MELJUN CORTES Automata Lecture Graphs 2

8/21/2019 MELJUN CORTES Automata Lecture Graphs 2

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Graphs

Theory of Computation (With Automata Theory)

* Property of STI

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GraphsIntroduction

Undirected Graphs

• An undirected graph is a finiteset of points and linesconnecting the points.

• The points in a graph are callednodes or vert ices , and the lines

are called edges or arcs .

For example:

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2 3

4 5

node

edge


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Graphs


* Property of STI

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• In this course, an undirected

graph will simply be called agraph.

• Each node in a graph is labeled

while each edge is identified

using a tuple (i , j ), where i and j are the nodes that the edge

connects.

For example:

Take note that edge (1, 3) can

also be referred to as edge (3,1).

1

2 3

4 5

edge

(1, 3)

edge(3, 5)


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Graphs


* Property of STI

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•

The number of edges connectedto a node is called the degree of

that node.

For example:

In this graph, node 1 has a

degree of 2 while node 4 has

a degree of 3.

No more than one edge is

allowed between any two nodes.

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4 5


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Graphs


* Property of STI

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•

A graph G can be described asbeing composed by a set of

nodes, N , and a set of edges, E .

G = (N , E )

Examples:

G = ({1, 2, 3, 4, 5}, {(1, 2), (1, 3),

(2, 4), (3, 4), (3, 5), (4, 5)})

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2 3

4 5

Graph G


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* Property of STI

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H = ({1, 2, 3, 4, 5, 6}, {(1, 2),

(1, 3), (2, 6), (3, 4), (3, 5),

(4,5)})

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2

3

6

5

4 Graph H


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Graphs


* Property of STI

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• A subgraph of graph G is a

graph whose set of nodes is asubset of that of G and whose

edges are the edges of G on the

corresponding nodes.

For example:

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2

3

6

5

4

3 5

4

Subgraph of

Graph G

Graph G


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Graphs


* Property of STI

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• A path in a graph is a sequence

of consecutive edges.

A path is a s imple path if there

are no repeated nodes.

The length of a path is thenumber of edges within the path.

For example:

The length of the path indicated

is 3.

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2

3

6

5

4

Path


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Graphs


* Property of STI

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•

If every two nodes in a graphhave a path between them, the

graph is called a connected

graph.

•

If there is at least two nodes in agraph that do not have a path

between them, then the graph is

called a disconnected graph .

For example:

1

2

3

6

5

4


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Graphs


* Property of STI

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• A path is a cyc le if it starts and

ends in the same node. Takenote that the choice of the

starting node is arbitrary.

A s imp le cyc le is a cycle that

contains at least three nodesand has no repeating nodes

except for the first and last node.

For example:

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2

3

6

5

4

A path which

is a cycle


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* Property of STI

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• A special kind of graph that is

connected and has no simple

cycles is called a tree .

For example:

1

2 3

54 6 7 8

9 10


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Graphs


* Property of STI

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Directed Graphs

• A directed graph is similar to an

undirected graph except that

edges are replaced by directed

edges .

In other words, lines are

replaced by arrows.

For example:

1

2

3

6

5

4

Directed GraphG


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Graphs


* Property of STI

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• An edge connecting two nodes

from node i to node j isrepresented as the ordered pair

(i , j ). While an edge connecting

two nodes from node j to node i

is represented as the ordered

pair ( j , i ).

Since these are ordered pairs,

edge (i , j ) is different from edge

( j , i ).

For example:

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2

3

6

5

4

Edge (2, 6)

Edge (6, 2)


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* Property of STI

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• The formal description of thegiven directed graph G would

then be:

G = ({1, 2, 3, 4, 5, 6}, {(1, 2),

(1, 3), (2, 6), (3, 1), (3, 4),

(4, 5), (5, 3), (6, 2)})

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2

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6

5

4

Directed Graph G


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Graphs


* Property of STI

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• In a directed graph, the number

of arrows pointing from a

particular node is the outdegree

of that node

The number of arrows pointing

to a particular node is the

indegree .

For example:

For the given directed graph

G, the outdegree of node 1 is

2 while its indegree is 1.

MELJUN CORTES Automata Lecture Graphs 2

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