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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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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) }

GraphA 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

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

From Goodrich and Tamassia (1998)

Paul

Linda

Ringo

Yoko

John

Applications of GraphsGraphs describe relationships

The Internet

Streets / Highways (Roadmaps)

Molecules

Flow Charts

Social Networks

Geometric Surfaces (CAD)

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 GraphsA 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 & Degreesw

u

u and v are adjacent

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

| E | ≤ | V | · (| V | – 1) / 2

| E | ≤ | V | · (| V | – 1)

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

PathA 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)}

d

f

g

b

c

f

e

C

3

C

2

a

C

d

e

1

b

c

ConnectivityG 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.

It is weakly connected if the underlying undirected graph is

connected.

c

f

a

b

e

d

Strong & Weak ConnectivityA 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 ConnectivityLet 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

Adjacency ListsIf 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| ).

Adjacency MatrixPreferred 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 TimeOperation Adjacency List Adjacency Matrix

Scan incident edges (deg(v)) (|V|)

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

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

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