9.0 graphs full set

8/6/2019 9.0 Graphs Full Set

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TERRIN [email protected]

ext 3714

Dept of Computational Science & Mathematics

FCSIT, UNIMAS

TMC1813TMC1813

DISCRETE MATHEMATICSDISCRETE MATHEMATICS[9.0[9.0 Introduction to Graphs]Introduction to Graphs]

Sem I 2010/2011


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DefinitionsDefinitions

def: A vertexvertexis a point, e.g. in 3-

space,

def: An edgeedge is a space curve

(without self-intersections) joining a

vertex to itself or to another vertex.Such vertices are called endpointsendpoints of

the edge.

def: Two vertices are adjacentadjacentif thereis an edge joining them.

def: A graphgraph is a collection ofverticesvertices (V) and edgesedges (E).

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Type of GraphsType of Graphs

def: A graph is simplesimple if

(1) no self-loops, and

(2) at most one undirectedundirected edgeedge

between any pair of vertices. (better

etymology: simplicial)

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A simple graphG = (V, E) consists ofV,a nonempty set ofvertices, and E, a setof unordered pairs of distinct elementsofVcalled edges.

A Graph

Asimple Graph


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Type of Graphs (Cont)Type of Graphs (Cont)

Graphs with multiple edges between pairs of

vertices allowed is called multigraphs.

Definition:

A multigraphmultigraph G = (V, E) consists of a set Vof

vertices, a set Eof edges, and a functionffrom E

to {{u, v} | u, v V, u { v}. The edges e1 and e2are called multiple or parallel edges iff(e1) =f(e2).

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Every simple graph isalso a multigraph.


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Edges from a vertex to itself, or loops, are not

allowed in multigraphs. Graphs with loops arecalledpseudographspseudographs.

Definition:

Apseudographpseudograph G = (V, E) consists of a set Vof

vertices, a set Eof edges, and a functionffrom E

to {{u, v} | u, v V}. An edge is a looploop iff(e) = {u,

v} = {u} for some u V

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Directed graphs are used to model direction of

edges between vertices. Directed graphs allowed loops of the same vertex

but multiple edges in the same direction

between two vertices are not.

Definition:

A directed graphdirected graph (V, E) consists of a set of

vertices Vand a set of edges Ethat are ordered

pairs of elements ofV.

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Directed multigraphs allow multiple directed

edges from a vertex to a second vertex.

Definition:

A directeddirected multigraphmultigraph G = (V, E) consists of a

set Vof vertices, a set Eof edges, and a functionffrom Eto {{u, v} | u, v V}. The edges e1 and e2are multiple edgesare multiple edges iff(e1) =f(e2).

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Summary of The Type of GraphsSummary of The Type of Graphs

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Type Edges Multiple EdgesAllowed LoopsAllowed

Simple Graph

Multigraph

Pseudograph

Directed Graph

Directed

Multigraph

Undirected

Undirected

Undirected

Directed

Directed

No

Yes

Yes

No

Yes

No

No

Yes

Yes

Yes


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Example: A Computer network graphExample: A Computer network graph

If multiple connection are possiblebetween two machines, we would

have a multigraph If a connection can only be one-way,

then we would have a directed

(multi)graph. 9

Computer 1 Computer 2

Computer 3

Computer 5Computer 4


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SummarySummary

Type of graphs Simple Graph

Multigraph

Pseudograph

Directed Graph

Directed Multigraph

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ext 3714


FCSIT, UNIMAS

9.2 Graph Terminology9.2 Graph Terminology

Sem I 2007/2008


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Example 1Example 1

What are the degrees of the vertices

in the following graph?

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b

a

c

f e

d

g

deg(a) =

deg(b) =

deg(c) =

deg(d) =

deg(e) =

deg(f) =

deg(g) =

2

4

4

3

1

4

0

Vertex of degree zero is calledisolated.

Vertex of only 1 degree iscalled pendant.

d is pendant and gis isolated.


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The Handshaking TheoremThe Handshaking Theorem

Theorem:

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

with e edges. Then

This theorem shows that the sum of

the degrees of the vertices of anundirected graph is even.

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2e = 7 deg(v)v V


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Example 2Example 2

How many edges are there in a graph

with seven vertices each of degreefour?

The sum of the degrees of the vertices is

4.7 = 28.

2e = 28, e = 14.

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TheoremTheorem

An undirected graph has an even number of vertices

with odd degrees.

In a directed graph, the in-degree of a

vertex v, denoted by deg(v), is

number of edges with v as theirterminal vertex.

The out-degree ofv, denoted by

deg+(v), is the number of edges with vas their initial vertex.

