4.1 Connectivity and Paths: Cuts and Connectivity - PowerPoint Presentation

Slide1

4.1 Connectivity and Paths: Cuts and Connectivity

This copyrighted material is taken from Introduction to Graph Theory, 2nd Ed., by Doug West; and is not for further distribution beyond this course.These slides will be stored in a limited-access location on an IIT server and are not for distribution or use beyond Math 454/553.

1Slide2

Connectivity of Graphs

2Motivating QuestionHow many vertices, or how many edges, can be deleted from a graph while keeping it connected?

Applications (vertex connectivity)

Robustness of supercomputers to failures of processor nodes

Sensor networks’ resistance to individual sensor failure

Applications (edge connectivity)Robustness of supercomputers to failures of wires/fiber opticsReliability of road networks with road closures/accidentsCommunication networks’ resistance to link failure

Contains copyrighted material from

Introduction to Graph Theory

by Doug West, 2

nd

Ed. Not for distribution beyond IIT’s Math 454/553. Slide3

Vertex Connectivity Examples

3

4.1.1. Definition.

A

separating set

or vertex cut of a graph G is a set SV

(

G

) such that

G

–S has more than one component. The connectivity of G, written κ(G), is the minimum size of a vertex set S such that G–S is disconnected or has only one vertex. A graph G is k-connected if its connectivity is at least k.Examples

Contains copyrighted material from Introduction to Graph Theory by Doug West, 2nd Ed. Not for distribution beyond IIT’s Math 454/553.

S

2-connected

S

1-connected

S

K

n

(

n

-1)-connected

S

K

m

,

n

min(

m

,

n

)-connected

0-connected

2

K

2

:

disconnected, so

0-connectedSlide4

Vertex Connectivity Examples

44.1.1. Definition. A separating set or vertex cut of a graph

G

is a set

S

V(G) such that G–S has more than one component. The connectivity of G, written κ

(

G

), is the minimum size of a vertex set

S

such that G–S is disconnected or has only one vertex. A graph G is k-connected if its connectivity is at least k.Contains copyrighted material from Introduction to Graph Theory by Doug West, 2nd Ed. Not for distribution beyond IIT’s Math 454/553.

K1

K2K3K4

Kn (n>3)C

4Cn (n>2)

Connectivity κ 0123

n-12

21-connected?N

YY

YY

YY

2-connected?NNYYYYY3-connected?NNNYYNNSlide5

Vertex Connectivity Examples

5

4.1.1. Definition.

A

separating set

or

vertex cut

of a graph

G

is a set

S



V

(

G

) such that

G–S has more than one component. The

connectivity of G, written κ

(G), is the minimum size of a vertex set S such that G

–S is disconnected or has only one vertex. A graph G is

k-connected if its connectivity is at least k.

Hypercubes QnContains copyrighted material from Introduction to Graph Theory by Doug West, 2nd Ed. Not for distribution beyond IIT’s Math 454/553. Sk = 22-connectedκ = 2

k

= 0

0-connectedκ = 0

k

= 1

1-connected

κ

= 1

S

k

> 2

??

κ

(Qk-1) = ??Qk-1Qk-1Slide6

Vertex-Connectivity of the Hypercube

64.1.3. Example. The hypercube Qk has connectivity

κ

(

Q

k)=k for all k≥0.Proof (By induction on k.)Base cases

k

=0,1 have

κ

(

Qk)=k by examples on previous slide.Induction step Let k≥2 and assume true for smaller k.The neighborhood of any v is a vertex cut, so κ(Qk)  k.View Qk as two copies of

Qk-1 plus a perfect matching M. Suppose S is a vertex cut for Qk

. Assume Qk–S leaves ≥1 vertex in Qk-1

and Q’k'-1, else |S| ≥ 2k-1

≥ k. Case 1 Both Qk-1

–S and Q’k-1–S are connected:Unless

S contains at least one endpoint of each edge of M, there is an edge between

Qk-1–S

and Q’k-1

–S, making Q

k–S connected. Therefore |

S| ≥ |M| =2k-1 ≥ k (since k ≥ 2).Contains copyrighted material from Introduction to Graph Theory by Doug West, 2nd Ed. Not for distribution beyond IIT’s Math 454/553. Qk-1Q’k-1Slide7

Vertex-Connectivity of the Hypercube

74.1.3. Example. The hypercube Q

k

has connectivity

κ

(Qk)=k for all k≥0.Proof

(By induction on

k

.)

