4.1 Connectivity and Paths: Cuts and Connectivity This copyrighted material is taken from...

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.

1

Connectivity of Graphs2

Motivating Question

How 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 nodesSensor 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, 2nd Ed. Not for distribution beyond IIT’s Math 454/553.

Vertex Connectivity Examples3

4.1.1. Definition. A separating set or vertex cut of a graph G is a set SV(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


S

2-connected

S

1-connected

S

Kn

(n-1)-connected

S

Km,n

min(m,n)-connected

0-connected

2K2:disconnected, so0-connected




K1 K2 K3 K4 Kn (n>3) C4 Cn (n>2)

Connectivity κ 0 1 2 3 n-1 2 2

1-connected? N Y Y Y Y Y Y

2-connected? N N Y Y Y Y Y

3-connected? N N N Y Y N N



Hypercubes Qn


S

k = 22-connected

κ = 2

k = 00-connected

κ = 0

k = 11-connected

κ = 1

S

k > 2??

κ(Qk-1) = ??

Qk-1 Qk-1

Vertex-Connectivity of the Hypercube6

4.1.3. Example. The hypercube Qk has connectivity κ(Qk)=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 Qk–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-1 Q’k-1


4.1.3. Example. The hypercube Qk 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 Qk–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, |SQk-1| ≥ k-1.

If |SQ’k-1| = 0, then Qk–S contains all of Q’k-1 and is thus connected.

Therefore |SQ’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-1 Q’k-1


Question

Does 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.


Minimum Size of a k-Connected Graph9

4.1.5. Theorem (Harary [1962a]) κ(Hk,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 2k<n using the Harary graph Hk,n

The result for odd k using the Harary graph Hk,n is similar.

Two remarks:

1. We always have κ(G)<n.

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


Minimum Disconnecting Sets are Edge Cuts10

4.1.7 Definition. A disconnecting set of edges is a set FE(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.


Connectivity and Min Degree for Simple Graphs11

4.1.9 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) }.


Connectivity and Min Degree for Simple Graphs12

4.1.9 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 most

the size of a minimum edge cut [S,V(G)–S],

and therefore κ(G) κ’(G).


S

G

V(G)–S

x

y

TTT

TT

(|T| bold edges)

Arbitrary Space in Whitney’s Inequalities13

4.1.10. 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) = 3

Important 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

F

Connectivity in 3-Regular Graphs14

4.1.11 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.


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 Bond15

A

B

S


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,H2

(1) H is an isolated vertex, a cut-edge, or a maximal 2-connected subgraph

(2) H1 cannot be properly contained in H2.

(3) H1Å H2=, or H1Å H2={v}, v a cut-vertex

(4) The blocks decompose G

4.1. Definition of a Block16


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