6 M131 Chapter9 Suha AlShaikh

M131-Chapter9 By Ms. Suha Al-Shaikh-Dammam

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A graph G =(V, E) where:

V: set of vertices (nodes)

E: set of edges.

Definition: A graph G=(V,E) consists of V, a non-empty set of vertices and E, a set of

edges. Each edge has either one or two vertices associated with it, called its end

points. An edge is said to connect its end point.

Remark: the computer can be modeled using a graph in which the vertices of the

graph represent the data centers, and the edges represent communication links.

Simple graphs: a graph in which each edge connects 2 different vertices and where

no 2 edges connect the same pair of vertices.

Note: in the simple graph each edge is associated to an unordered pair of vertices, and

no other edge is associated to this same edge.

We say that {u,v} is an edge in the simple graph if there is an edge associated to {u,v}

Multi graphs: graphs that may have multiple edges connecting the same vertices.

Where there are m different edges associated to the same unordered pair of vertices

{u,v}. we say that {u,v} is an edge of multiplicity m.

Chapter 9: Graphs

Section (9.1): Graph and Graph Models

Undirected Graphs:


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Pseudo graphs: graphs that may include loops and possibly multiple edges

connecting the same pair of vertices are sometimes called pseudo graphs.

Definition: a directed graph (or digraph) (V,E) consists of a non empty set of vertices

V and a set of directed edges E. each directed edge is associated with an order pair of

vertices (u,v) is said to start at u and end at v.

Simple directed graph: it is when a directed graph has no loops and has no multiple

directed edges .

Directed multi graphs: directed graphs that may have multiple directed edges from a

vertex to a second vertex .

When there are m directed edges, each associated to an ordered pair of vertices (u,v),

we say that( u,v) is an edge of multiplicity m.

directed Graphs:

Directed graphs: we obtain a directed graph when we assign a direction to each

edge in an undirected graph.

Note:A graph with both directed an undirected edges is called a mixed graph


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The Terminology for the various types of graphs is summarized in the following

Table:

Example: draw graph models, starting the type of graph (from table1) used to

represent air line routes where every day there are four flights from Boston to New

York, 2 flights from New York to Boston, 3 flights from New York to Miami, 2

flights from Miami to New York, one flight from New York to Detroit, 2 flights from

Detroit to New York, 3 flights from New York to Washington, 2 flights from

Washington to New York, and one flight from Washington to Miami, with

a) An edge between vertices representing cities that have a flight between them

(in either direction)

b) An edge between vertices representing cities for each flight that operate

between them (in either direction)

c) An edge between vertices representing cities for each flight that operate

between them (in either direction) plus a loop for a special sight seeing trip

that takes off and land in Miami.

d) An edge from a vertex representing a city where a flight starts to the vertex

representing the city where it ends

e) An edge for each flight from a vertex representing a city where the flight

begins to the vertex representing the city where the flight ends.

loops Multiple

edges

edges Table(1)

No No undirected Simple graph

No Yes Undirected Multi graph

Yes Yes Undirected Pseudo graph

No No Directed Simple directed graph

Yes Yes Directed Directed multi graph

yes yes Directed and

undirected

Mixed graph


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Solution:


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Graph Models:

1. Niche over lap Graph in Ecology. It is a simple graph because no loops or

multiple edges are needed in this model. In this model we connect two animals

if some of their food resource are the same.(refer to text book page 593).

2. Acquaintanceship graph: ( we use it to represent relation between people). It is

a simple graph to represent whether 2people know each other, that is whether

they are acquainted. (refer to text book page 593; read the other models from

the text book page 593, 594, and 595.


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Basic Terminology:

Definition: two vertices u and v in an undirected graph G are called adjacent

(neighbors) in G if u and v are end points of an edge in G.

If e is associated with {u,v}, the edge e is called incident with the vertex u and v. The

edge e is also said to connect u and v.

Definition: (gives how many edges are incident to a vertex)

The degree of a vertex in an undirected graph is the number of edges incident with it,

except that a loop at a vertex contributes twice to the degree of that vertex.

Remark: the degree of a vertex is denoted by deg (v).

Example: what are the degrees of the vertices in the following graphs G and H?

Solution:

(1) In undirected graph G: deg(a)=2, deg(b)=4

deg(c)=4, deg(f)=4

deg(d)=1, deg(e)=3

deg(g)=0 (isolated)

(2) In Undirected graph H: deg(a)=2, deg(b)=6

deg(e)=6, deg(c)=1 (pendant)

deg(d)=5

Remark:

1) a vertex of degree zero is called isolated.

2) A vertex is pendant if and only if it has degree one.

Section9.2: Graphs Terminology and Special Types of Graphs


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Theorem : (THE HANDSHAKING THEOREM)

let G=(V,E) be an undirected graph with e=|E| edges then:

Example: how many edges where in a graph with 10 vertices each of degree six?

Solution:

Remark: last theorem shows that the sum of the degrees of the vertices of an

undirected graph is even.

Theorem: an undirected graph has an even number of vertices of odd degree.

