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Consider the following directed graph. E B Which of the following are valid cycles? A, A, D A, D, C, A A, A, A, A, A C, E, B, C A E, D, C, B, E C, E, B

Consider the following directed graph. E B Which of the following are valid cycles? A, A, D A, D, C, A A, A, A, A, A C, E, B, C A E, D, C, B, E C, E, B

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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**Directed Graph Analysis** In this exercise, we are given a directed graph consisting of five nodes labeled A, B, C, D, and E interconnected with directed edges. A directed graph or digraph is a set of vertices connected by edges, where the edges have a direction associated with them. ### Graph Description: - Node E has directed edges pointing towards nodes B and D. - Node D has directed edges pointing towards nodes A and C. - Node A has a self-loop, indicating a directed edge from A to A. - Node C has directed edges pointing towards nodes B and A. - Node B does not have any outgoing edges. ### Cycle Validation: A cycle in a directed graph is a path that starts and ends at the same node with all edges following the designated directions. We need to determine which of the provided options represent a valid cycle in the given graph. **Options:** 1. A, A, D 2. A, D, C, A 3. A, A, A, A, A 4. C, E, B, C 5. E, D, C, B, E 6. C, E, B By analyzing the graph, you can determine which sequences form valid cycles. Remember, a cycle must start and end at the same node with continuous valid paths between them. ### Instructions: Evaluate each option to see if you can traverse the nodes in the specified order, beginning and ending at the same node, while following the directed edges. **Note:** When considering valid cycles, ensure that all transitions between nodes respect the arrow directions in the graph.
Transcribed Image Text:**Directed Graph Analysis** In this exercise, we are given a directed graph consisting of five nodes labeled A, B, C, D, and E interconnected with directed edges. A directed graph or digraph is a set of vertices connected by edges, where the edges have a direction associated with them. ### Graph Description: - Node E has directed edges pointing towards nodes B and D. - Node D has directed edges pointing towards nodes A and C. - Node A has a self-loop, indicating a directed edge from A to A. - Node C has directed edges pointing towards nodes B and A. - Node B does not have any outgoing edges. ### Cycle Validation: A cycle in a directed graph is a path that starts and ends at the same node with all edges following the designated directions. We need to determine which of the provided options represent a valid cycle in the given graph. **Options:** 1. A, A, D 2. A, D, C, A 3. A, A, A, A, A 4. C, E, B, C 5. E, D, C, B, E 6. C, E, B By analyzing the graph, you can determine which sequences form valid cycles. Remember, a cycle must start and end at the same node with continuous valid paths between them. ### Instructions: Evaluate each option to see if you can traverse the nodes in the specified order, beginning and ending at the same node, while following the directed edges. **Note:** When considering valid cycles, ensure that all transitions between nodes respect the arrow directions in the graph.
Expert Solution
Step 1: Cycle ( graph theory)

Only the start and end vertices of a graph are equivalent to a cycle in graph theory. Non-empty directed cycles in directed graphs have just the initial and end vertices equal. Graphs without cycles are acyclic. A directed acyclic graph has no directed cycles. A connected graph without cycles is a tree. 

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