Theorem: Let G = (V, E) be a directed graph. Then

16 deg(v) = deg

(v) = |E|v V v V


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nn--CubesCubes The n-dimensional cube, n-cube, denoted by Q

n, is the

graph that has vertices representing the 2n bit strings of

length n.

Two vertices are adjacent if and only if the bit string s

that they represent differ in exactly one position.

Q1 Q2

Q3

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0 1

00 01

10 11

000

100

011

110 111

010

101

001


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SubgraphSubgraph

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

graph H = (W, F) where W V and F E.

Example: S is a subgraph of K5

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a

c

e

d

b

a

c

e b

K5S


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SummarySummary

Adjacency and The Degree of a Vertex

The Handshaking Theorem

Complete Graph

Cycles Wheels

n-Cubes

Subgraph

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AdjacencyAdjacency LListist RRepresentationepresentation

DEF: An adjacency list fora vertexvof a

graph Gis a list containing each vertex wofGonce for each edge between vand w.

DEF: An adjacency list representation of a

graph is a table of all the adjacency lists.

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Adjacency ListsAdjacency Lists

Examples:

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Vertex AdjacentVertices

a

b

c

d

e

b, c, e

a

a,d,e

c,e

a,c,d

Initial Vertex Terminal

Vertex

a

b

c

d

e

b, e

b

a,d,

c,e

-

e d

a c

b

e d

a c

b


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Adjacency MatricesAdjacency Matrices

Suppose G = (V, E) is a simple graph

where |V| =n. The adjacency matrixA (or AG) of G is the n v n zero-one

matrix.

Matrix AG = [aij], where

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aij =1 if{vi , vj} is an edge of G0 otherwise


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AAdjacencydjacency MMatrixatrix RRepresentationepresentation

Adjacency matrix Adjacency matrix

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0 1 1 0 1

1 0 0 0 0

1 0 0 1 1

0 0 1 0 1

1 0 1 1 0

a b c d ea

b

c

d

e

0 1 0 0 1

0 1 0 0 0

1 0 0 1 0

0 0 1 0 1

0 0 0 0 0

a b c d ea

b

cd

e

e d

a c

e d

a c

bb


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Adjacency Matrices (Cont)Adjacency Matrices (Cont)

Can also be used to represent graphs

with loops and multiple edges(pseudographs) by letting each matrixelements be the number of links

(possibly >1) between the nodes. When multiple edges are present, the

adjacency matrix is no longer a zero-one matrix.

An entry of this matrix equals to thenumber of edges that are associatedto {ai, aj}.

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Adjacency Matrix for PseudographsAdjacency Matrix for Pseudographs

Example:

Find the adjacency matrix for the

following graph

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0 3 0 2

3 0 1 10 1 1 2

2 1 2 0

a

d c

b


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IsomorphismIsomorphism

Example:

Determine if the graphs G = (V, E) and H = (W, F)below are isomorphic. Give the possible

correspondence.

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The functionf(a) = 1,f(b) = 4,f(c) = 3,f(d) = 2 is a 1-to-1 and ontocorrespondence between Vand W. Thisalso preserves adjacency where

(a, b) = (1, 4); (a, b) = (1, 3) and soonHence, the graphs areisomorphicand the possible correspondenceis:

a-1, b-4, c-3 and d-2.

a

c

b

dG

1

3 4

2

H


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Graph Invariants under IsomorphismGraph Invariants under Isomorphism

Graphs that are not isomorphic are

called invariant. To be isomorphic, three properties

should be met.

Graph G1=(V1, E1) is isomorphic tograph G2=(V2, E2) iff:

|V1| = |V2|, and |E1| = |E2|.

The number of vertices with degreen

is thesame in both graphs.

For every proper subgraph g of graph G1, thereis a proper subgraph of graph G2 that isisomorphic to g.

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Isomorphism ExampleIsomorphism Example

If isomorphic, label the 2nd graph to

show the isomorphism, else identify

difference.

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ab

cd

e f

b

d

a

e

fc


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SummarySummary

Adjacency List Representation

Adjacency Lists

Adjacency Matrices

Adjacency Matrix Representation Adjacency Matrix for Pseudographs

Graph Isomorphism

Graph Invariants

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ext 3714


FCSIT, UNIMAS

9.4 Connectivity9.4 Connectivity

TMC 1813 Discrete Mathematics

Sem I 2009/2010


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8.4: Connectivity

In an undirected graph, apath of length n

from u to vis a sequence of adjacent edgesgoing from vertex u to vertex v.

A path is a circuitif it begins and ends at the

same vertex that is u=vand has length

greater then zero.A path or circuit is also called simple if it

does not contain the same edge more then

once.

A path traverses the vertices along it.


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8.4: Connectivity

a b c

d e f

a,d,c,f,e

isasimplepathoflength 4.


