Simple Graphs: if they remain connected k-Connectivity whenever - - PowerPoint PPT Presentation

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Nov 11, 2022 161 likes •236 views

Edge Connectedness Mathematics for Computer Science MIT 6.042J/18.062J Def: vertices v, w are k-edge connected Simple Graphs: if they remain connected k-Connectivity whenever fewer than k edges are deleted. Albert R Meyer, April 5, 2013

SLIDE 1

Albert R Meyer, April 5, 2013

Simple Graphs: k-Connectivity

k-connect.1

Mathematics for Computer Science MIT 6.042J/18.062J

Albert R Meyer, April 5, 2013

Def: vertices v, w are

k-edge connected if they remain connected whenever fewer than k edges are deleted.

k-connect.2

Edge Connectedness

Albert R Meyer, April 5, 2013

k-edge Connectedness

k-connect.3

1-edge connected

Albert R Meyer, April 5, 2013

k-edge Connectedness

k-connect.4

1-edge connected no path delete

SLIDE 2

Albert R Meyer, April 5, 2013

Edge Connectedness

k-connect.5

2-edge connected

Albert R Meyer, April 5, 2013

Edge Connectedness

k-connect.6

2-edge connected no path

Albert R Meyer, April 5, 2013

Edge Connectedness

k-connect.7

3-edge connected

Albert R Meyer, April 5, 2013

Edge Connectedness

k-connect.8

3-edge connected no path

SLIDE 3

Albert R Meyer, April 5, 2013

k-edge Connectedness

Def: A graph is

k-edge connected iff every two vertices are k-edge connected.

k-connect.9

Albert R Meyer, April 5, 2013

k-connect.10

Connectivity measures fault tolerance of a network: how many connections can fail without cutting off communication?

Edge Connectedness

Albert R Meyer, April 5, 2013

delete

k-edge Connectedness

k-connect.11

1-edge connected this whole graph is

Albert R Meyer, April 5, 2013

k-edge Connectedness

k-connect.12

2-edge connected this whole graph is

SLIDE 4

Albert R Meyer, April 5, 2013

k-vertex Connectedness

k-connect.13

k-vertex connectedness defined similarly

Albert R Meyer, April 5, 2013

k-vertex Connectedness

k-connect.14

k-vertex connected

IMPLIES

k-edge connected not conversely:

2-edge connected 1-vertex connected

Albert R Meyer, April 5, 2013

k-vertex Connectedness

Kn is the complete graph on n vertices. Kn is (n-1)-vertex connected.

k-connect.15

Albert R Meyer, April 5, 2013

k-vertex Connectedness

The n-dimensional

hypercube Hn V(Hn) ::= {0,1}n an edge IFF u,v differ in 1 place

u− v

k-connect.16

SLIDE 5

Albert R Meyer, April 5, 2013

k-vertex Connectedness

Hn is n-vertex

connected.

(class problem)

k-connect.17

Albert R Meyer, April 5, 2013

Menger’s Theorem

k-connected vertices will be connected by k non-overlapping paths

k-connect.18

SLIDE 6

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6.042J / 18.062J Mathematics for Computer Science

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