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

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

Dec 13, 2023 185 likes β€’524 views

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 = 10 9 = 10 12 5 6 Donor 2 Exchange Patient 2 Kidney

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SLIDE 1

CMU 15-251

Graphs: Basics

Teachers: Anil Ada Ariel Procaccia (this time)

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SLIDE 2

Zachary Karate Club

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SLIDE 3

Zachary Karate Club CLUB

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networkkarate.tumblr.com

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Facebook

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SLIDE 5

Facebook

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π‘œ = 109 𝑛 = 1012

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SLIDE 6

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Donor 2 Patient 2 Donor 1 Patient 1

Kidney Exchange

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Kidney Exchange

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SLIDE 8

World Wide Web

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SLIDE 9

9

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SLIDE 10

Types of graphs

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𝑏 𝑐 𝑑 𝑒 𝑏 𝑐 𝑑 𝑒 𝑏 𝑑 𝑐 𝑒

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SLIDE 11

Retronym

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Basic Definitions

  • 𝐻
  • π‘Š

π‘Š = π‘œ

  • 𝐹

𝐹 = 𝑛

  • {𝑣, 𝑀}

𝑣 β‰  𝑀

  • π‘Š = 𝑏, 𝑐, 𝑑, 𝑒
  • 𝐹 = { {𝑏, 𝑐}, {𝑏, 𝑑}, {𝑐, 𝑑}, 𝑑, 𝑒 }

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𝑏 𝑐 𝑑 𝑒

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Edge Cases

  • π‘Š = 1,2,3,4
  • 𝐹 = βˆ…

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The Null Graph

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SLIDE 15

The Null Graph

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  • Mr. Vertex’s Neighborhood
  • 𝑣, 𝑀 ∈ 𝐹 𝑣

𝑀

  • 𝑂(𝑣)

𝑣 𝑀 ∈ π‘Š 𝑣, 𝑀 ∈ 𝐹}

  • deg(𝑣)

𝑣 𝑂 𝑣

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𝑏 𝑐 𝑑 𝑒 𝑂 𝑐 = 𝑏, 𝑑 deg 𝑐 = 2

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  • π‘£βˆˆπ‘Š deg(𝑣) = 2𝑛
  • π‘£βˆˆπ‘Š deg(𝑣)
  • 2𝑛 ∎

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  • β€’
  • β€’
  • 2 + 2 + 3 + 1 = 2 β‹… 4
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SLIDE 18

Facebook, revisited

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π‘œ = 109 𝑛 = 1012

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Regular Graphs

  • 𝑒

𝑒

  • 1
  • 2

π‘Š = {𝑏, 𝑐, 𝑑, 𝑒}

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1 3 6 12 1

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3-regular graphs

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Connectedness

  • 𝐻

𝑣, 𝑀 ∈ π‘Š 𝑣 𝑀

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SLIDE 22

Connectedness

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SLIDE 23

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π‘œ = 1 𝑛 = 0 π‘œ = 2 𝑛 = 1 π‘œ = 3 𝑛 = 2 π‘œ = 4 𝑛 = 3

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SLIDE 24

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π‘œ βˆ’ 1 π‘œ

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  • π‘œ βˆ’ 1

π‘œ

  • 𝐻

𝑙 𝐻′ 𝐻 𝐻′ 𝑙 βˆ’ 1

  • π‘œ βˆ’ 1

∎

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Acyclic graphs

  • 𝐻

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𝑛 = π‘œ βˆ’ 1 β‡’ 𝐻 𝐻 β‡’ 𝑛 = π‘œ βˆ’ 1 𝐻 ⇔ 𝑛 = π‘œ βˆ’ 1

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Trees

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Graph Theory Haiku

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Hamiltonian Cycle *

  • 𝐻

𝑀 ∈ π‘Š

  • 𝐻

π‘œ β‰₯ 3 deg 𝑣 + deg 𝑀 β‰₯ π‘œ 𝑣, 𝑀 ∈ π‘Š 𝐻

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Proof *

  • 𝐻

𝐷

  • 𝐷

𝐷′

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Proof *

  • {𝑏, 𝑐}

𝐷

  • 𝑇

𝑂(𝑏) 𝐷

  • deg 𝑐 β‰₯ π‘œ βˆ’ deg(a)

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𝑐 𝑏

= π‘Š βˆ’ 𝑂 𝑏 = π‘Š βˆ’ S > |π‘Š βˆ– (𝑇 βˆͺ 𝑐 )|

  • 𝑐

𝑑 ∈ 𝑇

  • ∎

𝑑 𝑏

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Summary

  • 𝐻

, 𝐹 = π‘œ βˆ’ 1 ⇔

  • π‘£βˆˆπ‘Š deg(𝑣) = 2𝑛
  • 32