Unit 4.2(graphs)

Graphs: Introduction
 Applications of graphs
 Graph representations
 graph traversals
 Minimal Spanning Trees.
S. Durga Devi ,CSE,CBIT

Introduction to Graphs
 Another non linear data structure.
 In tree structure, there is a hierarchical relationship between
parent and children, that is, one parent and many children.
 On the other hand, in graph, the relationship is from many
parents to many children.
Definition-A graph G = (V,E) is composed of:
V: set of vertices(or nodes)
E: set of edges connecting the vertices in V
• An edge e = (u,v) is a pair of vertices
• Example:
a b
c
d e
V= {a,b,c,d,e}
E= {(a,b),(a,c),(a,d),(b,e),
(c,d),(c,e),(d,e)}

Graph terminology
• Digraph- A digraph is also called a directed graph. It is a graph G, such
that, G=<V,E>, where V is the set of all vertices and E is the set of ordered
pair of elements from V.
• Edges have directions
v= A,B,C,D
E=((A,B),(A,C),(A,D),(D,B))
A B
C
D
A B
C
D
These are directed graphs
G directed if edges ordered pairs (u,v)
 u v
G undirected-
if edges unordered pairs {u,v}
  S. Durga Devi ,CSE,CBIT

Adjacent vertices
If two nodes are connected by an edge, they are neighbors (and the nodes are adjacentto each other).
 The size of a graph is the number of nodes in it
 The empty graph has size zero (no nodes).
Loop-An edge that connects the vertex with itself
Degree of a Vertex- the number of edges connected to that vertex
• Fordirected graph,
– the in-degree of a vertex v is the number of edges incident on to the vertex.
– the out-degree of a vertex v is the number of edges emanating from that vertex.
Examples
3
2
3
3
0
1 33
0
1 2
3 4 5 6
2
3 3
1 1 1 1
directed graph
in-degree
out-degree
0
1
2
in:1, out: 1
in: 1, out: 2
in: 1, out: 0

 path: sequence of vertices v1,v2,. . .vk such that consecutive vertices vi and vi+1 are adjacent.
a
d e
c
b a b
c
d e
a b e d c
simple path: no vertex is repeated
a b
c
d e
b e c
Cycle - some path with distinct vertices except that the last
vertex is the same as the first vertex
a c d a
a b
c
d e
Parallel edges- if there are one or
More than one edges between
the same pair of vertices.

 A graph without cycles is called acyclicgraph.
 Sub graph is a graph with subset of vertices and edges of a graph.
 A graph is called a simple graph if it has no loops and no parallel edges.
 A graph is termed as weighted graph if all the edges in it are labeled with
some weights.
 A complete graph is a graph if every node in a graph is connected to
every othernode.
 Two vertices are said to be connected if there is a path from vi to vj or vj to
vi
 connected graph: any two vertices are connected by some path. if there is a
path from vi to vj and vj to vi.
 A directed graph is said to be strongly connected if for every pair of distinct
vertices vi,vj in G, there is a path from vi to vj and also from vj to vi.
 Weakly connected graph- if a graph is not strongly connected but it is
connected
S. Durga Devi
,CSE,CBIT

A
D
B C
Acyclic graph
A
C
BD
Strongly Connected
A B
C D
E
CompleteGraph
Weighted Graph
A
D
B C
Cyclic graph

0 0
1 2 3
1 2 0
1 2
3
(i) ii) (iii) (iv)
(a) Some of the sub graph of G1
0
1 2
3
G1
0 0
1
0
1
2
(i) (ii) (iii)
(b) Some of the sub graph of G2
0
1
2
G2
S. Durga Devi
,CSE,CBIT

More definitions : Loop
An edge that connects the vertex with itself
A B
C
D
Or an edge whose starting and end vertices are same

Even More Terminology
connected not connected
•connected graph: any two vertices are connected by some path
At least two vertices
are not connected in a
graph is called
disconnected graph.

0 0
1 2 3
1 2 0
1 2
3(i) (ii) (iii) (iv)
(a) Some of the subgraph of G1
0 0
1
0
1
2
0
1
2
(i) (ii) (iii) (iv)
(b) Some of the subgraph of G3
0
1 2
3
G1
0
1
2
G3
Subgraphs Examples

Simple graph
• Simple graph- A graph (digraph) if it does
not have any loops or parallel edges (edge
from v1 to v2 and v2 to v1) is called
simple graph.

