Graph Theory MCQ Quiz - Objective Question with Answer for Graph Theory - Download Free PDF

Last updated on Jun 4, 2025

Latest Graph Theory MCQ Objective Questions

Graph Theory Question 1:

Which of the following statements is/are TRUE for undirected graphs?

  1.  Sum of degrees of all vertices is odd for special case.
  2.  ​Number of odd degree vertices is even.
  3. Number of even degree vertices is always even.
  4. None of the above

Answer (Detailed Solution Below)

Option 2 :  ​Number of odd degree vertices is even.

Graph Theory Question 1 Detailed Solution

Data:

sum of degree in a graph = dsum

number of edges in a graph  = e

Formula:

By handshaking lemma:

dsum = 2 × e

Calculation

sum of odd degree + sum of even degree = 2× e 

sum of odd degree = 2× e - sum of even degree 

2 × e → is even

sum of even degree  → even

even - even = even

Therefore the sum of odd degrees is even and hence the number of odd degree vertices is even too.

Therefore option 1 is false

Graph Theory Question 2:

The largest number of faces in a simple connected maximal planar graph with 100 vertices is :

  1. 200 
  2. 198 
  3. 196 
  4. 96

Answer (Detailed Solution Below)

Option 3 : 196 

Graph Theory Question 2 Detailed Solution

Concept:

For a simple connected maximal planar graph with \( n \) vertices:

The graph is a triangulation, meaning every face (except possibly the outer one) is a triangle.

In such graphs, the number of edges \( e \) is given by:

\( e = 3n - 6 \)

Using Euler's formula:

\( n - e + f = 2 \)

Given:

\( n = 100 \)

So, number of edges:

\( e = 3(100) - 6 = 294 \)

Now, apply Euler's formula to find faces:

\( f = 2 - n + e = 2 - 100 + 294 = 196 \)

Final Answer:196

Graph Theory Question 3:

Which of the following Graph is/are planer?

qImage67a7495f20e8f5705240ed37

Choose the most appropriate answer from the options given below:

  1. A and C only
  2. B only
  3. A only
  4. A and B only

Answer (Detailed Solution Below)

Option 4 : A and B only

Graph Theory Question 3 Detailed Solution

The correct answer is A and B only.

key-point-image Key Points
  • A planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints.
  • Planar graphs follow Kuratowski's theorem, which states that a graph is planar if and only if it does not contain a subgraph that is a subdivision of K5 (complete graph on five vertices) or K3,3 (complete bipartite graph on six vertices).
  • To determine if a graph is planar, one can try to draw it in such a way that no edges cross each other, except at the vertices.
additional-information-image Additional Information
  • Graphs A and B are determined to be planar by verifying that they can be drawn without any edge crossings.
  • Graph C is not planar because it contains a subgraph that is homeomorphic to K5 or K3,3, which means it cannot be drawn without edge crossings.
  • Planar graphs are useful in various fields such as computer networking, geography, and circuit design, where planar embeddings help to minimize complexity and avoid intersections.
  • Graph theory provides various algorithms to check for planarity, such as the Hopcroft and Tarjan planarity testing algorithm.

Graph Theory Question 4:

Arrange the following graph on the basis of number of edges in increasing order [for n > 3]

A. Kn (Complete Graph)

B. Cn (Cycle graph)

C. Wn (Wheel graph)

D. Kn, n (Complete Bipartite Graphs)

E. Qn (n-cubes graph)

Choose the correct answer from the options given below:

  1. B, C, A, D, E
  2. B, A, C, D, E
  3. A, B, C, E, D
  4. E, D, C, A, B

Answer (Detailed Solution Below)

Option 1 : B, C, A, D, E

Graph Theory Question 4 Detailed Solution

- guacandrollcantina.com

The correct answer is Option 1.

key-point-image Key Points
  • Complete Graph (Kn): Has n(n−1)/2 edges.
  • Cycle Graph (Cn): Has n edges.
  • Wheel Graph (Wn): Has 2(n−1) edges.
  • Complete Bipartite Graph (Kn,n): Has n2 edges.
  • n-Cube Graph (Qn): Has n × 2n−1 edges.
additional-information-image Additional Information
  • Let us verify using n = 5:
Graph Formula Edges (n = 5)
Cn (Cycle) n 5
Kn (Complete) n(n−1)/2 10
Wn (Wheel) 2(n−1) 8
Kn,n (Complete Bipartite) 25
Qn (n-Cube) n × 2n−1 80
  • Now arranging based on increasing number of edges:
  • Cycle (5) < Wheel (8) < Complete (10) < Complete Bipartite (25) < n-Cube (80)
  • So, the correct answer is: B,C,A,D,E.

Graph Theory Question 5:

Match List I with List II

LIST - I

LIST - II

A.

Planar Graph

I.

Probabilistic Model

B.

