CMU 15-251 Graphs: Basics Teachers: Anil Ada Ariel Procaccia - PowerPoint PPT Presentation

CMU 15-251 Graphs: Basics Teachers: Anil Ada Ariel Procaccia (this time)

Zachary Karate Club 2

Zachary Karate Club CLUB networkkarate.tumblr.com 3

Facebook 4

Facebook 𝑜 = 10 9 𝑛 = 10 12 5

6 Donor 2 Exchange Patient 2 Kidney Patient 1 Donor 1

Kidney Exchange 7

World Wide Web 8

9

Types of graphs 𝑏 𝑏 𝑏 𝑐 𝑐 𝑐 𝑑 𝑑 𝑑 𝑒 𝑒 𝑒 10

Retronym 11

Basic Definitions 𝐻 • 𝑊 𝑊 = 𝑜 o 𝐹 𝐹 = 𝑛 o 𝑏 {𝑣, 𝑤} • 𝑐 𝑣 ≠ 𝑤 𝑑 • 𝑊 = 𝑏, 𝑐, 𝑑, 𝑒 𝑒 o 𝐹 = { {𝑏, 𝑐}, {𝑏, 𝑑}, {𝑐, 𝑑}, 𝑑, 𝑒 } o 12

Edge Cases • • 𝑊 = 1,2,3,4 o 𝐹 = ∅ o 13

The Null Graph 14

The Null Graph 15

Mr. Vertex’s Neighborhood 𝑣, 𝑤 ∈ 𝐹 𝑣 • 𝑏 𝑤 𝑐 • 𝑑 𝑂(𝑣) 𝑣 𝑤 ∈ 𝑊 𝑣, 𝑤 ∈ 𝐹} 𝑒 deg(𝑣) • 𝑂 𝑐 = 𝑏, 𝑑 𝑣 𝑂 𝑣 deg 𝑐 = 2 16

𝑣∈𝑊 deg(𝑣) = 2𝑛 • • • • • • o o 𝑣∈𝑊 deg(𝑣) • •• o • o 2𝑛 ∎ 2 + 2 + 3 + 1 = 2 ⋅ 4 17

Facebook, revisited 𝑜 = 10 9 𝑛 = 10 12 18

Regular Graphs 𝑒 • 𝑒 0 • • 1 2 • 𝑊 = {𝑏, 𝑐, 𝑑, 𝑒} 1 1 3 6 12 19

3-regular graphs 20

Connectedness 𝐻 𝑣, 𝑤 ∈ 𝑊 • 𝑣 𝑤 21

Connectedness 22

𝑜 = 1 𝑜 = 2 𝑜 = 3 𝑛 = 1 𝑛 = 2 𝑛 = 0 𝑜 = 4 𝑛 = 3 23

𝑜 − 1 𝑜 24

𝑜 − 1 • 𝑜 • 𝐻 𝑙 o 𝐻′ 𝐻 𝐻′ 𝑙 − 1 o 𝑜 − 1 ∎ 25

Acyclic graphs 𝐻 • 𝑛 = 𝑜 − 1 ⇒ 𝐻 𝐻 ⇒ 𝑛 = 𝑜 − 1 𝐻 ⇔ 𝑛 = 𝑜 − 1 26

Trees 27

Graph Theory Haiku 28

Hamiltonian Cycle * 𝐻 • 𝑤 ∈ 𝑊 𝐻 • 𝑜 ≥ 3 deg 𝑣 + deg 𝑤 ≥ 𝑜 𝑣, 𝑤 ∈ 𝑊 𝐻 29

Proof * 𝐻 • 𝐷 𝐷 • 𝐷′ 30

Proof * {𝑏, 𝑐} 𝐷 • 𝑇 • 𝑏 𝑏 𝑐 𝑂(𝑏) 𝐷 • deg 𝑐 ≥ 𝑜 − deg(a) = 𝑊 − 𝑂 𝑏 = 𝑊 − S 𝑑 > |𝑊 ∖ (𝑇 ∪ 𝑐 )| 𝑐 𝑑 ∈ 𝑇 • ∎ • 31

Summary • o o o o • 𝐻 , o 𝐹 = 𝑜 − 1 ⇔ 𝑣∈𝑊 deg(𝑣) = 2𝑛 o o 32