CORE – 7: Discrete Mathematical Structures – Unit-3

Definition of Graph − A graph (denoted as G = (V, E)) consists of a non-empty

set of vertices V and a set of edges E.

Example –Let us consider, a Graph

G=(V,E) where V={a,b,c,d} and
E={{a,b},{a,c},{a,d},{b,b},{b,c},{b,d}}

Degree of a Vertex − The degree of a vertex is the

number of edges incident on a vertex v. The self-loop is
counted twice. The degree of a vertex is denoted by
d(v).
Here, d(a)=3; d(b)=5; d(c) = 2; d(d)=2.

Degree of a Graph − the degree of a graph is the largest vertex degree of that
graph. For the above graph the degree of the graph is 5.

Pendant Vertex: A vertex with degree one is called a Pendant Vertex.

Pendant Edge: The only edge which is an incident with a pendant vertex is called
the Pendant Edge.

Odd Vertex: A vertex having degree odd is called an odd vertex.

Even Vertex: A vertex having a degree even is called an even vertex.

Adjacent Vertices: Two vertices are called adjacent if an edge links them. If there
is an edge (u, v), then we can say vertex u is adjacent to vertex v, and vertex v is
adjacent to vertex u.
Example: Consider the graph as shown
in fig:

Determine the following:

1. Pendant Vertices
2. Pendant Edges
3. Odd vertices
4. Even Vertices

Solution:
1. The vertex V5 is the pendant vertex.
2. The edge (V4,V5) or e5 is the pendant edge.
3. The vertices V3 and V5 are odd vertices.
4. The vertices V1, V2,and V4 are even vertices.

Manas Ku Mishra, Asst. Prof. of Comp. Sc., FM (A) College, BLS. Page 1 of 16
Cut Set: Consider a graph G = (V, E). A cut set for G is the smallest set of edges
such that the removal of the set disconnected the graph whereas the removal of any
proper subset of this set left a connected subgraph.

Cut Points or Cut Vertices: Consider a graph G = (V, E). A cut point for a
graph G is a vertex v such that G-v has more connected components than G. The
subgraph G-v is obtained by deleting the vertex v from graph G and also deleting
the entire edges incident on v.

Bridge (Cut Edges): Consider a graph G = (V, E). A bridge for a graph G, is an
edge e such that G-e has more connected components than G.

Example-1: Consider the graph shown in

the fig. Determine the cut set, cut vertices
and cut edges of the graph.

Solution: For this graph, the edge set

{(V1, V5), (V7, V5)} is a cut set.

V5 is a cut vertex of the graph.

There is no cut edge in this graph.

Example-2: Consider the graph shown in the fig.

Determine the cut set, cut vertices and cut edges of the
graph.
Solution: For this graph, the edge set {(c,d)} is a cut set.
c, d are the cut vertices of the graph.
The edge (c, d) is the cut edge in this graph.

The Handshaking Lemma − in a graph, the sum of all the degrees of all the
vertices is equal to twice the number of edges.
Types of Graphs:
1. Null Graph: A null graph has no edges or in
other words a graph having only isolated
vertex.

2. Undirected Graphs: An Undirected graph G consists of

a set of vertices, V and a set of edge E. The edge set
contains the unordered pair of vertices. If (u, v) ∈ E, then
we say u and v are connected by an edge where u and v
are vertices in the set V.

Manas Ku Mishra, Asst. Prof. of Comp. Sc., FM (A) College, BLS. Page 2 of 16
3. Multigraph: If in a graph multiple edges between the
same set of vertices are allowed, it is known as
Multigraph. In other words, it is a graph having at least one
loop or multiple edges.

4. Directed Graphs: A directed graph or digraph G is

defined as an unordered pair (V, E), where V is the set
of points called vertices and E is the set of edges. Each
edge in the graph G is assigned a direction and is
identified with an ordered pair (u, v), where u is the initial
vertex, and v is the end vertex. For a directed graph, the
number of edges coming into a vetex is called the
indegree of the vertex and the number of edges coming
out of a vertex is called as outdegree of the vertex. Here
indegree(3) = 2 and outdegree(3) = 0.

