Connectivity and Paths

報告人：林清池

Connectivity

A separating set of a graph G is a set such that G-S has more than one component.

The connectivity of 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.

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Example

.1)( nKn

}.,min{)( , nmK nm

.0)( 1 K

Hypercube

The K-dimensional cube is the simple graph whose vertices are the k-tuples with entries in and whose edges are the pairs of k-tuples that differ in exactly one position.

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kQ

111011

101001

110010

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The neighbors of one vertex in form a separating set, so . To prove , we show that every separating set has size at least .

Prove by induction on . Basis step: For , is a complete graph w

ith vertices and has connectivity .

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k

k

kQk )(

kQk )(

1k kQ1k

An example: Induction step: Let S be a vertex cut in Case 1: If Q-S is connected and Q’-S is

connected, then , for .

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 . We have .

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2k

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Harary graphs Given k <n, place n vertices

around a circle. If k is even, form by making each vertex adjacent to the nearest k/2 vertices in each direction around the circle.

nkH ,

nkH ,

8,4H

Harary graphs If k is odd and n is even, form

by making each vertex adjacent to the nearest (k-1)/2 vertices in each direction around the circle and to the diametrically opposite vertex.

nkH ,

nkH ,

8,5H

Harary graphs If k and n are both odd, index the

vertices by the integers modulo n. Construct form by adding the edges for

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nkH ,

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Harary graphs Theorem. , and hence

the minimum number of edges is a k-connected graph on n vertices is

kH nk )( ,

2/kn

Harary graphs Proof. (Only the even case k =2r. Pigeonhole)

Since , it suffices to prove 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.

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Harary graphs nkH ,

8,4H

u

v

Edge-Connectivity

A disconnecting set of a graph G is a set such that G-F has more than one component.

The edge-connectivity of 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.

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Edge-Connectivity

An edge cut is an edge set of the form where is a nonempty proper subset of and denotes SGV )(

SS,

S)(GV

S

Disconnecting set

S

S

Edge cut

Theorem

If G is a simple graph, then Proof: , trivial. Case 1: if every vertex of is adjacent to every

vertex of , then

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)()(' GG

S

S

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S S

Theorem Case 2: , with

: consist of all neighbors of in and

all vertices of with neighbors in .

is a separating set

picking the red edges yields |T| distinct

edges.

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Example

3)(,2)(',1)( GGG

Theorem

)(')( GG If G is a 3-regular graph, then

Proof: S

H 'H

Theorem

)(')( GG If G is a 3-regular graph, then

Proof: S

H 'H

Theorem

)(')( GG If G is a 3-regular graph, then

Proof: S

H 'H

Theorem

)(')( GG If G is a 3-regular graph, then

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 is a connected graph, then an

edge cut is a bound if and only if has exactly two components.

Proof: is a subset of . is

connected.

S

S

SSF ,

'F F

G

F FG

'FG

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Proposition If is a connected graph, then an

edge cut is a bound if and only if has exactly two components.

Proof: Suppose has more than two component. and are proper subsets of , so is not a bound.

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F FG

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B

A

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Definition A Block of a graph is a maximal conne

cted subgraph of G that has no cut-vertex.

A connected graph with on cut-vertex need not be 2-connected, since it can be or . 1K 2K

Proposition Two blocks in a graph share at most one

vertex.Proof:

Suppose for a contradiction. and 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.

2B1B

Definition The block-cutpoint graph of a graph G is a bip

artite graph H in which one partite set consists of the cut-vertices of G, and the other has a vertex for each block of G. We include as an edge of H if and only if .

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Algorithm

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Computing the blocks of a graph.

Algorithm

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Computing the blocks of a graph.

Algorithm

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Computing the blocks of a graph.

Algorithm

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Computing the blocks of a graph.

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 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.

)(, GVvu

Theorem Prove by induction on Basis step.

The graph G-uv is connected.

Induction step. Let w be the vertex before v on a shortest u,v pa

th;

),( vud

w vu

P

Q

Theorem

Case 1: if , 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.

)()(, QVPVvu

w vu

P

Q

zR

w vu

P

Q

zR

Expansion Lemma If G is a k-connected, and G’ is obtained from G b

y adding a new vertex y with at least k neighbors in G, then G’ is k-connected.

Case 1: if , then Case 2: if and , then Case 3: and lie in a single component of

, then

yG

Sy .1|| kS

SyN )(Sy .|| kS

y SyN )(

SG ' .|| kS

Theorem For a graph G with at least three vertices,

the following condition are equivalent. G is connected and no cut-vertex. For , there are internally disjoint x, y

paths.

For , there is a cycle through x and y.

, and every pair of edges in G lies on a

common cycle.

)(, GVyx

)(, GVyx

1)( G

Definition In a graph G, subdivision of an edge uv is the op

eration of replacing uv with a path u, w, v through a new vertex w.

u v u vw

Corollary If G is a 2-connected, then the graph G’ obtaine

d by subdividing an edge of G is 2-connected. Proof: It suffices to find a cycle through arbitrary edges e,f o

f 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 and , then … Case 3: if , then …

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},{},{ wvuwfe )(GEe

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

such that is a cycle and for

is an ear of .

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4P

0P iP

kPP 0 3P

2P 1P0P

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.

4P

3P

2P 1P0P

Theorem

4P

3P

2P 1P0P

4P

3P

2P 1P0P

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 G is a

decomposition such that is a

cycle and for is either an

(open) ear or a closed ear in .

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1i

4P

0P

iP

kPP 03P

2P 1P0P

5P

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.

Case 2: when adding a open ear , …

4P

3P

2P 1P0P

2P

Theorem Proof:

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0P

4P

3P

2P 1P0P

5P2P5P

3P

Theorem Proof:

4P

3P

2P 1P0P

Connectivity of Digraphs

A separating set of a digraph D is a set such that D-S is not strongly connected.

The connectivity of G, 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.

)(DVS

)(D

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 for

some . A diagraph is k-edge-

connected if every edge cut has at least k

edges.

The minimum size of an edge cut is the edge-

connected

).(' D

SS,

)(DVS

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 orientatio

n 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 , a set is an x,y separator or x, y-cut if G-S has no x, y-path.

Let be the minimum size of an x,y-cut. Let be the maximum size of a set of pairwise inter

nally disjoint x, y-paths. For , an X, Y-path is a graph having first vertex i

n X, last vertex in Y, and no other vertex in

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.YX )(, GVYX

),( yx

},{)( yxGVS

),( yx

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).,(),( yxyx

Example Although , it takes four edges to break

all w, z-paths, and there are four pairwise edge-disjoint w, z-paths.

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