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Connectivity and Paths
報告人：林清池
Connectivity



A separating set of a graph G is a set
S  V (G ) such that G-S has more than one
component.
The connectivity of G,  (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.
Example

 ( Kn )  n  1.

 ( K m ,n )  min{ m, n}.

 ( K1 )  0.
Hypercube

The K-dimensional cube Qk is the simple
graph whose vertices are the k-tuples with
entries in {0,1} and whose edges are the pairs
of k-tuples that differ in exactly one position.
011
010
Q3
110
001
000
111
100
101
 (Qk )  k


The neighbors of one vertex in Qk form a
separating set, so  (Qk ). Tok prove
, we
 (Qk )  that
k
show
every separating set has size at
k
least .
Prove by induction on .

Basis step: For k  1 , Qk is a complete graph
with k  1 vertices and has connectivity k .
An example:  (Qk )  k



Induction step: Let S be a vertex cut in Qk
Case 1: If Q-S is connected and Q’-S is
k 1
connected, then | S | 2  k , for k  2 .
Case 2: If Q-S is disconnected, which
means S has at least k-1 vertex in Q. And,
S must also contain a vertex of Q ' . We
have | S | k .
Q
Q'
Q  Q'  Qk 1
Harary graphs

H k ,n
Given k <n, place n vertices around a
circle. If k is even, form H k ,n by making
each vertex adjacent to the nearest k/2
vertices in each direction around the
circle.
H 4 ,8
Harary graphs

H k ,n
If k is odd and n is even, form H k ,n by
making each vertex adjacent to the
nearest (k-1)/2 vertices in each
direction around the circle and to the
diametrically opposite vertex.
H 5,8
Harary graphs

H k ,n
If k and n are both odd, index the
vertices by the integers modulo n.
Construct H k ,n form H k 1,n by adding the
edges i  i  (n  1) / 2 for 0  i  (n  1) / 2.
4
5
3
2
6
1
7
0
8
H 5, 9
Harary graphs

Theorem.  ( H k ,n )  k , and hence the
minimum number of edges is a kconnected graph on n vertices is kn / 2
Harary graphs
Proof. (Only the even case k =2r. Pigeonhole)






Since  (G )  k , it suffices to prove
 (G )  k .
Clockwise u,v paths and counterclockwise u,v
paths.
Let A and B be the sets of internal vertices on
these two paths.
One of {A, B} has fewer that k/2 vertices.
Thus, we can find a u,v path in G-S via the set A
or B in which S has fewer than k/2 vertices.
Harary graphs
H k ,n
u
H 4 ,8
v
Edge-Connectivity



A disconnecting set of a graph G is a
set F  E (G )such that G-F has more than
one component.
The edge-connectivity of G,  ' (G ) is
the minimum size of a disconnecting
set.
A graph G is k-edge-connected if
every disconnecting set has at least k
edges.
Edge-Connectivity

An edge cut is an edge set of the form
S, S 
where S is a nonempty proper
subset of V (G ) and S denotes V (G )  S
S
S
Disconnecting set
Edge cut
Theorem

If G is a simple graph, then  (G )   ' (G )   (G )
Proof:  ' (G )   (G ) , trivial.
Case 1: if every vertex of S is adjacent to every
vertex of S , then | S , S || S || S | n(G )  1   (G )
S
S
Theorem

Case 2: x  S , y  S with xy  E (G )
T : consist of all neighbors of x in S and all
vertices of S  {x}
 T
with neighbors in S .
is a separating set
picking the red edges yields |T| distinct edges.
  ' (G )  S , S  | T |  (G )
x
S
T
T
T
T
T
y
S
Example
 (G )  1,  ' (G )  2,  (G )  3
Theorem

If G is a 3-regular graph, then  (G )   ' (G )
Proof:
S
H
H'
Theorem

If G is a 3-regular graph, then  (G )   ' (G )
Proof:
S
H
H'
Theorem

If G is a 3-regular graph, then  (G )   ' (G )
Proof:
S
H
H'
Theorem

If G is a 3-regular graph, then  (G )   ' (G )
Proof:
S
H
H'
Definition


A Bond is a minimal nonempty edge
cut.
Here “minimal” means that no
proper nonempty subset is also an
edge cut.
Proposition

If G is a connected graph, then an edge
cut F is a bound if and only if G  F has
exactly two components.
Proof:
F ' is a subset of F . G  F ' is connected.
 
G  F'
F  S, S
S
S
Proposition
If G is a connected graph, then an edge
cut F is a bound if and only if G  F has
exactly two components.
Proof: Suppose G  F has more than two
component. B, B  and A, A are proper
subsets of F , so F is not a bound.

A
S
B
S
Definition


A Block of a graph is a maximal
connected subgraph of G that has no
cut-vertex.
A connected graph with on cut-vertex
need not be 2-connected, since it can
be K1 or K2 .
Proposition

Two blocks in a graph share at most one
vertex.
Proof:
Suppose for a contradiction. B1 and B2
have at least two common vertices.
Since the blocks have at least two common
vertices, deleting one singe vertex, what
remains is connected. A contradiction.
Definition

The block-cutpoint graph of a graph G is a
bipartite graph H in which one partite set
consists of the cut-vertices of G, and the other
has a vertex bi for each block Bi of G. We include
vbi as an edge of H if and only if v  Bi .
b
c
h
i
a
d
b5
e
x
g b1
f
j
b3
b2
a
e
x
b4
Algorithm
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Computing the blocks of a graph.
x
b
c
a
d
a
h
i
e
x
g
f
b
j
i
e
c
f
j
g
d
h
Algorithm

