Graph Theory MCQs

Are you diving into graph theory and need a solid resource?

You’re in the right place! We’ve compiled over 530 carefully selected MCQs covering all the crucial topics in graph theory. Whether you’re preparing for exams, brushing up for interviews, or just aiming to deepen your understanding, our extensive question bank is here to help you succeed.

Why Choose Our Graph Theory MCQs?

Our MCQs cover the full spectrum of graph theory, from basic concepts like trees and connectivity to advanced topics like network flows and graph algorithms. Each question is crafted to challenge your knowledge and ensure you’re ready for any academic or professional challenge.

Perfect for Students and Enthusiasts

This collection is ideal for anyone studying graph theory, making it a fantastic resource for students, researchers, and math enthusiasts alike. The questions align with common curricula, ensuring you’re always on the right track.

Syllabus Overview

Unit I: Introduction

• Brief History of Graph Theory
• Applications of Graph Theory
• Finite and Infinite Graphs
• Matrix Representations, Degree
• Operations on Graphs: Isolated Vertex, Pendant Vertex, Null Graph

Unit II: Trees, Connectivity and Planarity

• Spanning Trees
• Fundamental Circuits
• Spanning Trees in a Weighted Graph
• Cut Sets: Definition, Properties, Types of Cut Sets, Fundamental Circuits and Cut Sets
• Connectivity and Separability
• Network Flows
• Isomorphism: 1-Isomorphism and 2-Isomorphism
• Combinatorial and Geometric Graphs

Unit III: Matrices, Colouring and Directed Graphs

• Chromatic Number
• Chromatic Partitioning
• Chromatic Polynomial
• Matchings
• Coverings
• Four Color Problem
• Directed Graphs
• Types of Directed Graphs
• Digraphs and Binary Relations
• Directed Paths, Directed Circuits

Unit IV: Permutations and Combinations

• Fundamental Principles of Counting, Permutations and Combinations
• Binomial Theorem
• Combinations with Repetition
• Combinatorial Numbers
• Principle of Inclusion and Principle of Exclusion
• Derangements, Arrangements with Forbidden Positions

Unit V: Generating Functions

• Generating Functions
• Partitions of Integers
• Exponential Generating Functions
• Summation Operator
• Recurrence Relations: First Order and Second Order, Non-homogeneous Recurrence Relations
• Method of Generating Functions

Question: 1

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A minimum spanning tree (MST) of a connected, weighted graph G is:

1. Any subgraph of G that connects all nodes with the least number of edges.
2. A subgraph of G that connects all nodes with the minimum total edge weight.
3. A subgraph of G that contains all edges from the original graph.
4. A subgraph of G that only contains cycles.

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Correct Answer: B. A subgraph of G that connects all nodes with the minimum total edge weight.

Explanation: A minimum spanning tree (MST) is a subset of edges from the original graph that connects all nodes exactly once while minimizing the total weight of the edges in the tree. Option A focuses only on the number of edges, not their weight. Option C includes all edges, which might not be minimal. Option D describes a subgraph with only cycles, which is not a characteristic of MSTs.

Question: 2

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Depth-First Search (DFS) and Breadth-First Search (BFS) are two fundamental graph traversal algorithms. What is the MAIN difference in their exploration strategy?

1. DFS prioritizes exploring edges with the highest weight.
2. BFS prioritizes exploring all neighbors of a node before moving to the next level.
3. DFS can only be applied to directed graphs, while BFS works for both directed and undirected graphs.
4. BFS prioritizes exploring the deepest path first, while DFS explores outward levels first.

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Correct Answer: B. BFS prioritizes exploring all neighbors of a node before moving to the next level.

Explanation: BFS explores a graph in a layer-by-layer fashion, visiting all neighbors of a node before proceeding to the next level. DFS, on the other hand, explores a single path as deeply as possible until it reaches a dead end, then backtracks and explores another path. Option A is not applicable for unweighted graphs. Option C is incorrect as both algorithms work for directed and undirected graphs. Option D partially describes the exploration strategies but doesn't capture the key difference.

Question: 3

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A graph G is isomorphic to another graph H if:

1. They have the same number of nodes and edges.
2. They have the same node degrees but different structures.
3. There exists a one-to-one correspondence between their nodes, preserving adjacency relationships.
4. They represent the same underlying problem but with different notations.

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Correct Answer: C. There exists a one-to-one correspondence between their nodes, preserving adjacency relationships.