View

Qk as two copies of Qk-1 plus a perfect matching M. Suppose S is a vertex cut for Qk. Assume Q

k–S leaves ≥1 vertex in Qk-1 and Q’

k'-1, else |S| ≥ 2k-1 ≥

k. Case 2 At least one of Qk-1–

S and Q’k-1–S is disconnected, say Qk-1

–S:By induction, |SQk-1| ≥ k

-1.If |S

Q’k-1

| = 0, then Qk–S contains all of

Q’k-1 and is thus connected.Therefore |

SQ’k

-1| ≥ 1, and so |S| ≥ k.Combining the lower bound of k on the size of a vertex cut with the observation that removal of the size k neighborhood of a vertex disconnects Qk, we have κ(Qk)=k for all k≥0.Contains copyrighted material from Introduction to Graph Theory by Doug West, 2nd Ed. Not for distribution beyond IIT’s Math 454/553. Qk-1Q’k-1Slide8

Vertex-Connectivity of the Hypercube

8QuestionDoes there exist a vertex cut of size k

in the

k

-dimensional hypercube that cannot be expressed as the neighborhood of a single vertex?

This is a basic question in the area of isoperimetric problems in graphs.

Contains copyrighted material from

Introduction to Graph Theory

by Doug West, 2

nd

Ed. Not for distribution beyond IIT’s Math 454/553. Slide9

Minimum Size of a k

-Connected Graph94.1.5. Theorem (Harary [1962a])

κ

(

H

k,n)=k, and hence the minimum number of edges in a k-connected graph on n vertices is kn/2.

Proof outline.

If a graph

G

has

fewer than kn/2 edges, then δ(G)<k, and we can remove the neighbors of a vertex of minimum degree to demonstrate that G has connectivity less than k. The lower bound of kn/2 is sharp: We proved the result for even k and 2k

<n using the Harary graph Hk,nThe result for odd

k using the Harary graph Hk,n is similar.

Two remarks:We always have κ(G)<n.

When k=1, the bound is sharp when n=2 when G is K2.

Contains copyrighted material from Introduction to Graph Theory by Doug West, 2nd Ed. Not for distribution beyond IIT’s Math 454/553. Slide10

Minimum Disconnecting Sets are Edge Cuts

104.1.7 Definition. A disconnecting set of edges is a set

F



E

(G) such that G–F has more than one component. A graph is k-edge-connected if every disconnecting set has at least k edges. The

edge-connectivity

of

G

, written

κ’(G), is the minimum size of a disconnecting set (equivalently, the maximum k such that G is k-connected). Given S,T  V(G), we write [S,T] for the set of edges having one endpoint in S and the other in T

. An edge cut is an edge set of the form [S,V(G)–S] where S is a nonempty proper subset of

V(G). 4.1.8. Remark. Every edge cut is a disconnecting set.Every minimal disconnecting set is an edge cut:

For a disconnecting set F, let H be a component of G–F. Then [

V(H), V(G)–V(H

)] is an edge cut. Contains copyrighted material from Introduction to Graph Theory by Doug West, 2

nd Ed. Not for distribution beyond IIT’s Math 454/553. Slide11

Connectivity and Min Degree for Simple Graphs

114.1.9 Theorem. (Whitney [1932a]) If G

is a simple graph, then

κ

(G)  κ’(G)  δ

(

G

).

Proof.

Proof of κ’(G)  δ(G): The edges incident to a vertex of minimum degree are a disconnecting set.Proof of κ(G)  κ’(G):Let F be a minimum disconnecting set of

G of size κ’(G), which is therefore equal to an edge cut [S,V(

G)–S] by Remark 4.1.8.Case 1 Every vertex of S is adjacent to every vertex of V(G

)–S.Then κ’(G) = |[S,

V(G)–S]|  n–1, and n–

1  κ(G) we already knew.Case 2 There exist vertices x

 S and y

 V(

G)–S with xy

 E(G).

