Data Structure Unit 6: Graphs

Questions and Answers

What defines a directed graph?

All edges are two-way.

Edges can have variable weights.

The graph has no vertices.

All edges are one-way. (correct)

Which component(s) must be defined to represent a graph G?

A cycle of vertices and paths.

A finite set of vertices and edges. (correct)

A finite set of edges only.

Only labeled edges.

What is a cycle in a directed graph?

A path with more than one edge.

A sequence of vertices where all edges point away from the starting vertex.

A series of vertices that are not connected.

A path that starts and ends at the same vertex. (correct)

What is the in-degree of a vertex in a directed graph?

The number of edges that point to the vertex. Signup and view all the answers

Which of the following best describes a directed acyclic graph (DAG)?

A directed graph without any cycles. Signup and view all the answers

Which of the following is NOT a common representation of a graph?

Weighted List Signup and view all the answers

What does the weight of an edge in a graph typically represent?

A numerical value like distance or cost. Signup and view all the answers

In the context of graphs, what is a path?

A sequence of edges that connect vertices. Signup and view all the answers

What does the adjacency matrix of a graph represent?

A 2D array indicating connections between vertices. Signup and view all the answers

What is indicated by a graph being labeled?

Vertices contain identifying information. Signup and view all the answers

What distinguishes a directed graph from an undirected graph?

Directed graphs have edges pointing from one vertex to another. Signup and view all the answers

Which statement is true about cycles in graphs?

A cycle is formed when a path starts and ends at the same vertex. Signup and view all the answers

How does a graph differ from a tree in terms of connectivity?

Graphs can have disconnected sets of nodes while trees cannot. Signup and view all the answers

What do we call a fundamental part of a graph that represents a connection between two vertices?

An edge Signup and view all the answers

Which of the following correctly describes weights in graphs?

Weights provide additional information about vertices and/or edges. Signup and view all the answers

In the context of graphs, what does a vertex represent?

A fundamental part of a graph that can have a name and additional information. Signup and view all the answers

When modeling a real-world problem using graphs, which example can be considered appropriate?

Modeling the Internet as a graph with Web pages as nodes. Signup and view all the answers

Which of the following statements about edges in a graph is correct?

Edges show a relationship between two vertices. Signup and view all the answers

What does a value of adj[i][j] = 1 signify in an adjacency matrix?

There is an edge from vertex i to vertex j. Signup and view all the answers

Which statement about the adjacency matrix is true?

It allows for O(1) time queries to check the existence of edges. Signup and view all the answers

What is a characteristic of Depth-First Search (DFS) as applied to a graph?

DFS forms multiple depth-first trees based on discovery times. Signup and view all the answers

What do the time stamps d[u] and f[u] represent in Depth-First Search?

The time vertex u was discovered and the time vertex u was fully explored. Signup and view all the answers

What is the time complexity of the Depth-First Search algorithm?

Θ(V + E) Signup and view all the answers

In an adjacency list representation, how is a weighted graph typically represented?

By storing weights in nodes of linked lists. Signup and view all the answers

Which statement best describes the drawbacks of adjacency matrix representation?

It requires O(V^2) space even for sparse graphs. Signup and view all the answers

What does the color attribute indicate during the DFS traversal?

The status of the vertex in the search process. Signup and view all the answers

Which of the following accurately describes BFS while traversing a graph?

It explores vertices in a first-in-first-out order. Signup and view all the answers

What is the primary structure used to represent the adjacency list for a graph?

Array of linked lists Signup and view all the answers

What is the primary goal when interconnecting a set of pins using an undirected graph?

To minimize the total weight of connections Signup and view all the answers

Which algorithm is classified as a greedy algorithm when determining the minimum spanning tree?

Kruskal's Algorithm Signup and view all the answers

What criteria does Kruskal's Algorithm use to select an edge for the minimum spanning tree?

Only edges that do not create a cycle are selected Signup and view all the answers

What does the minimum spanning tree ensure when connecting a set of islands with bridges?

Each island is reachable from any other with minimal cost Signup and view all the answers

How does Kruskal's Algorithm determine if an edge can be added to the spanning tree?

It ensures the edge does not create a cycle Signup and view all the answers

Which of the following is NOT a characteristic of the edges selected by Kruskal's Algorithm?