Definition: when (u,v) is an edge of the graph G with directed edges, u is said to be

adjacent to v and v is said adjacent to v.

u→ the vertex u is called the initial vertex of (u,v)

v→ is called the terminal or end vertex of (u,v)

The initial vertex and the terminal vertex for a loop are the same.

Definition: in a graph with directed edges:

1) The in-degree of a vertex v, denoted by degˉ(v) is the number of edges with v

as their terminal vertex.

2) The out-degree of v, denoted by is the number of edges with v as

their initial vertex.

Example: find the in-degree and out-degree of each vertex in the graph G with

directed edges in the following directed graph G.

Solution: The In-degree in G

Vvve )deg(2

30.

302

60602

60)10(6)deg(

isedgesofnothehence

ee

V

)(deg V


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Theorem: let G=(V,E) be a graph with directed edges. Then

Refer to example4.

Some special simple graphs

Example(page601): complete graphs

The complete graph on n vertices denoted by , is the simple graph that contains

exactly one edge between each pair of distinct vertices.

The complete graph for n=1,2,3 and 4

For n=5 and 6 see the book page(601) .

Example: Cycles

Example: wheels;

we obtain the wheel when we add an a additional vertex to the cycle ,for 3n and

connect the new vertex to each of the n vertices in nC .

Read Example 8(page 602).

EVVVvVv

)(deg)(deg

nk

nW nC


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Bipartite Graphs:

A simple graph G is called bipartite if its vertex set V can be partitioned into 2

disjoint sets such that every edge in the graph connects a vertex in and a

vertex in . (That is no edge in G connect either two vertices in V1 or two vertices in

V2).

We call the pair a bipartition of the vertex set V of G.

Example:

Consider C6

C6 is bipartite because its vertex set can be partitioned into the 2 sets

and every edge of C6 connects a vertex in 1V and a vertex in 2V .

Example:

K3 is not bipartite .

Suppose

Since there is an edge between is not a bipartite graph .

21andVV 1V

2V

),( 21 VV

64225311 ,,,, VVVandVVVVV

},{},{ 32211 VVVVV

332 , KandVV


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Example: show that graph G is bipartite

Solution: since the vertex set of G , where

And every edge is a connection of so G is a bipartite.

Read (example 11 page 603).

Theorem: A simple graph is bipartite if and only if it is possible to assign one of 2

different colors to each vertex of the graph so that no 2 adjacent vertices are assigned

the same color.

Example: Complete Bipartite Graph :

Is the graph that has its vertex set partitioned into 2 subsets of m and n vertices

respectively. There is an edge between 2 vertices if and only if one vertex is in the

first subset and the other vertex in the second subset.

Read Examples:14,15 and 16

New graph from an old one:

(1) Sub graph H=(W,F) of G=(V,E) is when WV and FE.

A sub graph H of G is proper sub graph if H≠G.

(2) Union of two simple Graphs: ),(),( 222111 EVandGEVG is the simple

graph with vertex set 21 VV and edges set 21 EE , and denoted as 21 GG

21 VVV },,,{},,{ 21 gfecandVdbaV

21andVV


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• Graph representations:

– Adjacency lists.

– Adjacency matrices.

Example: Use adjacency lists to describe the following graph:

Solution: An adjacency list for a simple graph

Matrix A=[aij], where aij is 1 if {vi, vj} is an edge of G, 0 otherwise.

Example:

Section9.3: Graph Representations

Undirected: Adjacency Lists

Undirected: Adjacency Matrices

otherwise

GofedgeanisVVifa

ji

ij0

},{1


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Example:Draw a graph using the following adjacency matrix with respect to the

vertices: a,b,c,d .

Example: use an adjacency matrix to represent the pseudo graph.

the adjacency matrix using the ordering of vertices a,b,c,d is: :Solution

Remark: 1) the adjacency matrix of a simple graph is symmetric

2) The adjacency matrix for a directed graph does not have to be symmetric, because

there may not be an edge from when there is an edge from

3) Adjacency matrices can also be used to represent directed multi graphs

4) In the adjacency matrix for a directed multi graph the number of edges that

are associated to

5) When a simple graph contains few edges, that is, when it is sparse use adjacency

list.

6) When a simple graph is dense, that is when it contains more than half of all

possible edges use adjacency matrix to represent the graph

ij atoa ji atoa

ija

ji VV ,


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Incidence Matrices:

Another way to represent graphs is to use incidence matrices. Let G=(V,E) be an

undirected graph. Suppose that are the vertices and

Are the edges of G. then the incidence matrix with respect to this ordering of V and E

is the nXn matrix , where

Example page 614:

Represent the following graph with an incidence matrix:

1 row per node, listing the terminal nodes of each edge incident from that node.

Directed Adjacency Lists

nVVV ,...,, 21 neee ,...,, 21

ijmM

otherwise

Vwithincidenteedgewhenm

ij

ij0

1


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Paths: a path is a sequence of edges that begins at a vertex of a graph and travels

from vertex to vertex along edges of a graph.