5. Complete Graph: A complete graph G = (V, E) of n

vertices is a graph in which each vertex is connected to
every other vertex i.e., an edge exist between every
pair of distinct vertices. It is denoted by Kn. A
complete graph with n vertices will have n(n-1)/2 edges.

6. Regular Graph: A graph G is said to be regular, if all its

vertices have the same degree. In a graph, if the degree
of each vertex is ‘k’, then the graph is called a ‘k-regular
graph’. In this fig. the graph is 2-regular.

7. Cycle Graph: A simple graph with ‘n’ vertices (n >= 3)

and ‘n’ edges is called a cycle graph if all its edges form a
cycle of length ‘n’ and is denoted as Cn.
Each vertex of a cycle graph has degree 2. Hence it is also a 2-regular graph. A
graph with no cycles is called an acyclic graph.

Manas Ku Mishra, Asst. Prof. of Comp. Sc., FM (A) College, BLS. Page 3 of 16
8. Wheel Graph: A wheel graph is obtained from a cycle
graph Cn by adding a new vertex. That new vertex is called
a Hub which is connected to all the vertices of Cn. A wheel
graph is denoted as Wn where n is the number of edges
coming out of the Hub. Here is an example of W4 in this
fig.

9. Bipartite Graph: A simple graph G = (V, E) with

vertex partition V = {V1, V2} is called a bipartite graph if
every edge of E joins a vertex in V1 to a vertex in V2.
In general, a Bipertite graph has two sets of vertices, let
us say, V1 and V2, and if an edge is drawn, it should
connect any vertex in set V1 to any vertex in set V2 (no
edge in G connects either two vertices in V1 or two
vertices in V2.

10. Complete Bipartite Graph: A bipartite graph

G = (V, E) with partition V = {V1, V2} is said to be a
complete bipartite graph if every vertex in V1 is
connected to every vertex of V2 and is denoted by
Km, n where m is number of vertices in V1 and n is
number of vertices in V2.
Here is an example of K3, 3

11. Star Graph: A complete bipartite graph of the form K1, n-1 is a
star graph with n-vertices. A star graph is a complete bipartite
graph if a single vertex belongs to one set and all the remaining
vertices belong to the other set. Here is an example of K1, 4

12. Connected Graph: A graph is called connected

if there is a path from any vertex u to v.

Manas Ku Mishra, Asst. Prof. of Comp. Sc., FM (A) College, BLS. Page 4 of 16
13. Disconnected Graph: A graph is called
disconnected if there is no path between any two of its
vertices.

14. Connected Component: A subgraph of graph G is called the connected

component of G, if it is not contained in any bigger subgraph of G, which is
connected. It is defined by listing its vertices.
Example: In the above disconnected graph the connected components are {a,b}
and {c,d}.

15. Complementary Graph:

Complement
omplement or inverse of
a graph G is a graph H on the
same vertices such that two
distinct vertices of H are
adjacent if and only if they are
not adjacent in G.

(Graph G) (Complement of Graph G)

16. Labeled Graphs: A graph

G = (V, E) is called a labeled
graph if its edges
dges are labeled
with some name.. So, we can
write these labels in place of
an ordered pair in its edges
set.

17. Weighted Graphs: A graph G=(V, E) is called a

weighted graph if each edge of graph G is assigned a
positive number w called the weight of the edge e.
Representation of Graphs: There are two principal
cipal ways to represent a graph.
1. Adjacency matrix representation
2. Incidence matrix representation
(a) Representation of Undirected Graph:
1. Adjacency Matrix Representation
Representation: If an Undirected Graph G consists of n
vertices then the adjacency matrix of the graph is an n x n matrix A = [aij] and
defined by

1, if (vi,vj) is an edge i.e. vi is adjacent to vj

aij =
0, if there is no edge between vi and vj
If there exists an edge between vertex vi and vj, where i is a row and j is a column
then the value of aij=1.
If there is no edge between vertex vi and vj, then value of aij=0.
Example: Find the adjacency
matrix MA of graph G shown
in Fig:

