Graphs Lect2

8/14/2019 Graphs Lect2

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Graphs

Nitin Upadhyay

February 25, 2006


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Discussion

What is a Graph?

Applications of Graphs

Categorization

Terminology


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Discussion

Hamiltonian Cycle

Special Graph Structures

Graph Representations


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Hamiltonian Cycle

Hamiltonian Cycle: A Cycle that contains all

the vertices of the graph

1,4,5,3,2,1 is a Hamiltonian

Cycle

1

2

3

4

5


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Special Graph Structures

Special cases of undirected graph structures:

Complete graphs Kn

Cycles Cn Wheels Wn

n-Cubes Qn

Bipartite graphs Complete bipartite graphs Km,n


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Revisiting Paths and

cycles Apath is a sequence of nodesv1, v2, , vN such that (vi,vi+1)E for 0


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Complete Graphs

For any nN, a complete graph on n vertices,

Kn, is a simple graph with n nodes in which

every node is adjacent to every other node:

u,vV: uv{u,v}E.

K1 K2 K3 K4 K5 K6Note that Kn has edges.2

)1(1

1

=

=

nni

n

i


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Cycles

For any n3, a cycle on n vertices, Cn, is a

simple graph where V={v1,v2, ,vn} and

E={{v1,v2},{v2,v3},,{vn1,vn},{vn,v1}}.

C3 C4C5 C6 C7 C8

How many edges are there in Cn?


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Wheels

For any n3, a wheelWn, is a simple graph

obtained by taking the cycle Cn and adding

one extra vertex vhub and n extra edges

{{vhub,v1}, {vhub,v2},,{vhub,vn}}.

W3 W4 W5 W6 W7W8

How many edges are there in Wn?


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n-cubes (hypercubes)

For any nN, the hypercube Qn is a simple

graph consisting of two copies ofQn-1

connected together at corresponding nodes.

Q0 has 1 node.

Q0

Q1 Q2 Q3 Q4

Number of vertices: 2n. Number of edges:Exercise to try!


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Bipartite Graphs

Let G=(V,E) be a graph, G is said to be

bipartite graph:

If its vertex set V can be partitioned into two

nonempty disjoint subsets V1 and V2, calleda bipartition.

And edge connecting one of its end in V1 and

other with V2.


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Complete Bipartite Graphs

A complete bipartite graph is a bipartite graph

with partition V1 and V2 in which each vertex

of V1 is joined by an edge to each vertex of

V2. A complete bipartite graph with | V1 | = m and

| V2 | = n is denoted as Km,n.


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Subgraphs

A subgraph of a graph G=(V, E) is a graph

H=(W, F) where WVand FE.

G H


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Graph Representations

Adjacency-matrix representation

Incidence Matrix representation

Edge-set representation

Adjacency-set representation

Adjacency List


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Adjacency-matrixrepresentationAdjacency Matrix Representation

AB

G

E

F

D

C One simple way of

representing a graph

is the adjacency

matrix A 2-D array has a

mark at [i][j] if there

is an edge from node i

to nodej The adjacency matrix

is symmetric about

the main diagonal

ABC

DEFG

A B C D E F G


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Adjacency-matrixrepresentation Adjacency Matrix (A)

The Adjacency MatrixA=(ai,j) of a graph G=(V,E)

with n nodes is an nXn matrix

Each element of A is either0 or 1, depending onthe adjacency of the nodes.

1 if (i,j) E

0 otherwiseaij =


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Adjacency-matrixrepresentation Find the adjacency matrices of the following

graphs1

2

3

4

5

1

2

3

01100

10001

10011

00101

01110

011

100

010


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Adjacency-matrixrepresentationRepresentation

AB

G

E

F

D

C

ABC

DEFG

A B C D E F G

An adjacency matrixcan equally well be

used for digraphs

(directed graphs)

A 2-D array has a

mark at [i][j] if there

is an edge from node i

to nodej


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Incidence Matrices

Let G = (V, E) be an undirected graph and v1,

v2, , vn be the vertices and e1, e2, , em be

the edges ofG. Then the incidence matrix

with respect to this ordering ofVand Eis the

n x m matrix M = [mij], where

1 when edge ej is incident with vi,

mij =0 otherwise


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Incidence Matrix Example

Represent the graph shown below with an

incidence matrix.

e9

e1

e2

e5

e3

e7

e4

e6

e8

v1 v2 v3

v4 v6v5

xxxxxxxxx

xxxxxxxxx

xxxxxxxxx

011000100

000111101

000000011v1

v2

v3

v4

v5

v6

e1 e2 e3 e4 e5 e6 e7 e8 e9

For your exercise


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An edge-set representation uses a setof nodes and a setofedges The sets might be represented by, say, linked lists The set links are stored in the nodes and edges themselves

The only other information in a node is its element (that is, its

value)it does not hold information about its edges The only other information in an edge is its source and

destination (and attribute, if any) If the graph is undirected, we keep links to both nodes, but dont

distinguish between source and destination

This representation makes it easy to find nodes from an edge,but you must search to find an edge from a node This is seldom a good representation


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edge nodeSet = {A, B, C, D, E, F, G}

edgeSet = { p: (A, E),

q: (B, C), r: (E, B),

s: (D, D), t: (D, A),

u: (E, G), v: (F, E),

w: (G, D) }

AB

G

E

FD

Cpq

r

s

t

uv

w

Here we have a set of nodes,

and each node contains only its

element (not shown)

Each edge contains references to its source

and its destination (and its attribute, if any)


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Adjacency-setrepresentation An adjacency-set representation uses a set

of nodes Each node contains a reference to the set ofits

edges For a directed graph, a node might only know

about (have references to) its out-edges

Thus, there is not one single edge set, but

rather a separate edge set for each node Each edge would contain its attribute (if any) andits destination (and possibly its source)


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Adjacency-setrepresentation Adjacency

AB

G

E

FD

Cpq

r

s

t

uv

w

A { p }B { q }

C { }

D { s, t }

E { r, u }

F { v }

G { w }

p: (A, E)

q: (B, C)

r: (E, B)

s: (D, D)

t: (D, A)

u: (E, G)

v: (F, E)

w: (G, D)

Here we have a set of

nodes, and each node

refers to a set of edges Each edge contains references to its

source and its destination (and itsattribute, if any)


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Adjacency-setrepresentation If the edges have no associated attribute, there is no

need for a separate Edge class Instead, each node can refer to a set of its neighbors

In this representation, the edges would be implicit in the

connections between nodes, not a separate data structure

For an undirected graph, the node would have

references to all the nodes adjacent to it

For a directed graph, the node might have: references to all the nodes adjacent to it, or

references to only those adjacent nodes connected by an

out-edge from this node


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Adjacency-setrepresentation Adjacency

AB

G

E

F

D

C

A { E }

B { C }

C { }

D { D, A }

E { B, G }

F { E }

G { D }

Here we have a set ofnodes, and each node

refers to a set of other(pointed to) nodes

The edges are implicit


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Adjacency List

An Adjacency list is an array of lists, each list

showing the vertices a given vertex is

adjacent to

1

2

3

4

5


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Adjacency List

Adjacency Listof a Weighted Graph The weight is included in the list

1

2

39

4 5


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Questions

Questions ?

Documents

Graphs Lect2