries of Bug World
Country

6. Countries of Bug World
City

7. Countries of Bug World
City
City
City
City

8. Countries of Bug World
Tunnels

9. Bug An lives in this country
Countries of Bug World
Bug An lives in this country

10. Bug Adam lives in this one
Countries of Bug World
Bug Adam lives in this one

11. Can they figure out they live in different countries?
Countries of Bug World
Can they figure out they live in different countries?

12. Countries of Bug World
An’s Country
Jeremy’s Country

13. Same or Different?

14. Same or Different?

15. Same or Different?

16. Same or Different?

17. Same or Different?

18. Same or Different?

19. Same or Different?

20. Same or Different?

21. Countries of Bug World
Task 1:Devise a precise description of what it means for two countries to be “the same” as far as the bugs are concerned.

22. Countries of Bug World
Task 2:Julie-Bug says “My country has seven cities and nine tunnels.
One city has just one tunnel connected to it,
one city has five tunnels connected to it,
two cities have three tunnels connected to them, and
the other three cities have two connecting tunnels”
How many different countries can you draw that fit this description? How can you tell them apart?

23. Countries of Bug World
Graph Theory

24. Graph Theory
Countries
Graphs

25. Graph Theory
Countries
Cities
Graphs
Points/Vertices

26. Graph Theory Countries Cities Tunnels Graphs Points/Vertices
Lines/Edges

27. Graph Theory
Graph

28. Graph Theory
Vertex

29. Graph Theory
Vertices

30. Note: One Vertex, Many Vertices
Graph Theory
Vertex
Vertices
Note: One Vertex, Many Vertices

31. Graph Theory
Edges

32. Graph Theory
One Graph…

33. Graph Theory
One Graph…

34. Graph Theory
One Graph…

35. Graph Theory
One Graph…

36. Graph Theory
One Graph…

37. Graph Theory
One Graph…

38. Graph Theory
One Graph…

39. One Graph, Many Graph Diagrams
Graph Theory
One Graph, Many Graph Diagrams

40. Here’s another graph diagram of the same graph
Graph Theory
Here’s another graph diagram of the same graph

41. Graph Theory
Connected Graph

42. Graph Theory
Disconnected Graph

43. Disconnected graph is just a collection of connected components
Graph Theory
Disconnected Graph
Component
Component
Disconnected graph is just a collection of connected components

44. Diagrams are just one way to represent (some) graphs.
Graph Theory
Diagrams are just one way to represent (some) graphs.

45. Graph Theory Other examples:
People (vertices), know each other (edges)

46. Graph Theory Other examples:
Tennis players (vertices), played against (edges)

47. Graph Theory Other examples:
Countries (vertices), share a border (edges)

48. Graph Theory
New Rules

49. Graph Theory
New Rules
1. No Loops

50. Graph Theory
New Rules
1. No Loops
2. No multiple edges

51. Draw all the graphs with 4 vertices.
Task 3
Draw all the graphs with 4 vertices.

52. Task 3 Draw all the graphs with 4 vertices. • • • •
Here’s one with no edges

53. Draw all the graphs with 4 vertices.
Task 3
Draw all the graphs with 4 vertices. • • • • • • • •
Is there a different graph (not a graph diagram, but a graph), with only one edge?

54. Draw all the graphs with 4 vertices.
Task 3
Draw all the graphs with 4 vertices. • • • • • • • •
Same or different?

55. Task 3 Draw all the graphs with 4 vertices. • • • • • • • •
Find all the other graphs in our catalog.

56. Task 4
Draw all the graphs with 5 vertices. (Hint: there are between 30 and 40. Look for patterns.)

57. Task 5
Explain the patterns that we find while counting graphs with 1, 2, 3, etc. edges in graphs with n vertices.

58. Task 6
Device a simplest way to communicate a graph (not a graph diagram, just a graph) over the telephone.

59. Paths a b g c f e d

60. Paths a b g c f e d
(d, c, …)

61. Paths a b g c f e d
(d, c, b, …)

62. Paths a b g c f e d
(d, c, b, d)

63. Paths a b g c f e d
(f, e, d, b, c)

64. Paths a • b g • c f e d
(e, g, b, a, b, g)

65. Paths • • • • • • •
closed
not closed
not closed

66. Paths • • f
closed
simple
not closed
simple
not closed
not simple

67. Paths not closed not closed closed+simple simple not simple circuit llg
not closed
simple
not closed
not simple
closed+simple ll
circuit

68. Trees

69. Trees
Not a Tree
Tree

70. Trees
Draw a bunch of trees

71. Task 7 What is the relationship between the number of vertices
and the number of edges in a tree?
Why does this relationship hold?

72. Task 8 Draw all the trees with 5 vertices. Then all the ones with 6.
There are between 20 and 30 with 8 vertices. Find them.

73. Task 9 Which algorithm/method would you use if you had to find
all trees with a given number of vertices (say, 12)?

74. Task 10 Revisit the method we had to communicate a graph
over the phone. Do you have a better, simpler one
that would work just for trees?

