This document discusses graphs and graph data structures. It defines a graph as a pictorial representation of a set of objects connected by links, with the objects represented as vertices and the links as edges. It provides definitions and examples of basic graph terminology like vertices, edges, adjacency, and different types of graphs like directed vs undirected graphs. It also covers graph implementations using adjacency matrices and adjacency lists, as well as common graph algorithms like depth-first search and breadth-first search. Finally, it lists some applications of graphs like social networks, maps, and computer networks.

1. Data Structures & Algorithm
CSC-102
Lecture 12
GRAPH
Course Supervisor: Syeda Nazia Ashraf
1

2. Graph Data Structures
• A graph is a pictorial representation of a set of objects where some pairs of objects
are connected by links. The interconnected objects are represented by points
termed as vertices, and the links that connect the vertices are called edges.
• A graph G is an ordered pair of:
V: set of vertices and E: set of edges connecting the vertices in V
G = (V,E)
• An edge E = (u,v) is a pair of vertices
V = {A, B, C, D, E, F}
E = {{A,B}, {A,C}, {A,D}, {B,C}, {B,E}, {D,E}, {D,F}}

3. Vertices
• Each node of the graph is
represented as a vertex.
• In general a vertex
(plural: vertices or
vertexes) is a point where
two or more curves,
lines, or edges meet.
Graph Terminologies

4. Edges
• An edge is a particular type
of line segment joining two
vertices.
• An edge is a line segment
on the boundary, and is
often called a side.
• Edge represents a path
between two vertices or a
line between two vertices.
Graph Terminologies

5. Adjacency
• Two node or vertices are adjacent if they are connected
to each other through an edge.
• The state of being adjacent is known as Adjacency.
• If (u,v) is in the edge set we say u is adjacent to v (which
we sometimes write as u ~ v).
7 is adjacent from 5
or
5 is adjacent to 7
5 7
5 7
7 is adjacent from/to 5
or
5 is adjacent from/to 7
Graph Terminologies
B is adjacent to A, C is adjacent to B, and so on.
•The Vertices set = {1,2,3,4,5,6}
•The Edge set = {(6,4),(4,5),(4,3),(3,2),(5,2),(2,1),(5,1)}

6. Vertex labeled Graphs.
In a labeled graph, each vertex is labeled with some data in addition to the data that identifies the vertex.
Only the identifying data is present in the pair in the Edge set. This is similar to the (key,satellite) data
distinction for sorting.
•The Vertices set = {(2,Blue),(4,Blue),(5,Red),(7,Green),(6,Red),(3,Yellow)}
•The Edge set = {(2,4),(4,5),(5,7),(7,6),(6,2),(4,3),(3,7)}

7. Edge labeled Graphs
A Edge labeled graph is a graph where the edges are associated with labels. One can indicate this be making
the Edge set be a set of triples. Thus if (u,v,X) is in the edge set, then there is an edge from u to v with
label XEdge labeled graphs are usually drawn with the labels drawn adjacent to the arcs specifying the edges.
•The Vertices set = {Red,Green,Blue,White}
•The Edge set = {(red,white,{white,green}) ,(white,red,{blue}) ,(white,blue,{green,red}) ,(red,blue,{blue})
,(green,red,{red,blue,white}) ,(blue,green,{white,green,red})}

8. Undirected Graphs
• Undirected edge has no orientation (no arrow head)
• Undirected graph – all edges are undirected
U----------V
• Undirected edge represents an un ordered pair {U,V}
The order of vertices in E
is not important for
undirected graphs!!
E (Edge1) = {{A,B},{A,D},{B,C},{B,D}}
The adjacency relation is symmetric in an undirected
graph, so if u ~ v then it is also the case that v ~ u.

9. Directed Graphs or Diagraph
• Directed edge has an orientation (has an arrow head)
• Directed graph – all edges are directed
U-----------→V origin ------→destination
• Directed edge represents an ordered pair (U,V)
E(Graph2) = {(1,3) (3,1) (5,9) (9,11) (5,7)
(9,9) (11,1)}
The order of vertices in E
is important for
directed graphs!!
u is adjacent to v only if the pair (u,v) is in the Edge set.