Solution: Since graph G

consist of four vertices.
Therefore, the adjacency
matrix wills a 4 x 4 matrix.
The adjacency matrix is as
follows in fig:
2. Incidence Matrix Representation:
Representation If an Undirected Graph G consists of n
vertices and m edges, then
en the incidence matrix is an n x m matrix C = [cij] and is
defined by
1, if the vertex vi incident by ej
cij =
0, otherwise
There is a row for every vertex and a column for every edge in the incident matrix.
The number of ones in an incidence matrix of the undirected graph (without loops) is
equal to the sum of the degrees of all the vertices in a graph. It is also equal to twice
the number of edges.
Example: Consider
the undirected graph
G as shown in fig.
Find its incidence
matrix MI.
Solution: The
undirected graph
consists of four
vertices and five
edges.
Therefore, the incidence
dence matrix is an 4 x 5 matrix.
(b)Representation of Directed Graph:
1. Adjacency Matrix Representation
Representation: If a directed graph G consists of n vertices
then the adjacency matrix of a graph is an n x n matrix A = [aij] and defined by
1, if (vi,vj) is an edge i.e. vi is initial vertex and vj is final vertex
aij =
0, if there is no edge between vi and vj
If there exists an edge between vertex Vi and Vj, with Vi as initial vertex and Vj as a
final vertex, then the value of aij=1.
If there is no edge between vertex Vi and Vj, then the value of aij=0.
The number of ones in the adjacency matrix of a directed graph is equal to the
number of edges.
Example: Consider the
directed graph shown
in fig. Determine its
adjacency matrix MA.

Solution: Since the

directed graph G
consists of five
vertices. Therefore, the
adjacency matrix will
be a 5 x 5 matrix as
shown in th fig.
2. Incidence Matrix Representation: If a directed graph G consists of n vertices
and m edges, then the incidence matrix is an n x m matrix C = [cij] and defined by
1, vi is the initial vertex of edge ei
aij = -1,
1, vi is the final vertex of edge ei
0, if vi is not incident on ei
The number of ones in an incidence matrix is equal to the number of edges in the
graph.
Example: Consider
the directed graph G
as shown in fig. Find
its incidence matrix MI.
Solution: The
directed graph
consists of four
vertices and five
edges.
Therefore, the incidence matrix is a 4 x 5 matrix as shown in the fig.
(c) Representation of Multigraph (Represented
Represented only by adjacency matrix
representation)
(i) Adjacency matrix representation of multigraph: If a multigraph G consists of
vertices, then the adjacency matrix of graph is an n x n matrix A = [aij] and is
defined by
aij = N, where N is the number of edges between Vi and Vj
aij = 0 If there is no edge between vi and vj.
Example: V1 V2 V3 V4 V5
Consider the multigraph
shown in Fig, V1 0 3 0 1 0
Determine its adjacency
matrix. V2 3 0 0 0 2

