**Problem 5: Graph Theory Exploration**Let \( n \in \{3, 4, 5, 6, \ldots\} \) be fixed. Show that there are exactly two orientations of \( C_n \) with vertex set \( V = \{0, 1, 2, \ldots, n - 1\} \) that are strongly connected.### Explanation:To explore this problem, consider the cycle graph \( C_n \) which consists of \( n \) vertices and \( n \) edges, forming a closed loop. An orientation of a graph is a direction assigned to each edge, turning it into a directed graph.For \( C_n \) to be strongly connected:- There must be a directed path from any vertex to every other vertex.- This requirement is satisfied if all edges are oriented consistently in a circular manner (i.e., all clockwise or all counterclockwise).Thus, there are exactly two valid orientations of \( C_n \) that are strongly connected:1. All edges oriented in a clockwise direction.2. All edges oriented in a counterclockwise direction.This completes the demonstration of the two specific orientations fulfilling the criteria of being strongly connected.

Now consider a DFS through a bridgeless connected graph. Clearly, we will visit each vertex. And…

Answered: Let n E {3,4,5,6, ...} be fixed. Show…

Let n E {3,4,5,6, ...} be fixed. Show that there are exactly two {0,1,2, ...,n – 1} that are orientations of C, with vertex set V = strongly connected.

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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Question

**Problem 5: Graph Theory Exploration**

Let \( n \in \{3, 4, 5, 6, \ldots\} \) be fixed. Show that there are exactly two orientations of \( C_n \) with vertex set \( V = \{0, 1, 2, \ldots, n - 1\} \) that are strongly connected.

### Explanation:

To explore this problem, consider the cycle graph \( C_n \) which consists of \( n \) vertices and \( n \) edges, forming a closed loop. An orientation of a graph is a direction assigned to each edge, turning it into a directed graph.

For \( C_n \) to be strongly connected:
- There must be a directed path from any vertex to every other vertex.
- This requirement is satisfied if all edges are oriented consistently in a circular manner (i.e., all clockwise or all counterclockwise).

Thus, there are exactly two valid orientations of \( C_n \) that are strongly connected:
1. All edges oriented in a clockwise direction.
2. All edges oriented in a counterclockwise direction.

This completes the demonstration of the two specific orientations fulfilling the criteria of being strongly connected.

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