Define T = ( N

(x)  (V(G)–S) )  {z  S–{x} : N(x)  (V(G)–S)  }. Contains copyrighted material from Introduction to Graph Theory by Doug West, 2nd Ed. Not for distribution beyond IIT’s Math 454/553. Slide12

Connectivity and Min Degree for Simple Graphs

124.1.9 Theorem. (Whitney [1932a]) If

G

is a simple graph, then

κ(G)  κ’(

G

)



δ(G).Proof. Proof of κ(G)  κ’(G):Case 2 There exist vertices x

 S and y  V(G)–

S with xy  E(G).Define

T = ( N(x)  (

V(G)–S) ) 

{z  S–{x} :

N(x

)  (V

(G)–S

)  

}.T is a vertex cut because all

x,y-paths wouldwould have to cross through T.The edges FT both incident to T and in the edgecut [S,V(G)–S] are a disconnecting set.Every vertex of T has at least one neighbor, so|[S,V(G)–S] |  |FT|  |T|.We have found a vertex cut T with size at mostthe size of a minimum edge cut [S,V(G)–S],and therefore κ(G)  κ’(G). Contains copyrighted material from Introduction to Graph Theory by Doug West, 2nd Ed. Not for distribution beyond IIT’s Math 454/553.

S

G

V

(

G

)–

S

x

y

T

T

T

T

T

(|

T

| bold edges)Slide13

Arbitrary Space in Whitney’s Inequalities

134.1.10. Whitney’s inequalities

κ

(

G

)  κ’(G)  δ(G)

can be made arbitrarily, and simultaneously, weak.

The following graph has

κ

(G) = 1, κ’(G) = 2, δ(G) = 3Important note: A 1-vertex graph has κ

’(G) = , so n(G)=1 is excluded.

Contains copyrighted material from Introduction to Graph Theory by Doug West, 2nd Ed. Not for distribution beyond IIT’s Math 454/553.

S

FSlide14

Connectivity in 3-Regular Graphs

144.1.11 Theorem If G is a 3-regular graph, then

κ

(

G

) = κ’(G).Important note: The graph G = is excluded from

Thm

. 4.1.11.

(3 parallel edges between two vertices)

There is no 3-regular 1-vertex graph, so we do know all 3-regular graphs satisfy

κ(G)  κ’(G)  δ(G) = 3.First, n(G) > 1 since no 1-vertex graph is 3-regular. There are two cases for a minimum vertex cut

S.Case 1 n(G–S) = 1: Then G has the complete graph

Kn as a spanning subgraph. This is only possible if n=2 and G = ,or if

n=4 and G = K4, which has κ(G) = κ

’(G) = 3.We prove Theorem 4.1.11 by assuming n(G–S)

> 1.

Contains copyrighted material from

Introduction to Graph Theory by Doug West, 2nd Ed. Not for distribution beyond IIT’s Math 454/553. Slide15

Definition

A bond is a minimal nonempty edge cut

[

S

,

S] is a cut, but not a bond. [B,B] is a bond.

4.1. Definition of a Bond

15

A

B

S

Contains copyrighted material from

Introduction to Graph Theory

by Doug West, 2

nd

Ed. Not for distribution beyond IIT’s Math 454/553. Slide16

Defn

A block H of a graph G is a maximal subgraph of G with no cut vertex.

Properties of blocks of a simple graph

G

; distinct blocks

H,H1,H2H is an isolated vertex, a cut-edge, or a maximal 2-connected subgraph

H

1

cannot be properly contained in

H

2.H1Å H2=, or H1Å H2={v}, v a cut-vertexThe blocks decompose G4.1. Definition of a Block

16

Contains copyrighted material from

Introduction to Graph Theory

by Doug West, 2

nd

Ed. Not for distribution beyond IIT’s Math 454/553.