Solution: V3 0 0 0 1 1
Since the multigraph
consists of five vertices, V4 1 0 1 1 0
therefore the adjacency
V5 0 2 1 0 1
matrix will be a 5 x 5
matrix.
The adjacency matrix of the multigraph is as shown in the above figure.
Isomorphism:
Graphs G(V, E) and G*
(V*,E*) are said to be
isomorphic if there
exists one to one
correspondence i.e.
f:V→V* such that {u, v}
is an edge of G if and
only if {f(u), f(v)} is an
edge of G* i.e. if u,v are
adjacent then f(u) and
f(v) are also adjacent. (In In the above figure graph G and graph G' are iosmerphic
So, f must be a bijective graphs.
graphs.)
function.
Euler paths and circuits:
 An Euler path is a path that uses every edge of a graph exactly once.
 An Euler circuit is a circuit that uses every edge of a graph exactly once.
 An Euler path starts and ends at different vertices.
 An Euler circuit starts and ends at the same vertex.
Theorem – “A connected multigraph (and simple graph) with at least two vertices
has a Euler circuit if and only if each of its vertices has an even degree.”
degree
Proof of the above statement is that every time a circuit passes through a vertex, it
adds twice to its degree. Since it is a circuit, it starts and ends at the same vertex,
which makes it contribute one degree when the circuit starts and one when it ends.
In this way, every vertex has an even degree.
Theorem – “A connected multigraph (and simple graph) has an Euler path but not
an Euler circuit if and only if it has exactly two vertices of odd degree.”
Proof: Since a path may start and end at different vertices, the vertices where the
path starts and ends are allowed to have odd degrees.
Example – Which graphs
shown in the fig. have an Euler
path or Euler circuit?
Solution – G1 has two vertices
of odd degree a and d and the
rest of them have even degree.
So this graph has an Euler path
but not an Euler circuit.
The path starts and ends at the vertices of odd degree. The path is
is- a,c,d,a,b,d.G2
has four vertices all of even degree, so it has a Euler circuit. The circuit is-
a,d,b,a,c,d,a.
Hamiltonian paths and circuits:
Hamilonian Path – A simple path in a graph that passes through every vertex
exactly once is called a Hamiltonian path.
Hamilonian Circuit – A simple circuit in a graph that passes through every
vertex exactly once is called a Hamiltonian circuit.
Unlike Euler paths and circuits, there are no simple necessary and sufficient criteria
to determine if there are any Hamiltonian paths or circuits in a graph. But there are
certain criteria which rule out the existence of a Hamiltonian circuit in a graph, such
as- if there is a vertex of degree one in a graph then it is impossible for it to have a
Hamiltonian circuit. There are certain theorems which give sufficient but not
necessary conditions for the existence of Hamiltonian graphs.
Example-1: Does the following Example-2: Does a Hamiltonian path or
graph have a Hamiltonian Circuit? circuit exist on the graph below?

Solution- We can see that once we travel to

vertex E there is no way to leave without
Solution- No the above graph does returning to C, so there is no possibility of a
not have a Hamiltonian circuit as Hamiltonian circuit. If we start at vertex E we
there are two vertices with degree can find several Hamiltonian paths, such as
one in the graph. ECDAB and ECABD
Trees: a tree is an undirected,
connected and acyclic graph.
In other words, a connected
graph that does not contain any
cycle is called a tree.
Binary Trees: If the outdegree
of every node is less than or
equal to 2, in a directed tree
than the tree is called a binary
tree.
Basic Terminology:
Root: A binary tree has a unique
node called the root of the tree.
Left Child: The node to the left of the root is called its left child.
Right Child: The node to the right of the root is called its right child.
Parent: A node having a left child or right child or both are called the parent of the
nodes.
Siblings: Two nodes having the same parent are called siblings.
Leaf: A node with no children is called a leaf.
Descendant: A node is called descendant of another node if it is the child of the
node or child of some other descendant of that node. All the nodes in
the tree are descendants of the root.
Left Subtree: The subtree whose root is the left child of some node is called the left
subtree of that node.
Right Subtree: The subtree whose root is the right child of some node is called the
right subtree of that node.
Level of a Node: The level of a node is its distance from the root. The level of root
is defined as zero. The level of all other nodes is one more than its
parent node. The maximum number of nodes at any level N is 2N.
Depth or Height of a tree: The depth or height of a tree is defined as the maximum
number of nodes in a branch of a tree. This is more than the maximum
level of the tree, i.e., the depth of root is one. The maximum number of
nodes in a binary tree of depth d is 2d+1 -1.
External Nodes: The nodes which have no children are called external nodes or
terminal nodes.
Internal Nodes: The nodes which have one or more children are called internal
nodes or non-terminal nodes.

Manas Ku Mishra, Asst. Prof. of Comp. Sc., FM (A) College, BLS. Page 10 of 16
Traversing Binary Trees: Example: Determine the preorder,
Traversing means visiting each node postorder and inorder traversal of the
of the tree exactly once. There are following binary tree.
three traversal methods of binary
trees.
1. Preorder Traversal
2. Postorder Traversal
3. Inorder Traversal

1. Preorder Traversal: The preorder

traversal of a binary tree is a recursive
process. The preorder traversal of a
tree is
 Visit the root of the tree.
 Traverse the left subtree in
preorder.
 Traverse the right subtree in
preorder.

2. Postorder Traversal: The postorder traversal of a binary tree is a recursive

process. The postorder traversal of a tree is
 Traverse the left subtree in postorder.
 Traverse the right subtree in postorder.
 Visit the root of the tree.