Bipartite Graph

II.

Deterministic Model

C.

PERT

III.

4-Colorable

D.

CPM

IV.

2-Colorable


Choose the correct answer from the options given below:

  1. A - IV; B - III; C - I; D - II
  2. A - III; B - IV; C - II; D - I
  3. A - II; B - IV; C - I; D - III
  4. A - III; B - IV; C - I; D - II
  5. A - III; B - II; C - I; D - IV

Answer (Detailed Solution Below)

Option 4 : A - III; B - IV; C - I; D - II

Graph Theory Question 5 Detailed Solution

The correct answer is A - III; B - IV; C - I; D - II

Key Points

  • A. Planar Graph - III. 4-Colorable.
    • The four-color theorem states that any map in a plane can be colored using four-colors in such a way that regions sharing a common boundary (other than a single point) do not share the same color.
    • F1 Savita ENG 22-12-23 D 16
  • B. Bipartite Graph - IV. 2-Colorable.
    • A bipartite graph can be divided into two sets of vertices such that all edges connect a vertex in one set to a vertex in the other set. Therefore, they are 2-colorable.
    • F1 Savita ENG 22-12-23 D 17
  • C. PERT (Program Evaluation and Review Technique) - I. Probabilistic Model.
    • PERT is a statistical tool, used in project management, designed to analyze and represent the tasks involved in completing a given project. It uses probabilistic time estimates.
  • D. CPM (Critical Path Method) - II. Deterministic Model.
    • CPM is a method used in project management for planning and scheduling the different steps in the project. It's a deterministic model because it assumes a fixed time duration for each task.

Top Graph Theory MCQ Objective Questions

Which of the given frequency polygons represents the following frequency distribution?

Class

4 - 8

8 - 12

12 – 16

16 - 20

20 – 24

Frequency

2

4

5

3

2

  1. F1 SSC Arbaz 1-3-24 D1
  2. F1 SSC Arbaz 1-3-24 D2
  3. F1 SSC Arbaz 1-3-24 D3
  4. F1 SSC Arbaz 1-3-24 D4

Answer (Detailed Solution Below)

Option 1 : F1 SSC Arbaz 1-3-24 D1

Graph Theory Question 6 Detailed Solution

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Option 1 is the correct option as the coordinates plotted in the graph are same as the values mentioned in the table.

The number of vertices in polyhedron which has 30 edges on 12 faces is

  1. 12
  2. 15
  3. 20
  4. 24

Answer (Detailed Solution Below)

Option 3 : 20

Graph Theory Question 7 Detailed Solution

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Given:

The edges of the polyhedron = 30

The faces of the polyhedron = 12

Formula used:

Euler's formula for the polyhedron

F + V = E + 2      (Where F = The face of the polyhedron, V = The vertices of the polyhedron, and E = The edge of the polyhedron)

Calculation:

Let be assume the vertices of the polyhedron is V

⇒ 12 + V = 30 + 2

⇒ V = 32 - 12 = 20

∴ The required result will be 20.

Let G be a connected planar graph with 10 vertices. If the number of edges on each face is three, then the number of edges in G is ________. 

Answer (Detailed Solution Below) 24

Graph Theory Question 8 Detailed Solution

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Data:

number of faces = |F|

number of vertices  = |V| = 10

edges covering each face = 3

Formula:

According to Euler’s formula : 

|V| - |E|  + |F| = 2

Calculation

edges on each face is three

∴ 2 |E| = 3 |F|       (every edge is shared by 2 faces)

\(|F| = \frac{2}{3} |E|\)

\(10 - E + \frac{2}{3}|E|=2\)

|E| = 24

the number of edges in is 24

The following pie chart shows the population of 5 states of India in 2010. What is the population of Rajasthan if the total population of these 5 states is 16200000?

F3 Shubhav.V Madhulisha 08-04-21 G1

  1. 4725000
  2. 540000
  3. 8100000 
  4. 810000 

Answer (Detailed Solution Below)

Option 3 : 8100000 

Graph Theory Question 9 Detailed Solution

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Given:

Population in states in 2010 depicted in a pie- chart.

Total population = 16200000

Calculation:

Rajasthan has 180° of population.

So, 16200000 × ( 180 / 360 ) 

= 8100000

∴ Rajasthan's population in year 2010 is 8100000

Let G = (V, E) be a directed graph where V is the set of vertices and E the set of edges. Then which one of the following graphs has the same strongly connected components as G?

  1. G1 = (V, E1) where E1 = {(u, v)|(u, v) ∉ E}
  2. G2 = (V, E2) where E2 = {(u, v)|(v, u) ∈ E}
  3. G3 = (V, E3) where E3 = {(u, v)| there is a path of length ≤ 2 from u to v in E}
  4. G4 = (V4, E) where V4 is the set of vertices in G which are not isolated

Answer (Detailed Solution Below)

Option 2 : G2 = (V, E2) where E2 = {(u, v)|(v, u) ∈ E}

Graph Theory Question 10 Detailed Solution

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In a directed graph G Strongly connected will have a path from each vertex to every other vertex.