Representation of a Graph
There are two ways of representinga graph in memory:
SequentialRepresentation by means of Adjacency Matrix.
Linked Representation by means of Linked List.

Adjacency Matrix
A B C D E
A 0 1 1 1 0
B 1 0 1 1 0
C 1 1 0 1 1
D 1 1 1 0 1
E 0 0 1 1 0
 Adjacency Matrix is a bit matrix which containsentries of only 0 and 1
 Since it stores the information of whether the adjacency vertices are present
or not.
 The connected edge between two vertices is represented by 1 and absence
of edge is represented by 0.
 This representation uses a square matrix of order n x n, where n is the
number of vertices in the graph.
 Adjacency matrix of an undirected graph is symmetric.
A
B C
D E

Linked List Representation
A
B
C
D
N E
B C D NULL
A B E NULLD
A C D NULL
A B E NULLC
C D NULL
A
B C
D E
 It saves the memory.
 The number of lists dependson the number of vertices in the graph.
 The header node in each list maintains a list of all adjacent vertices of a
node.

Undirected Graph Adjacency List Adjacency Matrix

Directed Graph Adjacency List Adjacency Matrix

Comparison among various representations
• Adjacency matrix representation always requires n x n matrix
with n vertices, regardless of the number of edges. But from
the manipulation point of view, this representation is the best.
• Insertion, deletion and searching are concerned, in some cases,
linked list representation has an advantage, but overall
performance is consider, matrix representation of graph is
more powerful than all

Graph Traversal Techniques
There are two standardgraph traversaltechniques:
Depth-FirstSearch (DFS)
Breadth-FirstSearch (BFS)
 Traversing a graph means visiting all the vertices in the graph exactly
once.
 DFS and BFS traversals result an acyclic graph.
 DFS and BFS traversal on the same graph do not give the same order of
visitof vertices.
 The order of visiting nodes depends on the node we have selected first
and hence the order of visitingnodes may not be unique.

Depth-First Search
 DFS is similarto the in order traversal of a tree
 Stack is used to implement the Depth- First-Search.
 DFS follows the followingrules:
1. Select an unvisitednode X, visitit, and treat as the current
node
2. Find an unvisitedadjacentof the current node as Y, visitit,
and make it the new current node;
3. Repeat step 1 and 2 till there is a ‘dead end’. dead end means
a vertex which do not have adjacent nodes or no more nodes
can be visited.
4.aftercoming to a dead end we backtrackingalong to its parent
to see if it has another adjacent vertex other than Y and then
repeat the steps 1,2,3. until all vertices are visited.

Depth-First Search
 A stack can be used to maintain the track of all paths from any
vertex so as to help backtracking.
 Initially starting vertex will be pushed onto the stack.
 To visit a vertex, we are to pop a vertex from the stack, and
then push all the adjacent vertices onto it.
 A list is maintained to store the vertices already visited.
 When a vertex is popped, whether it is already visited or not
that can be known by searching the list.
 If the vertex is already visited, we will simply ignore it and we
will pop the stack for the next vertex to be visited.
 This procedure will continue till the stack not empty

Breadth First Traversal:
The breadth first traversal is similar to the pre-order traversal of a binary tree.
The breadth first traversal of a graph is similar to traversing a binary tree
level by level (the nodes at each level are visited from left to right).
All the nodes at any level, i, are visited before visiting the nodes at level i + 1.
To implement the breadth first search algorithm, we use a queue.
BFS follows the following rules:
1. Select an unvisited node x, visit it, have it be the root in a BFS tree being formed. Its level
is called the current level.
2. From each node x in the current level, visit all the unvisited neighbors of x. The newly
visited nodes from this level form a new level that becomes the next current level.
3. Repeat step 2 until no more nodes can be visited.
4. If there are still unvisited nodes, repeat from Step 1.