3. Inorder Traversal: The inorder traversal of a binary tree is a recursive process.

The inorder traversal of a tree is
 Traverse the left subtree in inorder.
 Visit the root of the tree.
 Traverse the right subtree in inorder.
Solution:
Preorder 1 2 3 4 5 6 7 8 9 10 11
Postorder 3 5 4 2 7 10 9 11 8 6 1
Inorder 3 2 5 4 1 7 6 9 10 8 11

Manas Ku Mishra, Asst. Prof. of Comp. Sc., FM (A) College, BLS. Page 11 of 16
Spanning Tree: A spanning tree is a subset of Connected Graph G, which has all
the vertices covered that does not have cycles. By this definition, we can draw a
conclusion that every connected and undirected Graph G has at least one spanning
tree.
Minimum Spanning Tree: Suppose G is a connected weight graph i.e., each edge
of G is assigned a non-negative number called the weight of edge, then any
spanning tree T of G is assigned a total weight obtained by adding weights of the all
the edges in T. A minimum spanning tree of G is a tree whose total weight is as
small as possible.
Kruskal's Algorithm to find a minimum spanning tree:
1. Input the given connected weighted graph G with n vertices whose minimum
spanning tree T, we want to find.
2. Order all the edges of the graph G according to increasing weights.
3. Initialize T with all the vertices of G.
4. Go on adding edges of the graphs G in T which does not form a cycle from the
ordered list in step-2 until n-1 edges are added, where n is number of vertices
in the graph G.
Example: Determine the minimum spanning tree of the weighted graph shown in
the following figure.
Solution:
Edges Weights Added or Not
(B, E) 2 Added
(C, D) 3 Added
(A, D) 4 Added
(C, F) 4 Added
(B, C) 5 Added
(E, F) 5 Not added
(A, B) 6 Not added
(D, E) 6 Not added
(A, F) 7 Not added
Using kruskal's algorithm arrange all the edges of the weighted graph in increasing
order and initialize spanning tree T with all the six vertices of G. Now start adding
the edges of G in T which do not form a cycle and having minimum weights until five
edges are not added as there are six vertices.

Manas Ku Mishra, Asst. Prof. of Comp. Sc., FM (A) College, BLS. Page 12 of 16
 Step1: Step2: Step3:

Step4: Step5:

Step6: Edge (A, B), (D, E) and (E, F) are discarded

because they will form the cycle in a graph.

So, the minimum spanning tree form in step 5 is

output, and the total cost is 18.

Graph Traversal Algorithm: Traversing the graph means examining all the
nodes or vertices of the graph. There are two standard graph traversal methods:
 Breadth First Search
 Depth First Search

Breadth First Search (BFS) Algorithm: Breadth first search is a graph

traversal algorithm that starts traversing the graph from any node and explores all
the neighbouring nodes until the goal is reached. In each iteration, it selects the
nearest unexplored node. The algorithm follows the same process for each of the
nearest node until it finds the goal. The algorithm ensures that each node is visited
exactly once and no node is visited twice. It uses the data structure QUEUE.
Algorithm
 Step 1: SET STATUS = 1 (ready state) for each node in G
 Step 2: Enqueue the starting node A and set its STATUS = 2 (waiting state)
 Step 3: Repeat Steps 4 and 5 until QUEUE is empty or the goal is reached

Manas Ku Mishra, Asst. Prof. of Comp. Sc., FM (A) College, BLS. Page 13 of 16
  Step 4: Dequeue a node N. Process it and set its STATUS = 3 (processed
state).
 Step 5: Enqueue all the neighbours of N that are in the ready state (whose
STATUS = 1) and set their STATUS = 2 (waiting state)
[END OF LOOP]
 Step 6: EXIT

Example: Consider the graph G shown in the following fig. calculate the
minimum path p from node A to node E by using BFS algorithm. Given that each
edge has a length of 1.

