Graph

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# Graph - PowerPoint PPT Presentation

V = { a, b, c, d, e, f, g, h, i, j, k, l }. E = { ( a, b ) , ( a, e ) , ( b, e ) , ( b, f ) , ( c, j ) , ( c, g ) , ( c, h ) , ( d, h ) , ( e, j ) , ( g, k ) , ( g, l ) , ( g, h ) , ( i, j ) }. Graph. A graph G is a set V ( G ) of vertices ( nodes ) and a set E ( G ) of

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## PowerPoint Slideshow about 'Graph' - jacob

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Presentation Transcript

V = { a, b, c, d, e, f, g, h, i, j, k, l }

E = { (a, b), (a, e), (b, e), (b, f), (c, j), (c, g), (c, h), (d, h), (e, j),

(g, k), (g, l), (g, h), (i, j) }

Graph

A graphG is a set V(G) of vertices (nodes) and a set E(G) of

edges which are pairs of vertices.

a

b

c

d

e

f

g

h

i

j

k

l

TW 45

BOS

NW 35

ORD

SFO

DL 247

JFK

AA 903

DL 335

UA 877

AA 1387

UA 120

MIA

AA 49

AA 523

LAX

DFW

AA 411

A Directed Graph

In a directed graph (digraph), edges are ordered pairs.

From Goodrich and Tamassia (1998)

George

Paul

Linda

Ringo

Yoko

John

Applications of Graphs

Graphs describe relationships

The Internet

Molecules

Flow Charts

Social Networks

Circuits

Parts in an Assembly

A pseudograph is a multigraph that allows self-loops (edges

from a vertex to itself).

2

1

3

5

6

4

More General Graphs

A multipgraphallows multiple edges between two vertices.

a

c

b

d

f

deg(w) = 1

e

2

degree(u) = 2

e

1

incident on

u and v

c

source

b

a

e

destination

d

Edges & Degrees

w

u

endpoint

v

in-degree(b) = 3

out-degree(b) = 4

deg(v) = 2 | E |

v V

4

2

K has ( ) = 6 edges

4

If G is directed:

indeg(v) = outdeg(v) = | E |

v V

v V

Handshaking Theorem

Let G = (V, E) be an undirected simple graph.

| E | ≤ | V | · (| V | – 1) / 2

| E | ≤ | V | · (| V | – 1)

a

b

c

d

Simple path:

a, e, k, p, l, q

m, h, d, c, g

e

f

h

g

(no repeated

vertices)

m

j

k

l

p

q

n

o

Path

A path of length k is a sequence v , v , …, v of vertices such

that (v , v ) for i = 0, 1, …, k – 1 is an edge of G.

0 1 k

i i+1

b, c, d not a path

Non-simple path:

a, b, e, f, g, b, g, l

Cycle

A cycle is a path that starts and ends at the same vertex.

A simple cycle has no repeated vertices.

a

b

c

d

e

f

h

g

m

j

k

l

k, j, n, k, p, o,k

is not simple.

p

q

n

o

Subgraph

A subgraphH of G

is a graph;

its edges and vertices are subsets of those of G.

a

b

c

d

e

f

h

g

m

j

k

l

p

q

n

o

V(H) = {b, d, e, f, g, h, l, p, q} E(H) = {(b, e), (b, g), (e, f), (d, h), (l, p), (l, q)}

a

d

f

g

b

c

f

e

C

3

C

2

a

C

d

e

1

b

c

Connectivity

G is connected if there is a path between every pair of vertices.

If G is not connected, the maximal connected subgraphs are the

connected componentsof G.

connected.

c

f

a

b

e

d

Strong & Weak Connectivity

A directed graph is strongly connected if every two vertices

are reachable from each other.

b

a

e

f

d

c

If G is connected, then | E | ≥ | V | – 1.

If G is a tree, then | E | = | V | – 1.

If G is a forest, then | E | ≤ | V | – 1.

Property of Connectivity

Let G = (V, E) be an undirected graph.

1

2

2

3

5

2

1

3

1

3

3

1

2

4

5

4

3

5

5

4

5

1

3

4

If G is directed, the total length of all the adjacency lists is | E |.

If G is undirected, the total length is 2 | E |.

Space requirement: (|V| + |E|). Preferable representation.

A = (a )

ij

2

1 if (i, j)  E(G)

a =

1

3

ij

0 otherwise

1 2 3 4 5

5

4

• 0 1 1 0 1
• 1 0 1 0 0
• 1 1 0 1 1
• 0 0 1 0 1
• 1 0 1 1 0

2

Space: (|V| ).

Preferred when the graph is small or dense.

of u and v

(min(deg(u), deg(v)) (1)

2

Space (|V|+|E|) (|V| )

Lecture notes modified over Dr. Fernandez-Baca’s

Operation Time

Scan incident edges (deg(v)) (|V|)

Scan outgoing edges (outdeg(v)) (|V|)

Scan incoming edges (indeg(v)) (|V|)