Explanation: Two graphs are isomorphic if there exists a bijective function (one-to-one and onto) that maps nodes in one graph to nodes in the other, while also maintaining the edge connections between those nodes. Option A is a necessary but not sufficient condition. Options B and D describe non-isomorphic graphs.

Question: 4

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In a directed graph, the in-degree of a node refers to:

1. The total number of edges connected to the node.
2. The number of edges leaving the node.
3. The number of edges entering the node.
4. The difference between the in-degree and out-degree.

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Correct Answer: C. The number of edges entering the node.

Explanation: In a directed graph, edges have a direction. The in-degree of a node refers to the number of edges that point towards that node. Option A considers both incoming and outgoing edges. Option B refers to the out-degree, and option D involves the difference between these two values.

Question: 5

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Which of the following statements is TRUE about a complete graph K_n?

1. It has n edges for every node.
2. It has (n-1) edges for every node.
3. It only has cycles as subgraphs.
4. It cannot be a planar graph.

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Correct Answer: B. It has (n-1) edges for every node.

Explanation: A complete graph K_n has an edge between every pair of distinct nodes. Since there are n nodes, each node will have connections to (n-1) other nodes. Option A overestimates the number of edges, while C and D are properties that may or may not hold true depending on the value of n.

Question: 6

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The development of graph theory has been driven by:

1. Purely theoretical applications in abstract mathematics.
2. Practical problems arising from various fields like science and engineering.
3. Both theoretical and practical motivations.
4. The need for efficient algorithms in data processing.

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Correct Answer: C. Both theoretical and practical motivations.

Explanation: Graph theory has evolved due to both the pursuit of theoretical understanding of graphs and their properties, as well as the need to solve practical problems in various fields. The Königsberg Bridge Problem is a classic example of this interplay.

Question: 7

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Which of the following is NOT a significant contribution of graph theory to computer science?

1. Modeling computer networks and communication protocols.
2. Developing algorithms for routing and navigation systems.
3. Analyzing social networks and information flow.
4. Representing and manipulating chemical structures.

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Correct Answer: D. Representing and manipulating chemical structures.

Explanation: While graph theory has applications in representing chemical structures using graph isomorphism, it's not as widely used compared to the other options. Options A, B, and C are major areas where graph theory plays a crucial role in computer science.

Question: 8

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The Four Color Problem, a famous unsolved problem in graph theory for centuries, asks:

1. If its possible to find a Hamiltonian cycle in a directed graph.
2. If there exists a planar graph that requires more than four colors to ensure no adjacent regions share the same color.
3. What the maximum number of edges a graph with n vertices can have.
4. How to efficiently determine the shortest path between two nodes in a weighted graph.

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Correct Answer: B. If there exists a planar graph that requires more than four colors to ensure no adjacent regions share the same color.

Explanation: The Four Color Problem, finally proven in 1976, asks if four colors are sufficient to color any planar graph (a graph that can be drawn on a plane without edges intersecting) such that no adjacent regions share the same color. Options A, C, and D deal with other graph theory problems.

Question: 9

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Which mathematician is credited with formally defining graphs and introducing the terminology 'vertex' and 'edge'?

1. Leonhard Euler
2. Gottfried Wilhelm Leibniz
3. Blaise Pascal
4. Joseph Louis Lagrange

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Correct Answer: A. Leonhard Euler

Explanation: Leonhard Euler, in his 1736 paper 'Seven Bridges of Königsberg,' is considered to have formally defined graphs and introduced the terms 'vertex' (now commonly called node) and 'edge.' While other mathematicians like Leibniz, Pascal, and Lagrange made significant contributions to mathematics, they are not directly credited with the formalization of graph theory.

Question: 10

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The Königsberg Bridge Problem, which is considered a founding problem in Graph Theory, involved:

1. Finding the shortest path between two points on a map.
2. Coloring a map such that no bordering countries share the same color.
3. Determining if a knights tour on a chessboard is possible.
4. Calculating the number of edges in a complete graph.

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Correct Answer: C. Determining if a knight's tour on a chessboard is possible.

Explanation: The Königsberg Bridge Problem asked if it was possible to walk across all seven bridges of Königsberg (now Kaliningrad, Russia) exactly once and return to the starting point. This problem led to the development of Eulerian circuits and laid the foundation for graph theory. While options A, B, and D deal with concepts in graph theory, they are not directly related to the Königsberg Bridge Problem.

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