10. Un weighted Graphs
A weighted Graph with all edges, having weight = 1 unit
Shortest path?

11. Weighted Graphs
A graph in which each edge carries a value i.e. labelled edge
•The Vertices set = {1,2,3,4,5}
•The Edge set = {(1,4,5) ,(4,5,58) ,(3,5,34) ,(2,4,5) ,(2,5,4)
,(3,2,14) ,(1,2,2)}

12. Possible Routs from City A to City D
Shortest rout is green through B and C, adding weights to calculate total cost.
Directed Weighed Graphs

13. Directed Acyclic Graphs (DAGs)
A Dag is a directed graph without cycles. They appear as special cases in CS applications all the time.
•The Vertices set = {1,2,3,4,5,6,7,8}
•The Edge set = {(1,7) ,(2,6) ,(3,1),(3,5) ,(4,6) ,(5,4),(5,2) ,(6,8) ,(7,2),(7,8)}

14. Connected Graph: There is a path between each two vertices.
Disconnected Graph: There are at least two vertices not
connected by a path.
Connected and Disconnected Graphs
Disconnected Graphs

15. Path
• Path represents a
sequence of edges
between two vertices.
Graph Terminologies
ABCD represents a path from A to D.

16. Simple Path And Cycle
A simple path is a path in which all vertices,
except possibly the first and the last, are distinct
i.e. No repeated Vertices
A cycle is a simple path in which the first and
the last vertices are the same
Graph Terminologies

17. •Complete graph: a graph in which every vertex is directly connected to every other vertex
• Length: The number of edges in the path.
e.g., a path from 1 to 4:
<1, 2, 3, 4>
1 2
3 4
Graph Terminologies

18. Graph Terminologies
Degree of Vertex
• In a Diagraph, the number of edges coming out of a Vertex is called
the out-degree of that Vertex
• Number of edges coming in, is the in-degree
• In an undirected graph, we just talk of degree of a Vertex.
• It is the number of edges incident on the Vertex.
In and out degrees of
Vertex

19. Incident Edge
• In an undirected graph, we say that an edge is incident on a Vertex if
a vertex is an end point of the edge

20. Basic Operations
• Following are basic primary operations of a Graph −
• Add Vertex − Adds a vertex to the graph.
• Add Edge − Adds an edge between the two vertices of the graph.
• Display Vertex − Displays a vertex of the graph.

21. Graph Implementation
• Array-based
• Linked-list-based

22. Array-based implementation
• Mathematical graphs can be represented in data structure. We can
represent a graph using an array of vertices and a two-dimensional
array of edges.
• Use a 1D array to represent the vertices
• Use a 2D array (i.e., adjacency matrix) to represent the edges

23. Array-based implementation (cont’d)
Array-based implementation (cont’d)

24. Linked-list-based implementation
• Use a 1D array to represent the vertices
• Use a list for each vertex v which contains the
vertices which are adjacent from v (adjacency
list)

25. Linked-list-based implementation (cont’d)

26. Graph Searching:
Problem: find if there is a path between two vertices of the graph.
Methods:
Depth-First-Search (DFS) : Once a possible path is found, continue
the search until the end of the path
• Breadth-First-Search (BFS) : Start several paths at a time, and
advance in each one step at a time

27. Depth-first Search:
Visit all neighbors of a
neighbor before visiting
your other neighbors
Breadth-First Search:
Pick each child of S in
turn and discover
their vertices
adjacent to that child.
1. Pick a source vertex S to start.
2. Discover the vertices that are adjacent to S.
Done when all children have been discovered and examined.
How it works?

28. Data Structure - Depth First Traversal
• Main idea:
• Travel as far as you can down a path
• Back up as little as possible when you reach a "dead end" (i.e., next vertex has
been "marked" or there is no next vertex)
• DFS uses a stack !
• As in the example given above, DFS algorithm traverses from A to B to C to
D first then to E, then to F and lastly to G. It employs the following rules.
• Rule 1 − Visit the adjacent unvisited vertex. Mark it as visited. Display it.
Push it in a stack.
• Rule 2 − If no adjacent vertex is found, pop up a vertex from the stack. (It
will pop up all the vertices from the stack, which do not have adjacent
vertices.)
• Rule 3 − Repeat Rule 1 and Rule 2 until the stack is empty.