The Depth First Search Tree Order : A, B, E, D, C
A
B
E
DC
A
B
E
DC
The Breadth First Search Tree Order : A, B, C, D, E
A
B
E
DC
Graph

BFS Traversal Order
A B D E C G F H IA B C F E G D H I
DFS Traversal Order
A B C
D E F
G H I
A B C
D E F
G H I

Practice problems
v1
v2
v6
v3
v7
v5
v8
v4
Dfs-
Bfs-

Applications of Graphs
 Electronic circuits
Printed circuit board
Integrated circuit
 Transportation networks
Highway network
Flight network
 Computer networks
Local area network
Internet
Web
 Databases
Entity-relationship diagram

A spanningtree of a graph is just a sub graph that containsall the vertices and
is a tree( without any cycles in a graph)
A graph may have many spanning trees.
o
r
o
r
o
r
Some SpanningTrees from GraphAGraphA
Spanning Trees

Minimum spanning tree
• Take an example that connect four cities in India like
vijayawada, hyderabad, banglore and mumbai by a graph
Hyderabad
Mumbai
Banglore vijayawada
270km
980km
580km
1000km
Find a short distance to cover all four cities
700km
650

Minimum Spanning Trees
The minimum spanning tree for a given graph is the spanning tree of
minimum cost for that graph.
5
7
2
1
3
4
2
1
3
WeightedGraph Minimum Spanning Tree
Algorithms to find Minimum SpanningTrees are:
Kruskal‘s Algorithm
Prim‘s Algorithm

Kruskal’s algorithm
Kruskal’s algorithm finds the minimum cost spanning tree
by selecting the edges one by one as follows.
1. Draw all the vertices of the graph.
2. Select the smallest edge from the graph and add it into
the spanning tree (initially it is empty).
3. Select the next smallest edge and add it into the
spanning tree.
4. Repeat the 2 and 3 steps until that does not form any
cycles.

C
FE
A B
D
5
64
3
4
2
1 2
3
2
C
FE
A B
D
3
2
1 2
2
Minimum Spanning Tree for the abovegraph is:

Walk-Through
Consider an undirected, weight graph
5
1
A
H
B
F
E
D
C
G 3
2
4
6
3
4
3
4
8
4
3
10

Sort the edges by increasing edge weight
edge dv
(D,E) 1
(D,G) 2
(E,G) 3
(C,D) 3
(G,H) 3
(C,F) 3
(B,C) 4
5
1
A
H
B
F
E
D
C
G 3
2
4
6
3
4
3
4
8
4
3
10 edge dv
(B,E) 4
(B,F) 4
(B,H) 4
(A,H) 5
(D,F) 6
(A,B) 8
(A,F) 10

Select first |V|–1 edges which do not
generate a cycle
edge dv
(D,E) 1 
(D,G) 2
(E,G) 3
(C,D) 3
(G,H) 3
(C,F) 3
(B,C) 4
5
1
A
H
B
F
E
D
C
G 3
2
4
6
3
4
3
4
8
4
3
10 edge dv
(B,E) 4
(B,F) 4
(B,H) 4
(A,H) 5
(D,F) 6
(A,B) 8
(A,F) 10

generate a cycle
edge dv
(D,E) 1 
(D,G) 2 
(E,G) 3
(C,D) 3
(G,H) 3
(C,F) 3
(B,C) 4
5
1
A
H
B
F
E
D
C
G 3
2
4
6
3
4
3
4
8
4
3
10 edge dv
(B,E) 4
(B,F) 4
(B,H) 4
(A,H) 5
(D,F) 6
(A,B) 8
(A,F) 10

generate a cycle
edge dv
(D,E) 1 
(D,G) 2 
(E,G) 3 
(C,D) 3
(G,H) 3
(C,F) 3
(B,C) 4
5
1
A
H
B
F
E
D
C
G 3
2
4
6
3
4
3
4
8
4
3
10 edge dv
(B,E) 4
(B,F) 4
(B,H) 4
(A,H) 5
(D,F) 6
(A,B) 8
(A,F) 10
Accepting edge (E,G) would create a cycle

generate a cycle
edge dv
(D,E) 1 
(D,G) 2 
(E,G) 3 
(C,D) 3 
(G,H) 3
(C,F) 3
(B,C) 4
5
1
A
H
B
F
E
D
C
G 3
2
4
6
3
4
3
4
8
4
3
10 edge dv
(B,E) 4
(B,F) 4
(B,H) 4
(A,H) 5
(D,F) 6
(A,B) 8
(A,F) 10

generate a cycle
edge dv
(D,E) 1 
(D,G) 2 
(E,G) 3 
(C,D) 3 
(G,H) 3 
(C,F) 3
(B,C) 4
5
1
A
H
B
F
E
D
C
G 3
2
4
6
3
4
3
4
8
4
3
10 edge dv
(B,E) 4
(B,F) 4
(B,H) 4
(A,H) 5
(D,F) 6
(A,B) 8
(A,F) 10

