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Find reflexive, symmetric and transitive closure graphs for following graphs

Find reflexive, symmetric and transitive closure graphs for following graphs

Database System Concepts
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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**Title: Understanding Reflexive, Symmetric, and Transitive Closures in Graph Theory** **Introduction to Graph Closures:** Graph theory is a significant area of study in mathematics and computer science, involving nodes (or vertices) and edges. In studying relations represented as graphs, we often explore how to find the reflexive, symmetric, and transitive closures of a given graph. Below are examples and explanations of these closures based on provided graphs. **Original Graph Analysis:** 1. **Graph Explanation:** - **Vertices**: a, b, c, d - **Edges**: - Self-loop at vertex a - Directed edge from a to b - Directed edge from a to c - Directed edge from c to a - Isolated vertex d This graph portrays directed connections among the vertices, with vertex d being isolated and lacking connections with any other vertices. **Graph Transformations:** 2. **Reflexive Closure:** To convert this graph into its reflexive closure, we need to add self-loops on all vertices not already possessing them. This means ensuring every vertex has a path leading to itself. 3. **Symmetric Closure:** Symmetric closure involves adding edges to make the graph symmetric. For every directed edge from vertex x to vertex y, there needs to be a corresponding edge from y to x, if not already present. In this graph: - Add an edge from b to a - Add an edge from c to b 4. **Transitive Closure:** The transitive closure of a graph includes additional edges to represent the reachability of vertices. If there is a direct path from vertex x to y and from y to z, then there should be a direct edge from x to z. In the given graph: - Add edges to represent all indirect paths as direct paths. **Modified Graph:** - **Vertices**: a, b, c, d - **Edges**: - Ensure self-loop for all vertices: a, b, c, d - Add necessary missing edges to make the graph transitive and symmetric. **Conclusion:** Understanding these closures helps in clarifying the properties of a relation represented by a graph. Reflexive closures ensure every element relates to itself, symmetric closures ensure bidirectional connections, and transitive closures illustrate full reachability. These properties have diverse applications in database
Transcribed Image Text:**Title: Understanding Reflexive, Symmetric, and Transitive Closures in Graph Theory** **Introduction to Graph Closures:** Graph theory is a significant area of study in mathematics and computer science, involving nodes (or vertices) and edges. In studying relations represented as graphs, we often explore how to find the reflexive, symmetric, and transitive closures of a given graph. Below are examples and explanations of these closures based on provided graphs. **Original Graph Analysis:** 1. **Graph Explanation:** - **Vertices**: a, b, c, d - **Edges**: - Self-loop at vertex a - Directed edge from a to b - Directed edge from a to c - Directed edge from c to a - Isolated vertex d This graph portrays directed connections among the vertices, with vertex d being isolated and lacking connections with any other vertices. **Graph Transformations:** 2. **Reflexive Closure:** To convert this graph into its reflexive closure, we need to add self-loops on all vertices not already possessing them. This means ensuring every vertex has a path leading to itself. 3. **Symmetric Closure:** Symmetric closure involves adding edges to make the graph symmetric. For every directed edge from vertex x to vertex y, there needs to be a corresponding edge from y to x, if not already present. In this graph: - Add an edge from b to a - Add an edge from c to b 4. **Transitive Closure:** The transitive closure of a graph includes additional edges to represent the reachability of vertices. If there is a direct path from vertex x to y and from y to z, then there should be a direct edge from x to z. In the given graph: - Add edges to represent all indirect paths as direct paths. **Modified Graph:** - **Vertices**: a, b, c, d - **Edges**: - Ensure self-loop for all vertices: a, b, c, d - Add necessary missing edges to make the graph transitive and symmetric. **Conclusion:** Understanding these closures helps in clarifying the properties of a relation represented by a graph. Reflexive closures ensure every element relates to itself, symmetric closures ensure bidirectional connections, and transitive closures illustrate full reachability. These properties have diverse applications in database
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