29. Step Traversal Description
1.
Initialize the stack.
2.
Mark S as visited and put it
onto the stack. Explore any
unvisited adjacent node
from S. We have three nodes
and we can pick any of them.
For this example, we shall take
the node in an alphabetical
order.
3.
Mark A as visited and put it
onto the stack. Explore any
unvisited adjacent node from
A. Both Sand D are adjacent
to A but we are concerned for
unvisited nodes only.
Data Structure - Depth
First Traversal

30. 4.
Visit D and mark it as visited
and put onto the stack. Here,
we have B and C nodes, which
are adjacent to D and both are
unvisited. However, we shall
again choose in an
alphabetical order.
5.
We choose B, mark it as
visited and put onto the stack.
Here Bdoes not have any
unvisited adjacent node. So,
we pop Bfrom the stack.
6.
We check the stack top for
return to the previous node
and check if it has any
unvisited nodes. Here, we
find D to be on the top of the
stack.
Data Structure -
Depth First
Traversal

31. 7.
Only unvisited adjacent node
is from D is C now. So we
visit C, mark it as visited and
put it onto the stack.
As C does not have any unvisited adjacent node so we keep popping the stack
until we find a node that has an unvisited adjacent node. In this case, there's
none and we keep popping until the stack is empty.
Data Structure - Depth First Traversal

32. Depth-first Depth First Search in Tree
searching  A depth-first search (DFS)
explores a path all the way to
a leaf before backtracking and
exploring another path.
 For example, after searching
A, then B, then D, the search
backtracks and tries another
path from B.
 Node are explored in the
order A B D E H L M N I
O P C F G J K Q.
 N will be found before J.
L M N O P
G
Q
H J
I K
F
E
D
B C
A

50. • Main idea:
• Look at all possible paths at the same depth before you go at a deeper
level
• Back up as far as possible when you reach a "dead end" (i.e., next
vertex has been "marked" or there is no next vertex)
• BFS uses a queue !
• As in the example given above, BFS algorithm traverses from A to B to E
to F first then to C and G lastly to D. It employs the following rules.
Rule 1 − Visit the adjacent unvisited vertex. Mark it as visited. Display it.
Insert it in a queue.
Rule 2 − If no adjacent vertex is found, remove the first vertex from the
queue.
Rule 3 − Repeat Rule 1 and Rule 2 until the queue is empty.
Data Structure - Breadth First Traversal

51. Step Traversal Description
1.
Initialize the queue.
2.
We start from
visiting S(starting node), and
mark it as visited.
3.
We then see an unvisited
adjacent node from S. In this
example, we have three nodes
but alphabetically we
choose A, mark it as visited
and enqueue it.
Data Structure -
Breadth First
Traversal

52. 4.
Next, the unvisited adjacent
node from S is B. We mark it
as visited and enqueue it.
5.
Next, the unvisited adjacent
node from S is C. We mark it
as visited and enqueue it.
6.
Now, S is left with no unvisited
adjacent nodes. So, we
dequeue and find A.
Data Structure -
Breadth First
Traversal

53. 7.
From A we have D as
unvisited adjacent node. We
mark it as visited and enqueue
it.
At this stage, we are left with no unmarked (unvisited) nodes. But as per the
algorithm we keep on dequeuing in order to get all unvisited nodes. When
the queue gets emptied, the program is over.
Data Structure - Breadth First Traversal

54. Breadth First Search in Tree
 A breadth-first search (BFS)
explores nodes nearest the
root before exploring nodes
further away.
 For example, after searching
A, then B, then C, the search
proceeds with D, E, F, G.
 Node are explored in the
order A B C D E F G H I J K L
M N O P Q.
 J will be found before N.
L M N O P
G
Q
H J
I K
F
E
D
B C
A

72. Applications of Graph
•Networks (computer network, communication networks, social network,
information network, biological networks, transportation networks(roads,
flight communications)).
•Flow charts
• Driving Map
• Electronic Circuits
•Tasks in some project (some of which should be completed before others),
so edges correspond to prerequisites.
•States of an automaton / program.
• Representing family histories
•Nodes could represent positions in a board game, and edges the moves that
transform one position into another.
•Nodes could represent computers (or routers) in a network and weighted
edges the bandwidth between them.
•Nodes could represent towns and weighted edges road distances between
them, or train journey times or ticket prices.

73. Applications of Graph
Computer networks
Circuits
Schedules
Hypertext
Maps