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Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Transcribed Image Text:**Title: Understanding Graph Theory: Acyclic Graphs and Depth-First Search** **Introduction:** In graph theory, one of the key concepts is determining whether a graph is acyclic, meaning it does not contain any cycles. A powerful tool for analyzing graphs is Depth-First Search (DFS), an algorithm that traverses or searches through graphs. **Key Concept:** A graph is acyclic if and only if a Depth-First Search (DFS) finds no back edges. **Explanation:** - **Depth-First Search (DFS):** This is a traversal method that explores as far down one branch as possible before backtracking. Starting from a selected root node, DFS visits and marks each node as visited before recursing through each adjacent node. - **Back Edge:** During a DFS, a back edge is an edge that points from a vertex to one of its ancestors in the DFS tree or forest. The presence of a back edge indicates the existence of a cycle in the graph. **Theorem:** - **Acyclic Graphs and DFS:** - If DFS finds no back edges during its traversal, the graph is acyclic. - Conversely, if the graph is acyclic, performing DFS will result in no back edges being found. **Conclusion:** Understanding the relationship between DFS and acyclic graphs is crucial for many applications, such as detecting cycles in algorithms, optimizing workflows, or designing efficient networks. This concept forms a cornerstone of deeper graph theory studies and practical applications in computer science.

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