generate a cycle
edge dv
(D,E) 1 
(D,G) 2 
(E,G) 3 
(C,D) 3 
(G,H) 3 
(C,F) 3 
(B,C) 4
5
1
A
H
B
F
E
D
C
G 3
2
4
6
3
4
3
4
8
4
3
10 edge dv
(B,E) 4
(B,F) 4
(B,H) 4
(A,H) 5
(D,F) 6
(A,B) 8
(A,F) 10

generate a cycle
edge dv
(D,E) 1 
(D,G) 2 
(E,G) 3 
(C,D) 3 
(G,H) 3 
(C,F) 3 
(B,C) 4 
5
1
A
H
B
F
E
D
C
G 3
2
4
6
3
4
3
4
8
4
3
10 edge dv
(B,E) 4
(B,F) 4
(B,H) 4
(A,H) 5
(D,F) 6
(A,B) 8
(A,F) 10

generate a cycle
edge dv
(D,E) 1 
(D,G) 2 
(E,G) 3 
(C,D) 3 
(G,H) 3 
(C,F) 3 
(B,C) 4 
5
1
A
H
B
F
E
D
C
G 3
2
4
6
3
4
3
4
8
4
3
10 edge dv
(B,E) 4 
(B,F) 4
(B,H) 4
(A,H) 5
(D,F) 6
(A,B) 8
(A,F) 10

generate a cycle
edge dv
(D,E) 1 
(D,G) 2 
(E,G) 3 
(C,D) 3 
(G,H) 3 
(C,F) 3 
(B,C) 4 
5
1
A
H
B
F
E
D
C
G 3
2
4
6
3
4
3
4
8
4
3
10 edge dv
(B,E) 4 
(B,F) 4 
(B,H) 4
(A,H) 5
(D,F) 6
(A,B) 8
(A,F) 10

generate a cycle
edge dv
(D,E) 1 
(D,G) 2 
(E,G) 3 
(C,D) 3 
(G,H) 3 
(C,F) 3 
(B,C) 4 
5
1
A
H
B
F
E
D
C
G 3
2
4
6
3
4
3
4
8
4
3
10 edge dv
(B,E) 4 
(B,F) 4 
(B,H) 4 
(A,H) 5
(D,F) 6
(A,B) 8
(A,F) 10

generate a cycle
edge dv
(D,E) 1 
(D,G) 2 
(E,G) 3 
(C,D) 3 
(G,H) 3 
(C,F) 3 
(B,C) 4 
5
1
A
H
B
F
E
D
C
G 3
2
4
6
3
4
3
4
8
4
3
10 edge dv
(B,E) 4 
(B,F) 4 
(B,H) 4 
(A,H) 5 
(D,F) 6
(A,B) 8
(A,F) 10

generate a cycle
edge dv
(D,E) 1 
(D,G) 2 
(E,G) 3 
(C,D) 3 
(G,H) 3 
(C,F) 3 
(B,C) 4 
5
1
A
H
B
F
E
D
C
G
2
3
3
3
edge dv
(B,E) 4 
(B,F) 4 
(B,H) 4 
(A,H) 5 
(D,F) 6
(A,B) 8
(A,F) 10
Done
Total Cost =  dv = 21
4
} not
considere
d

Prim’s algorithm
Prim’s algorithm finds the minimum cost spanning tree by
selecting the edges one by one as follows.
1. All vertices are marked as not visited
2. Any vertex v you like is chosen as starting vertex and is
marked as visited (define a cluster C).
3. The smallest- weighted edge e = (v,u), which connects one
vertex v inside the cluster C with another vertex u outside of C,
is chosen and is added to the MST.
4. The process is repeated until a spanning tree is formed.

Walk-Through
Initialize array
K dv pv
A F  
B F  
C F  
D F  
E F  
F F  
G F  
H F  
4
25
A
H
B
F
E
D
C
G 7
2
10
18
3
4
3
7
8
9
3
10
2

4
25
A
H
B
F
E
D
C
G 7
2
10
18
3
4
3
7
8
9
3
10
Start with any node, say D
K dv pv
A
B
C
D T 0 
E
F
G
H
2

4
25
A
H
B
F
E
D
C
G 7
2
10
18
3
4
3
7
8
9
3
10
Update distances of adjacent,
unselected nodes
K dv pv
A
B
C 3 D
D T 0 
E 25 D
F 18 D
G 2 D
H
2