75. Trees in Graphs

76. Trees in Graphs
A Spanning Tree

77. Trees in Graphs Can you find another one?
How can we count how many there are in a given graph?

78. Regions in Graphs

79. Regions in Graphs 1

80. Regions in Graphs 1 2

81. Regions in Graphs 1 2 4 3

82. Regions in Graphs 1 5 2 4 3
The Outside region

83. (Regions are aka Faces)
Regions in Graphs 1 5 2 4 3
(Regions are aka Faces)

84. Draw several graphs and record the following:
Task 11
Draw several graphs and record the following:
Number of vertices (v)
Number of edges (e)
Number of faces (f)
Number of vertices in Spanning Tree (A)
Number of edges in Spanning Tree (B)
Number of edges not in Sp. Tree (C)

85. Task 11

86. Task 11 • • • • •
v=8 • • •

87. Task 11 3 1 2 4 5
v=8
e=11 11 6 9 8 10 7

88. Task 11 5 1
v=8
e=11
f=5 2 4 3

89. Task 11 • • •
v=8
e=11
f=5
A=8 • • • • •

90. Task 11 7
v=8
e=11
f=5
A=8
B=7 1 2 3 6 4 5

91. Task 11
v=8
e=11
f=5
A=8
B=7
C=4

92. Task 11 Draw several graphs and record the following:
Number of vertices (v)
Number of edges (e)
Number of faces (f)
Number of vertices in Spanning Tree (A)
Number of edges in Spanning Tree (B)
Number of edges not in Sp. Tree (C)
Do only connected graphs w/o crossing edges

93. What patterns did you find? Explain why they work.
Task 12
What patterns did you find? Explain why they work.

94. Use your findings to show
Task 13
Use your findings to show
v – e + f = 2

95. Euler’s Formula
v – e + f = 2

96. Euler’s Formula
“Oiler’s”
v – e + f = 2

97. A graph has 7 vertices and 9 edges. How many faces should it have?
Task 13
A graph has 7 vertices and 9 edges. How many faces should it have?
Try to draw a graph with 7 v’s and 9 e’s without the requisite number of faces.

98. Euler’s Formula? • • • •

99. Euler’s Formula? • •
v=4
e=6
f=5 • •

100. Euler’s Formula? • •
v=4
e=6
f=5 • •
4 – ≠ 2 ???

101. Euler’s Formula? • •
v=4
e=6
f=5 • •
“Fake” regions

102. Euler’s Formula? 4 • • • •
v=4
e=6
f=4 3 1 2 • • • •
“Fake” regions

103. Euler’s Formula? v=4 e=6 f=4 4 – 6 + 4 = 2 “Fake” regions • • • • • •3 1 2 • • • •
4 – = 2
“Fake” regions

104. Planar version of same graph
Euler’s Formula? • • • • • • • •
Non-planar graph
Planar version of same graph

105. Euler’s Formula Non-planar graph Planar version of same graph• • • • • • • •
Non-planar graph
Planar version of same graph
Euler’s only works on planar graphs

106. Do all graphs have a planar version?
Important Question
Do all graphs have a planar version?
If not, which ones do? • • • • • • • •

107. Task 14
What happens with Euler’s Formula when a graph is NOT connected? Can you fix it?

108. n vertices - all connected to each other
K Graphs
n vertices - all connected to each other

109. K Graphs • • • • K4

110. How many edges does K5, K6, Kn have?
Task 15
How many edges does K5, K6, Kn have?

111. Draw K5, both in non-planar, and in planar versions.
Task 16
Draw K5, both in non-planar, and in planar versions.

112. Task 16 Draw K5, both in non-planar, and in planar versions.
You are failing at this, aren’t you?

113. Task 16 Draw K5, both in non-planar, and in planar versions.
Let’s see if you can prove it’s not possible

114. Let’s focus on this face
Orders of Faces
Let’s focus on this face

115. Orders of Faces It has 4 edges as boundary
we call this the order of the face

116. Orders of Faces
o(f)=4

117. Orders of Faces
o(f)=3

118. Task 17 Claim 1: Sum of all orders of all faces = 2 x number of edges
Draw a bunch of graphs and see if true. Why is it true?

119. Task 18 Claim 2: 3 x number of faces ≤ 2 x number of edges
is true for all (planar) graphs.
Try to prove me wrong, find which graphs have 3f=2e,
and see if you can understand why my claim is true.

120. Task 18 Claim 2: 3 x number of faces ≤ 2 x number of edges
is true for all (planar) graphs.
aka, Every face is at least a triangle

121. Task 18 Claim 2: 3 x number of faces ≤ 2 x number of edges
is true for all (planar) graphs.
aka, Every face is at least a triangle
(and each edge has two faces - so it’s counted twice)

122. Task 19 Proving that K5 is not planar:
Use Euler’s Formula and 3f≤2e to prove it.
(Hint: What do we know about K5?)

123. Task 20 Draw a planar version of the following graphs,
or prove that it’s not possible to do so:
v=7, e=17
v=8, e=12
v=7, e=15