Solution:
The algorithm uses two queues, namely QUEUE1 and QUEUE2. QUEUE1 holds all
the nodes that are to be processed while QUEUE2 holds all the nodes that are
processed and deleted from QUEUE1.
Let’s start examining the graph from Node A.
1. Add A to QUEUE1 and NULL to QUEUE2.
So, QUEUE1 = {A} QUEUE2 = {NULL}
2. Delete the Node A from QUEUE1 and insert all its neighbours. Insert Node A
into QUEUE2
So, QUEUE1 = {B, D} QUEUE2 = {A}
3. Delete the node B from QUEUE1 and insert all its neighbours. Insert node B
into QUEUE2.
So, QUEUE1 = {D, C, F} QUEUE2 = {A, B}
4. Delete the node D from QUEUE1 and insert all its neighbours. Since F is the
only neighbour of it which has been inserted, we will not insert it again. Insert
node D into QUEUE2.
So, QUEUE1 = {C, F} QUEUE2 = { A, B, D}
5. Delete the node C from QUEUE1 and insert all its neighbours. Add node C to
QUEUE2.
So, QUEUE1 = {F, E, G} QUEUE2 = {A, B, D, C}

Manas Ku Mishra, Asst. Prof. of Comp. Sc., FM (A) College, BLS. Page 14 of 16
 6. Remove F from QUEUE1 and add all its neighbours. Since all of its
neighbours have already been added, we will not add them again. Add node F
to QUEUE2.
So, QUEUE1 = {E, G} QUEUE2 = {A, B, D, C, F}
7. Remove E from QUEUE1, all of E's neighbours has already been added to
QUEUE1 therefore we will not add them again. All the nodes are visited and
the target node i.e. E is encountered into QUEUE2.
So, QUEUE1 = {G} QUEUE2 = {A, B, D, C, F, E}
Now, backtrack from E to A, using the nodes available in QUEUE2.
The minimum path will be A → B → C → E.
Depth First Search (DFS) Algorithm: Depth first search (DFS) algorithm
starts with the initial node of the graph G, and then goes to deeper and deeper until
we find the goal node or the node which has no children. The algorithm, then
backtracks from the dead end towards the most recent node that is yet to be
completely unexplored. It uses the data structure STACK.
Algorithm
 Step 1: SET STATUS = 1 (ready state) for each node in G
 Step 2: Push the starting node A on the stack and set its STATUS = 2 (waiting
state)
 Step 3: Repeat Steps 4 and 5 until STACK is empty or the goal is reached
 Step 4: Pop the top node N. Process it and set its STATUS = 3 (processed
state)
 Step 5: Push on the stack all the neighbours of N that are in the ready state
(whose STATUS = 1) and set their STATUS = 2 (waiting state)
[END OF LOOP]
 Step 6: EXIT
Example: Consider the graph G along with its adjacency list, given in the figure
below. Calculate the order to print all the nodes of the graph starting from node H,
by using depth first search (DFS) algorithm.

Manas Ku Mishra, Asst. Prof. of Comp. Sc., FM (A) College, BLS. Page 15 of 16
Solution:
1. Push H onto the stack, STACK : H
2. POP the top element of the stack i.e. H, print it and push all the neighbours
of H onto the stack that are is ready state, Print H STACK: A
3. Pop the top element of the stack i.e. A, print it and push all the neighbours
of A onto the stack that are in ready state, Print A Stack: B, D
4. Pop the top element of the stack i.e. D, print it and push all the neighbours
of D onto the stack that are in ready state, Print D Stack: B, F
5. Pop the top element of the stack i.e. F, print it and push all the neighbours of
F onto the stack that are in ready state, Print F Stack: B
6. Pop the top of the stack i.e. B and push all the neighbours,
Print B Stack: C
7. Pop the top of the stack i.e. C and push all the neighbours,
Print C Stack: E, G
8. Pop the top of the stack i.e. G and push all its neighbours,
Print G Stack: E
9. Pop the top of the stack i.e. E and push all its neighbours,
Print E Stack: ∅
Hence, the stack now becomes empty and all the nodes of the graph have been
traversed. So, the printing sequence of the graph will be:
H→A→D→F→B→C→G→E

Manas Ku Mishra, Asst. Prof. of Comp. Sc., FM (A) College, BLS. Page 16 of 16