4
25
A
H
B
F
E
D
C
G 7
2
10
18
3
4
3
7
8
9
3
10
Select node with minimum
distance
K dv pv
A
B
C 3 D
D T 0 
E 25 D
F 18 D
G T 2 D
H
2

4
25
A
H
B
F
E
D
C
G 7
2
10
18
3
4
3
7
8
9
3
10
unselected nodes
K dv pv
A
B
C 3 D
D T 0 
E 7 G
F 18 D
G T 2 D
H 3 G
2

4
25
A
H
B
F
E
D
C
G 7
2
10
18
3
4
3
7
8
9
3
10
distance
K dv pv
A
B
C T 3 D
D T 0 
E 7 G
F 18 D
G T 2 D
H 3 G
2

4
25
A
H
B
F
E
D
C
G 7
2
10
18
3
4
3
7
8
9
3
10
unselected nodes
K dv pv
A
B 4 C
C T 3 D
D T 0 
E 7 G
F 3 C
G T 2 D
H 3 G
2

4
25
A
H
B
F
E
D
C
G 7
2
10
18
3
4
3
7
8
9
3
10
distance
K dv pv
A
B 4 C
C T 3 D
D T 0 
E 7 G
F T 3 C
G T 2 D
H 3 G
2

4
25
A
H
B
F
E
D
C
G 7
2
10
18
3
4
3
7
8
9
3
10
unselected nodes
K dv pv
A 10 F
B 4 C
C T 3 D
D T 0 
E 2 F
F T 3 C
G T 2 D
H 3 G
2

4
25
A
H
B
F
E
D
C
G 7
2
10
18
3
4
3
7
8
9
3
10
distance
K dv pv
A 10 F
B 4 C
C T 3 D
D T 0 
E T 2 F
F T 3 C
G T 2 D
H 3 G
2

4
25
A
H
B
F
E
D
C
G 7
2
10
18
3
4
3
7
8
9
3
10
unselected nodes
K dv pv
A 10 F
B 4 C
C T 3 D
D T 0 
E T 2 F
F T 3 C
G T 2 D
H 3 G
2
Table entries unchanged

4
25
A
H
B
F
E
D
C
G 7
2
10
18
3
4
3
7
8
9
3
10
distance
K dv pv
A 10 F
B 4 C
C T 3 D
D T 0 
E T 2 F
F T 3 C
G T 2 D
H T 3 G
2

4
25
A
H
B
F
E
D
C
G 7
2
10
18
3
4
3
7
8
9
3
10
unselected nodes
K dv pv
A 4 H
B 4 C
C T 3 D
D T 0 
E T 2 F
F T 3 C
G T 2 D
H T 3 G
2

4
25
A
H
B
F
E
D
C
G 7
2
10
18
3
4
3
7
8
9
3
10
distance
K dv pv
A T 4 H
B 4 C
C T 3 D
D T 0 
E T 2 F
F T 3 C
G T 2 D
H T 3 G
2

4
25
A
H
B
F
E
D
C
G 7
2
10
18
3
4
3
7
8
9
3
10
unselected nodes
K dv pv
A T 4 H
B 4 C
C T 3 D
D T 0 
E T 2 F
F T 3 C
G T 2 D
H T 3 G
2
Table entries unchanged

4
25
A
H
B
F
E
D
C
G 7
2
10
18
3
4
3
7
8
9
3
10
distance
K dv pv
A T 4 H
B T 4 C
C T 3 D
D T 0 
E T 2 F
F T 3 C
G T 2 D
H T 3 G
2

4
A
H
B
F
E
D
C
G
2
3
4
3
3
Cost of Minimum Spanning
Tree =  dv = 21
K dv pv
A T 4 H
B T 4 C
C T 3 D
D T 0 
E T 2 F
F T 3 C
G T 2 D
H T 3 G
2
Done

C
FE
A B
D
3
2
1 2
2
C
FE
A B
D
5
64
3
4
2
1 2
3
2
Minimum Spanning Tree for the abovegraph is:
A B C D E F
A - 5 4 6 2 -
B 5 - - 2 - 3
C 4 - - - 3 -
D 6 2 - - 1 2
E 2 - 3 1 - 4
F - 3 - 2 4 -
Adjacency matrix

1
3
4
65
2
6
3
6
5
5
1
6
4 2
5
Exercise:

Unit 4.2(graphs)

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Unit 4.2(graphs)