If the direction of the edges is reverse, then also graph is strongly connected components as G

Option 2: G2 = (V, E2) where E2 = {(u, v)|(v, u) ∈ E}

In this option G2, edges are reversed and hence it is a strongly connected components as similar to G

So, changing the direction of all the edges, won't change the SCC.

Graph G is obtained by adding vertex s to K3,4 and making s adjacent to every vertex of K3,4. The minimum number of colours required to edge-colour G is ______.

Answer (Detailed Solution Below) 7

Graph Theory Question 11 Detailed Solution

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Edge coloring for a graph refers to assigning colors to the graph edges in a way that no two incident edges have same color.

A K3,4 graph looks like

F1 R.S Madhu 13.05.20 D2

To edge color this graph properly, we will consider every edge from vertex set of max(u, v). In our case, max(3, 4) will be 4.

F1 R.S Madhu 13.05.20 D3

Hence from u1, 4 edges of different colors (Red, Black, blue, Green) are incident on v1, v2, v3, v4.

Similarly, 4 edges of different colors (Red, Green, Black, Blue) are incident on v1, v2, v3, v4.

Now adding a new vertex s which is adjacent to all the vertices will require 7 different colors of edges. But we can still use Red, Green, Brown, Blue and 3 other colors.

Hence, maximum colors required for edge-coloring in a planar graph is directly related to the maximum degree of the graph, therefore, MAX (3, 4, 7) = 7

Important Point:

The question is asking about edge coloring and not about vertex coloring.

Let G be an undirected complete graph on n vertices, where n > 2. Then, the number of different Hamiltonian cycles in G is equal to

  1. n!
  2. (n - 1)!
  3. 1
  4. \(\frac{{\left( {n - 1} \right)!}}{2}\)

Answer (Detailed Solution Below)

Option 4 : \(\frac{{\left( {n - 1} \right)!}}{2}\)

Graph Theory Question 12 Detailed Solution

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A simple circuit in a graph G that passes through every vertex exactly once is called a Hamiltonian circuit.

In Hamiltonian cycle is a Hamiltonian circuit in which initial and final vertex are same.

In an undirected complete graph on n vertices, there are n permutations are possible to visit every node. But from these permutations, there are: n different places (i.e., nodes) you can start; two (clockwise or anticlockwise) different directions you can travel. So, any one of these n! cycles is in a set of 2n cycles which all contain the same set of edges.

So, there are = \(\frac{{n!}}{{2n}} = \frac{{\left( {n - 1} \right)!}}{2}\) distinct Hamilton cycles.

In a connected graph, a bridge is an edge whose removal disconnects a graph. Which one of the following statements is true?

  1. A tree has no bridges
  2. A bridge cannot be part of a simple cycle
  3. Every edge of a clique with size ≥ 3 is a bridge (A clique is any complete sub graph of a graph)
  4. A graph with bridges cannot have a cycle

Answer (Detailed Solution Below)

Option 2 : A bridge cannot be part of a simple cycle

Graph Theory Question 13 Detailed Solution

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(A) FALSE:

e.g.

The only edge in the above tree is bridge.

(B) TRUE:

If an edge is the part of the cycle than its removal will not disconnect the graph.

(C) FALSE:

e.g. y28

Here no edge of the clique is a bridge

(D) FALSE:

e.g. y29

In network topology, the property between two graphs so that both have got same Incidence matrix is known as:

  1. Isomorphism
  2. Tree
  3. Polymorphism
  4. Tree Compliment

Answer (Detailed Solution Below)

Option 1 : Isomorphism

Graph Theory Question 14 Detailed Solution

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Concept:-

Isomorphic graphs:-

If two graphs have some numbers of nodes and the some numbers of branches and they look different but have some incidence matrix called isomorphic graphs

F1 Tapesh 20.5.21 Pallavi D1

F1 Tapesh 2.6.21 Pallavi D1

This is the same incidence matrix for both the graphs

Additional Information

Tree:- Tree is a connected subgraph of a given graph, which contains all the nodes of a graph, but then should in no any loop in the subgraph called a tree.

F1 Tapesh 20.5.21 Pallavi D2

In an undirected connected planar graph G, there are eight vertices and five faces. The number of edges in G is ______

Answer (Detailed Solution Below) 11

Graph Theory Question 15 Detailed Solution

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Data

Number of vertices = V = 8

Number of faces = 5

Number of edges = E

Euler's formula:

V + F = E + 2

Calculation

⇒ E = V + F - 2

⇒ E = 8 + 5 - 2

⇒ E = 11 

∴ Number of edges is 11

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