Definition: let n be a non negative integer and G an undirected graph. A path of length

n from u to v in G is a sequence of n edges, of G such that is associated

with , and so on, with associated with

Remarks: 1) the path is a circuit if it begins and ends at the same vertex, that is, if

u=v, and has length greater than zero.

2) A path or circuit is simple if it doesn’t contain the same edge more than once.

3) When the graph is simple, we denote this path by its vertex sequence

},,,{ 20 nxxx

Example: consider the following simple graph.

1) a,d,c,f,e is a simple path of length4. {a,d},{d,c},{c,f},{f,e} are all edges.

2) d,e,c,a is not a path, because {e,c} is not an edge.

3) b,c,f,e,b is a circuit of length4. because {b,c},{c,f},{f,e},{e,b} are all edges and

this path begins and ends at b.

Directed: Adjacency Matrices

Section9.4: Connectivity

nee ,.....,1 1e

},{ 10 xx ne

vxanduxwherexx nnn 01 },{


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4) a,b,c,d,a,b is a path of length 5. which is not simple because it contains the edge

{a,b} twice.

Example: does each of these lists of vertices form a path in the following graph?

Which paths are simple which are circuits? What are the length of those that are

paths?

a) a,e,b,c,b b) a,e,a,d,b,c,a c) e,b,a,d,b,e d) c,b,d,a,e,c

Solution: a) {a,e},{e,b},{b,c},{c,b} are all edges so it is a path of length 4. not a

circuit, not simple (because of {b,c}, {c,b})

b) {a,e},{e,a},{a,d},{d,b},{b,c}, there is no edge {c,a} so it os not a path

c) {e,b} there is no path between {b,a} so it is not a path

d) {c,b},{b,d},{d,a},{a,e},{e,c} is a circuit of length 5 it is simple circuit.

Definition: let n be a non negative integer and G a directed graph. A path of length n

from u to v in G is a sequence of edges of G such that is associated

with is associated with and so on, with associated with

Remarks: 1) when there are no multiple edges in the directed graph this path is

denoted by its vertex sequence

2) A path of length greater than zero that begins and ends at the same vertex is called

a circuit or cycle.

3) A path or circuit is called simple if it doesn’t contain the same edge more than

once.

Read examples 2,3 and 4 page 623.

neee ,...,, 21 1e

210 ),,( exx ),( 21 xx ne

vxanduxwherexx nnn 01 ),,(

nxxx ,...,, 10


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When is there always a path between 2 vertices in the graph?

Definition: an undirected graph is called connected if there is a path between every

pair of distinct vertices of the graph.

Remark: any 2 computers in the network can communicate if and only if the graph of

this network is connected.

Example: consider the following graphs:

G1 is connected because for every pair of distinct vertices there is a path between

them.

The graph G2 is not connected because there is no path between a and d, or c and d,

or b and d.

Theorem: there is a simple path between every pair of distinct vertices of a connected

undirected graph.

Remark: 1) a connected component of a graph G is a connected sub graph of G that

is not a proper sub graph of another connected sub graph of G.

2) A graph G that is not connected has 2 or more connected components that are

disjoint and have G as their union.

3) Sometimes the removal of a vertex and all edges incident with it produces a sub

graph with more connected components than in the original graph such vertices are

called cut vertices.

4) An edge whose removal produces a graph with more connected components than in

the original graph is called a cut edge or bridge.

Note:[the removal of a cut vertex from a connected graph produces a sub graph that is

not connected]

Connectedness In Undirected Graphs:


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Example: find the cut vertices and cut edges in graph G

remark: (remove a,h,g,h gives 1 components)

Solution: the cut vertices of G are b,c, and e. the cut edges are {a,b}, {c,e}

Removing either one of these edges or vertices disconnect G

Definition: a directed graph is strongly connected if there is a path from a to b and

from b to a whenever a and b are vertices in the graph.

Definition: a directed graph is weakly connected if there is a path between every 2

vertices in the underlying undirected graph.

Remarks:

1) a directed graph is weakly connected if and only if there is always a path

between 2 vertices when the direction of the edges are disregarded.

2) Any strongly connected directed graph is also weakly connected.

Example: are the directed graphs G and H strongly connected? Are they weakly

connected?

Solution: the graph G is strongly connected there is a path between any 2 vertices in

G. hence G is also weakly connected.

Connectedness in Directed Graph:


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The graph H is not strongly connected, because there is no directed path from a to b.

H is weakly connected.

Example: determine whether each of these graphs is strongly connected and if not.

Whether it is weakly connected.

Solution: a) the graph G is weakly connected, because there is no path from a to b.

b) The graph H is weakly connected, because there is a path from e to c, but there is

no path from c to e.

Remark: the sub graphs of a directed graph G that are strongly connected but not

contained in larger strongly connected sub graphs are called the strongly connected

components or strong components of G.

Example:

the graph H has 3 strongly components the vertex a; the vertex e; and the graph

consisting of the vertices b,c, and d and edges (b,c), (c,d) and (d,b).

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6 M131 Chapter9 Suha AlShaikh

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