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Let D be a finite digraph. If id(x) >0 for every x element of V(D), then D contains a cycle.

Let D be a finite digraph. If id(x) >0 for every x element of V(D), then D contains a cycle.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section: Chapter Questions
Problem 21RE
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**Understanding the Relationship Between In-Degree and Cycles in Directed Graphs** Consider a finite directed graph (digraph) \(D\). If the in-degree \(id(x)\) of each vertex \(x\) in the set of vertices \(V(D)\) is greater than 0, then it implies that every vertex in \(D\) has at least one incoming edge. Under this condition, we can conclude that the digraph \(D\) contains at least one cycle. **Key Concepts:** 1. **Directed Graph (Digraph)**: - A graph where the edges have a direction, represented by arrows pointing from one vertex to another. 2. **In-Degree (id(x))**: - The number of edges directed into a vertex \(x\). 3. **Cycle**: - A path in which the starting and ending vertices are the same, and no vertices are repeated along the path except for the start/end vertex. **Implication**: - If every vertex \(x\) in \(D\) has \(id(x) > 0\), there are no vertices which are isolated from incoming connections. This condition ensures that there is a cycle within the digraph, meaning there is a closed loop path that can be formed within \(D\). Understanding these principles is important for analyzing the structure and properties of directed graphs, especially in fields such as graph theory, computer science, and network analysis.
Transcribed Image Text:**Understanding the Relationship Between In-Degree and Cycles in Directed Graphs** Consider a finite directed graph (digraph) \(D\). If the in-degree \(id(x)\) of each vertex \(x\) in the set of vertices \(V(D)\) is greater than 0, then it implies that every vertex in \(D\) has at least one incoming edge. Under this condition, we can conclude that the digraph \(D\) contains at least one cycle. **Key Concepts:** 1. **Directed Graph (Digraph)**: - A graph where the edges have a direction, represented by arrows pointing from one vertex to another. 2. **In-Degree (id(x))**: - The number of edges directed into a vertex \(x\). 3. **Cycle**: - A path in which the starting and ending vertices are the same, and no vertices are repeated along the path except for the start/end vertex. **Implication**: - If every vertex \(x\) in \(D\) has \(id(x) > 0\), there are no vertices which are isolated from incoming connections. This condition ensures that there is a cycle within the digraph, meaning there is a closed loop path that can be formed within \(D\). Understanding these principles is important for analyzing the structure and properties of directed graphs, especially in fields such as graph theory, computer science, and network analysis.
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