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6/12/2011

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8.4: Connectivity8.4: Connectivity

b,c,f,e,b isa

circuit oflength 4.

a b c

d e f


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Paths in Directed Graphs

Same as in undirected graphs, but the path must go inthe direction of the arrows.

For ease of reference the definitions are summarized inthe following table.

Repeated Edge? Repeated Vertex? Start and Ends at

Same Point?

Path yes allowed allowed

Simple path no no no

Circuit yes allow

ed yes

Simple circuit no First and last only yes


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Directed ConnectednessDirected Connectedness

A directed graph is strongly connectediff there is

a directed path from a to b for any two verticesa and b.

It is weakly connectediff the underlyingundirectedgraph (i.e., with edge directionsremoved) is connected.

Note stronglyimplies weaklybut not vice-versa.

Any strongly connected directed graph is alsoweekly connected.


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Directed ConnectednessDirected ConnectednessAre the directed graphs G and H

shown in figure 5 strongly

connected? Are they weeklyconnected

aa aabb bb

cc cc

dd ddee ee

GG HHStrongly andweakly connected Weakly connected


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Paths & IsomorphismPaths & Isomorphism

Note that connectedness, and the

existence of a circuit or simple circuitof length kare graph invariants with

respect to isomorphism.


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Counting Paths w Adjacency MatricesCounting Paths w Adjacency Matrices

Let A be the adjacency matrix of graph

G.

The number of paths of length kfrom

vito vjis equal to (Ak)i,j.

The notation (M)i,jdenotes mi,jwhere

[mi,j]=M.


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Counting Paths w Adjacency MatricesCounting Paths w Adjacency Matrices

Example

How many paths of length 4 are there from

aa to ddin the simple graph a below?

aa bb

ccdd


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ext 3714


FCSIT, UNIMAS

9.5 Euler & Hamilton Paths9.5 Euler & Hamilton Paths

TMC 1813 Discrete Mathematics

l & il hl & il h


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Euler & Hamilton PathsEuler & Hamilton Paths

An EEuler circuituler circuitin a graph G is a simple circuit

containing every edge ofG.

An EEuler pathuler path in G is a simple path containing

every edge ofG.

A HamilHamiltton circuiton circuitis a circuit that traverses each

vertex in G exactly once.

A HamilHamiltton pathon path is a path that traverses each

vertex in G exactly once.


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H il Ci iH il Ci i


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Hamiltons CircuitHamiltons Circuit

Which of these graph has a Hamilton circuit and

which one has a Hamilton path?

aa bb

cc

dd

ee

aa bb

ccdd

G1G1G2G2

HamiltonHamilton

circuitcircuit

HamiltonHamilton

pathpath


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E l P th Th


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Euler Path Theorems

Theorem:Theorem: A connected multigraph has an Euler

circuit iff each vertex has even degree.Proof:

() The circuit contributes 2 to degree of each node.

() By construction using algorithm on p. 636

TheoremTheorem: A connected multigraph has an

Euler path (but not an Euler circuit) iff it has

exactly 2 vertices of odd degree.

One is the start, the other is the end.


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E l Ci it Al ithE l Ci it Al ith


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Euler Circuit AlgorithmEuler Circuit Algorithm

Begin with any arbitrary node.

Construct a simple path from it till youget back to start.

Repeat for each remaining subgraph,

splicing results back into original

cycle.

H il V R dH il V R d hh W ld P lW ld P l


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59

Hamiltons Voyage RoundHamiltons Voyage Round--thethe--World PuzzleWorld Puzzle

Can we traverse all the vertices of a

dodecahedron, visiting each once?`

Dodecahedron

puzzle

Equivalent

graph

Pegboard version

H ilt i P th ThH ilt i P th Th


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Hamiltonian Path TheoremsHamiltonian Path Theorems

Diracs theoremDiracs theorem: If (but not only if) G

is connected, simple, has nu3 vertices,and vdeg(v)un/2, then G has a

Hamilton circuit.

Ores corollaryOres corollary: IfG is connected,simple, has n3 nodes, and

deg(u)+deg(v)n for every pair u,vof

non-adjacent nodes, then G has aHamilton circuit.

HAMHAM CIRCUIT i NPCIRCUIT i NP l tl t


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HAMHAM--CIRCUIT is NPCIRCUIT is NP--completecomplete

Let HAM-CIRCUIT be the problem:

Given a simple graph G, does G contain aHamiltonian circuit?

This problem has been proven to be

NP-complete! This means, if an algorithm for solving it

in polynomial time were found, it could

be used to solve allNP problems inpolynomial time.


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FIN!FIN!


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FIN!FIN!

Tutorial Following week!

9.4 pg.629

4, 8, 14, 20, 24

9.5 pg.643

8, 14, 18, 24, 30,

32, 38, 42, 44

9.0 graphs full set

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