Computing the blocks of a graph.
x
b
c
a
h
i
a
e
j
g
i
e
d
x
f
f
j
g
h
Algorithm

Computing the blocks of a graph.
x
b
c
a
h
i
a
e
j
g
e
d
x
f
j
i
Algorithm

Computing the blocks of a graph.
x
b
c
a
h
i
a
e
j
g
e
d
x
f
j
Definition

Two paths from u to v are internally
disjoint if they have no common internal
vertex.
Theorem

G is 2-connected if and only if for each for
each pair u, v V (G )
there exist internally
disjoint u,v paths in G.
Proof: Since for every pair u,v, G has
internally disjoint u,v paths, deletion of one
vertex cannot make any vertex
unreachable from any other.
Theorem

Prove by induction on d (u, v )
Basis step.
The graph G-uv is connected.
Induction step.
Let w be the vertex before v on a shortest u,v
path;
P
u
w
Q
v
Theorem

Case 1: if u, v V ( P)  V (Q ) , done.

Case 2: G-w is connected and contains a
u,v path R. If R avoids P or Q, done.

Case 3: Let z be the last vertex of R.
P
u
Q
R
P
z
w
v
u
Q
R
z
w
v
Expansion Lemma
If G is a k-connected, and G’ is obtained from G by

adding a new vertex y with at least k neighbors in G,
then G’ is k-connected.

Case 1: if y  S, then | S | k  1.

Case 2: if y  S and N ( y )  S , then | S | k .

Case 3: y and N ( y )  S lie in a single component
of G'S , then | S | k .
G
y
Theorem
For a graph G with at least three vertices, the

following condition are equivalent.

G is connected and no cut-vertex.

For x, y  V (G ), there are internally disjoint x, y paths.

For x, y  V (G ), there is a cycle through x and y.

 (G )  1 , and every pair of edges in G lies on a common
cycle.
Definition

In a graph G, subdivision of an edge uv is the
operation of replacing uv with a path u, w, v
through a new vertex w.
u
v
u
w
v
Corollary
If G is a 2-connected, then the graph G’ obtained

by subdividing an edge of G is 2-connected.
Proof: It suffices to find a cycle through arbitrary edges e,f
of G’. Since G is 2-connected, any two edges of G lie on a
common cycle.

Case 1: if a cycle through them in G uses uv, then replace the
edge uv with a path u,w,v.

Case 2: if e  E (G ) and f  {uw, wv}, then …

Case 3: if {e, f }  {uw, wv}, then …
Definition

An ear of a graph G is a maximal path whose
internal vertices have degree 2 in G.

An ear decomposition of G is a decomposition
P0 ,, Pk such that P0 is a cycle and Pi for i  1
is an ear of P0   Pk .
P
3
P0
P2
P4
P1
Theorem

A graph is 2-connected if and only if it has an ear
decomposition.
Proof:  Since cycles are 2-connected, it suffices to show
that adding an ear preserves 2-connectedness. Trivial.

P3
P0
P2
P4
P1
Theorem
P3
P0
P2
P1
P3
P4
P0
P2
P4
P1
Definition

An close ear in a graph G is a cycle C such that
all vertices of C expect one have degree 2 in G

An
close-ear
decomposition
of
decomposition P0 ,, Pk such that P0
and
G
is
a
is a cycle
Pi for i  1 is either an (open) ear or a
closed ear in P0   Pk .
P3
P0
P5
P2
P4
P1
Theorem

A graph is 2-edge-connected if and only if it has
an closed-ear decomposition.
Proof:  G is 2-edge-connected if and only if every edge
lies on a cycle.
Case 1: when adding a closed ear, Trivial.
P3
Case 2: when adding a open ear P2 , …
P0
P2
P4
P1
Theorem
Proof: 

P3
P0
P5
P2
P4
P3
P1
P0
P5
P2
P4
P1
Theorem

Proof: 
P3
P0
P2
P4
P1
Connectivity of Digraphs



A separating set of a digraph D is a set
S  V (D ) such that D-S is not strongly
connected.
The connectivity of G,  (D ) is the minimum
size of a vertex set S such that D-S is not
strong or has only one vertex.
A graph G is k-connected if its connectivity is
at least k.
Edge-Connectivity of Digraphs

For vertex sets S, T in a digraph D, let [S,T] denote the
set of edges with tail in S and head in T.

An edge cut is an edge set of the form
S, S 
for
some   S  V (D ) . A diagraph is k-edge-connected
if every edge cut has at least k edges.

The minimum size of an edge cut is the edgeconnected
 ' ( D ).
Proposition

Adding a directed ear to a strong digraph
produces a larger strong digraph.
Theorem

A graph has a strong orientation if and only if it is
2-edge-connected.
Proof:If G has a cut-edge xy oriented from x to y in an
orientation D, then y cannot reach x in D.
 1.) G has a closed-ear decomposition.
2.) Orient the initial cycle consistently to
obtain a strong diagraph.
3.) Directing new ear consistently.
Definition




Given x, y V (G ) , a set S  V (G )  {x, y}
is an x,y
separator or x, y-cut if G-S has no x, y-path.
Let
 ( x, y ) be the minimum size of an x,y-cut.
Let  ( x , y ) be the maximum size of a set of pairwise
internally disjoint x, y-paths.
For X , Y  V ,(Gan
) X, Y-path is a graph having first vertex
in X, last vertex in Y, and no other vertex in
X Y.
Remark

An x, y-cut must contain an internal vertex of
every x, y-path, and no vertex can cut two
internally disjoint x,y-paths. Therefore, always
 ( x, y )   ( x, y ).
Example

Although  ( w, z )  3 , it takes four edges to
break all w, z-paths, and there are four
pairwise edge-disjoint w, z-paths.
x
w
z
y
 ( x, y )   ( x, y )  4
 ( w, z )   ( w, z )  3