4.1 Connectivity and Paths: Cuts and Connectivity
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Connectivity of Graphs
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 nodes
Sensor networks’ resistance to individual sensor failure
Applications (edge connectivity)
Robustness of supercomputers to failures of wires/fiber optics
Reliability of road networks with road closures/accidents
Communication 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.
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Vertex Connectivity Examples
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.
Kn
Km,n
Examples
S
S
S
2-connected
S
1-connected
0-connected
(n-1)-connected
min(m,n)-connected
2K2:
disconnected, so
0-connected
Contains copyrighted material from Introduction to Graph Theory by Doug West, 2nd Ed. Not for distribution beyond IIT’s Math 454/553.
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Vertex Connectivity Examples
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.
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
Contains copyrighted material from Introduction to Graph Theory by Doug West, 2nd Ed. Not for distribution beyond IIT’s Math 454/553.
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Vertex Connectivity Examples
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 Qn
Qk-1
S
k=0
0-connected
κ=0
k=1
1-connected
κ=1
Qk-1
S
k=2
2-connected
κ=2
k>2
??
κ(Qk-1) = ??
Contains copyrighted material from Introduction to Graph Theory by Doug West, 2nd Ed. Not for distribution beyond IIT’s Math 454/553.
Vertex-Connectivity of the Hypercube
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.
Qk-1
Q’k-1
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).
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Vertex-Connectivity of the Hypercube
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. Qk-1
Q’k-1
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, |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.
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Vertex-Connectivity of the Hypercube
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.
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Minimum Size of a k-Connected Graph
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 2k<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.
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Minimum Disconnecting Sets are Edge Cuts
4.1.7 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-edgeconnected if every disconnecting set has at least k edges. The edgeconnectivity 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.
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Connectivity and Min Degree for Simple Graphs
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)  }.
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Connectivity and Min Degree for Simple Graphs
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 would
G
would have to cross through T.
x
The edges FT both incident to T and in the edge
T
cut [S,V(G)–S] are a disconnecting set.
T
T
Every vertex of T has at least one neighbor, so
T
|[S,V(G)–S] |  |FT|  |T|.
T
We have found a vertex cut T with size at most
y
the size of a minimum edge cut [S,V(G)–S],
S
V(G)–S
and therefore κ(G)  κ’(G).
(|T| bold edges)
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Arbitrary Space in Whitney’s Inequalities
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
F
S
Important note: A 1-vertex graph has κ’(G) = , so n(G)=1 is
excluded.
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Connectivity in 3-Regular Graphs
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.
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4.1. Definition of a Bond
Definition A bond is a minimal nonempty edge cut
A
S
B
[S,S] is a cut, but not a bond. [B,B] is a bond.
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4.1. Definition of a Block
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 2connected 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
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