Edges are connected to non-existing vertices Signup and view all the answers

What is the result of applying Kruskal's Algorithm to a graph?

A minimum spanning tree for the graph Signup and view all the answers

What aspect is primarily considered by the government when approving bridge constructions between islands?

The total cost of construction Signup and view all the answers

What best describes the complexity of the while-loop in breadth-first search?

O(V + E) Signup and view all the answers

Which of the following statements about spanning trees is true?

An edge in a spanning tree is called a branch. Signup and view all the answers

What is the primary goal of a minimum spanning tree (MST)?

To connect all vertices with the lowest possible edge weight. Signup and view all the answers

How many edges does a spanning tree for a graph with |V| vertices contain?

|V| - 1 edges Signup and view all the answers

Which condition must be true for constructing a minimum spanning tree?

The graph must be connected and undirected. Signup and view all the answers

In breadth-first search (BFS), what happens when a vertex 'u' is dequeued?

It changes its color to gray. Signup and view all the answers

Which statement accurately represents how edges are treated in undirected graphs during BFS?

Each edge is examined twice. Signup and view all the answers

What is a common application of spanning trees?

Designing efficient routing algorithms. Signup and view all the answers

What is the relationship between BFS and the discovery of all vertices in a graph?

BFS discovers vertices level by level. Signup and view all the answers

What is the outcome if a vertex's color remains white during BFS?

The vertex has never been encountered. Signup and view all the answers

Study Notes

Graphs

Graphs represent relationships between pairs of items and are crucial in algorithm studies.
Composed of vertices (nodes) and edges (connections), graphs can be directed or undirected.
Directed graphs (digraphs) have edges that point from one vertex to another, while undirected graphs connect vertices symmetrically.
Graphs can have weights, adding values to vertices or edges to indicate costs or distances.

Types of Graphs

Graphs may contain disconnected sets of nodes and cycles, differentiating them from trees.
Examples:
- Graph (a) - Contains disconnected nodes.
- Graph (b) - Features multiple cycles.
- Graph (c) - Represents a tree with no cycles.

Key Components

Vertex (Node): Fundamental graph element that can hold names (keys) and additional information (payload).
Edge (Arc): Connects two vertices, showing their relationship; can be directed or undirected and may have weights.

Representation of Graphs

Represented as G = (V, E), where V is the vertex set and E is the edge set.
Edges are represented as tuples (u, v) with optional weights.
Subgraph: A portion of a graph defined by a subset of vertices and edges.

Graph Representations

Adjacency Matrix: 2D array where entry adj[i][j] indicates an edge from vertex i to vertex j. It’s symmetric for undirected graphs and can store weights.
- Pros: Easier implementation, O(1) edge removal and queries.
- Cons: Uses O(V²) space, inefficient for sparse graphs.
Adjacency List: Array of linked lists representing adjacent vertices. Edge weights can be included in linked lists.

Graph Traversal Algorithms

Depth-First Search (DFS):
- Explores vertices deeper before backtracking.
- Utilizes timestamps to track discovery and finishing times of vertices.
- Running time: Θ(V + E).
Breadth-First Search (BFS):
- Explores all neighbors at the present depth prior to moving on.
- Also uses timestamps but initializes with all vertices colored white.
- Running time: O(V + E).

Spanning Trees

A spanning tree includes all vertices with minimal edges, thus forming a tree from the original graph.
Spanning trees are significant in network design and various optimization problems.
Minimum Spanning Tree (MST): The spanning tree that minimizes the total edge weight.
- Essential for efficiently connecting points (like cities or network nodes) while minimizing costs.

Algorithms for MST

Kruskal's Algorithm: A greedy approach that selects edges based on weight and checks if they connect existing components without forming a cycle.
Prim's Algorithm: Another method for finding the MST, starting with a single vertex and growing the tree by adding the lowest weight edges connecting to it.

Applications of Graphs

Widely used for representing networks such as transportation, social networks, and logistical frameworks.

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Description

Explore the key concepts of graphs in Data Structures through this quiz. Understand the different types of graphs, their components, and how they can model relationships. This unit delves into directed and undirected graphs and their significance in algorithm study.