Graphs Lect4

8/14/2019 Graphs Lect4

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Graph

Nitin Upadhyay

March 01, 2006

Bits-Pilani Goa campus


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Discussion

Counting Path Example

ConnectednessCut Vertices and Cut Edges

General Graphs

Planar Graphs Planar Embedding

Plane Graph Regions

Plane Graph Degree


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Counting Paths and Adjacency Matrices

Let A be the adjacency matrix of graph G.

The number of paths of length kfrom vi to vj

is equal to (Ak)i,j. (The notation (M)i,j denotes

mi,j where [mi,j] = M.)


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Counting Paths Example

How many paths of length 4 are there from 1

to 5 in the graph?

1 2

3

4 5

e1

e2e3

e4

e5e6


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Adjacency Matrix

1 2

3

4 5

e1

e2e3

e4

e5

e6

The number

of paths of length4 from a to dis the

(1, 5)th entry ofA4.

Since

0 1 0 1 0

1 0 1 0 10 1 0 1 1

1 0 1 0 0

0 1 1 0 0

3

9 3 11 1 6

3 15 7 11 8

11 7 15 3 8

1 11 3 9 6

6 8 8 6 8

The (1, 5)th entry ofA4

is 6, indicating that thereare six different paths of

length 4 between 1 and

5.

1:{1-4-1-2-5}

2:{1-2-1-2-5}

3:{1-4-3-2-5}

4:{1-2-5-2-5}

5:{1-2-3-2-5}

6:{1-2-5-3-5}


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Counting Paths Example

How many paths of length 4 are

there from a to d in the right

graph? The adjacency matrix of the

graph is

Hence, the number

of paths of length

4 from a to d is the

(1, 4)th entry ofA4.

Since

There are 8 paths of length

4 from a to d.

a b

cd

=

0110

1001

1001

0110

A

=

8008

0880

0880

8008

4A


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Connectedness

An undirected graph isconnectediff there is a pathbetween every pair ofdistinct vertices in the graph.

E.g., Graph G1 is connected,since for every pair ofdistinct vertices, there is apath between them.

However, G2 is notconnected. E.g., no path inG2 between vertices a and c.

a b c

d fe

a b c

d fe


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Connectedness

Theorem: There is a simple path between

every pair of distinct vertices of a connectedundirected graph.

Connected component: A graph that is notconnected is the union of two or more

connected subgraphs, each pair of which hasno vertex in common. These disjointconnected subgraphs are called theconnected components of the graph.


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Connected Component Example

The graph His the

union of three disjoint

connected subgraph

H1, H2, and H3. These

three subgraphs are the

connectedcomponents

ofH.

a

c

d

f

e

b gh

H1

H2

H3

H


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Cut Vertices and Cut Edges

A cut vertexorcut edge

separates 1 connected

component into 2 ifremoved.

E.g., The cut vertices ofG

are b, c, and e, since

removing one of these

vertices (and its adjacentedges) disconnects the

graph. The cut edges are

[a, b] and [c, e].

a

c

g

b he

d f


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Directed Connectedness

A directed graph is strongly connectediff

there is a directed path from any vertex toany other vertex in the graph.

It is weakly connectediff the underlying

undirectedgraph (i.e., with edge directionsremoved) is connected.


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Directed Connectedness Example

Are the directed graphs G and H strongly

connected or weakly connected?

a b

c

de

a b

c

de

G H


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Directed Connectedness Example

G is strongly connectedsince there is a directed

path between any twovertices. Hence, G is alsoweakly connected. His notstrongly connected since, forinstance, theres nodirected path from a to b. H

is however weaklyconnected since there is apath between any twovertices with all directionsremoved.

ab

c

de

ab

c

de

G H


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General Graphs

Generally graphs drawn on a piece of paper

permit edges to intersect at points other thanvertices.

These points of intersection are called

crossovers.

And the intersecting edges are said to crossover each other.


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Example Crossovers

Here this graph exhibits three crossovers:

{b, e} crosses over {a, d} and {a, c}. {b, d} crosses over {a, c}


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Planar Graphs

A Graph G is said to be planar if it can be

drawn on a plane without any crossovers.

A graph is non planar if there is no way to

convert it into planar.


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Example conversion

Non planar to planar

c c


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Planar embedding

Graphs can be represented such that no two

edges of the graph intersect except possiblyat a vertex to which they both incident.

A drawing of a geometric representation of a

graph on any surface such that no edges

intersect is called embedding. An graph G is planar if there exists a graph

isomorphic to G that is embedded in a plane.


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Plane Graph Regions

A plane graph G can be thought of as dividing

the plane into regions orfaces.

The regions are the connected portions of the

plane remaining after all the curves and pointsof the plane corresponding, respectively, to

edges and vertices of G have been deleted.


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Plane Graph Regions

A plane graph partitions the plane into

regions of G.

There is exactly one region whose area is

infinite, known as exteriororinfinite region.

Every other region is an interior region.


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Plane Graph Regions

The boundary of a region is the subgraph

formed by the vertices and edgesencompassing that region.

If the boundary of the exterior region of a

plane graph is a cycle, that cycle is knownas the maximal cycle of the graph.


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Plane Graph Regions

The degree of a region is the number of

edges in a (closed) walk encloses it. A bridge belongs to the boundary of only

one region, thus it contributes to the size of

the boundary twice.

The sum of the degrees of all the regions in aplane graph is twice the size if the graph.


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Region Example

Graph has six interior regions and one exterior

regions.a

eb c

d

f

g

h

i

j

Interior regions- 1:{a-b-c-a} 3:{a-c-d-e-a} 5:{g-h-j-g} 6:{g-h-i-f-g}

2:{b-c-d-b} 4:{j-h-i-j}

Exterior region- 1:{a-b-d-e-f-i-j-g-f-e-a}


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Degree example

In the graph there exists four interior regions

of degree 3 and two interior regions ofdegree 4.The degree of exterior region is 10.

a

eb c

d

f

g

h

i

j


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Degree example

The sum of the degrees (of the regions) is

30, and the graph has 15 edges.

a

eb c

d

f

g

h

i

j


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Example 1

Find out the interior and exterior regions,

degree of each region and total degree of theplanar graph G.

a b c d

h g f e

Interior regions:

1:{a-b-g-h-a} 2:{b-c-f-g-b}

3:{c-d-e-f-c} degree: {4}

Exterior region

1: {a-b-c-d-e-f-g-h-a}

Degree: {8}

Total degree: 8 + (4 * 3) = 20

Number of edges: 20/2 = 10


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Example 2

Find out the interior and exterior regions,

degree of each region and total degree of theplanar graph G.Interior regions:

1:{b-d-c-b} 2:{e-f-g-e}

degree: {3}

Exterior region

1: {a-b-c-d-e-f-g-e-d-b-a}

Degree: {10}

Total degree: 10 + (2 * 3) = 16c

b

a

d e

f

g


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Graphs Lect4

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