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Construct a graph with vertices P, Q, R, S, T, U, V that has an Euler path, the degree of R is 1 and the degree of U is 3. What is the edge set?

Construct a graph with vertices P, Q, R, S, T, U, V that has an Euler path, the degree of R is 1 and the degree of U is 3. What is the edge set?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Problem Statement:** Construct a graph with vertices \( P, Q, R, S, T, U, V \) that has an Euler path, the degree of \( R \) is 1 and the degree of \( U \) is 3. **Question:** What is the edge set? **Explanation:** In graph theory, an Euler path is a path that visits every edge of a graph exactly once. For an Euler path to exist in a graph, the graph must have exactly 0 or 2 vertices of odd degree. Given that \( R \) has degree 1 (which is odd) and \( U \) has degree 3 (which is odd), we need to ensure these are the only vertices with odd degrees in our graph, and the rest of the vertices should have even degrees. Please describe the edge connections to form a valid Euler path while satisfying the given degree conditions for vertices \( R \) and \( U \). **Potential Solution:** The edge set can be formed as follows: - Connect \( R \) to \( Q \) (since \( R \) must have exactly one edge connected to it) - \( U \) needs to have three connections. Let’s connect \( U \) to \( P \), \( Q \), and \( V \). To ensure that the graph is Eulerian, the remaining vertices should form a connected path. Here’s an example edge set that satisfies the conditions: - \( (R, Q) \) - \( (Q, U) \) - \( (U, P) \) - \( (U, V) \) - \( (V, S) \) - \( (S, T) \) - \( (T, P) \) This structure has: - Vertex \( R \) with degree 1 (ensures the Euler path condition) - Vertex \( U \) with degree 3 (satisfies the degree condition for \( U \) and ensures the Euler path condition) All vertices connect to form a path visiting every edge exactly once with precisely two vertices of odd degree, ensuring the possibility of an Euler path.
Transcribed Image Text:**Problem Statement:** Construct a graph with vertices \( P, Q, R, S, T, U, V \) that has an Euler path, the degree of \( R \) is 1 and the degree of \( U \) is 3. **Question:** What is the edge set? **Explanation:** In graph theory, an Euler path is a path that visits every edge of a graph exactly once. For an Euler path to exist in a graph, the graph must have exactly 0 or 2 vertices of odd degree. Given that \( R \) has degree 1 (which is odd) and \( U \) has degree 3 (which is odd), we need to ensure these are the only vertices with odd degrees in our graph, and the rest of the vertices should have even degrees. Please describe the edge connections to form a valid Euler path while satisfying the given degree conditions for vertices \( R \) and \( U \). **Potential Solution:** The edge set can be formed as follows: - Connect \( R \) to \( Q \) (since \( R \) must have exactly one edge connected to it) - \( U \) needs to have three connections. Let’s connect \( U \) to \( P \), \( Q \), and \( V \). To ensure that the graph is Eulerian, the remaining vertices should form a connected path. Here’s an example edge set that satisfies the conditions: - \( (R, Q) \) - \( (Q, U) \) - \( (U, P) \) - \( (U, V) \) - \( (V, S) \) - \( (S, T) \) - \( (T, P) \) This structure has: - Vertex \( R \) with degree 1 (ensures the Euler path condition) - Vertex \( U \) with degree 3 (satisfies the degree condition for \( U \) and ensures the Euler path condition) All vertices connect to form a path visiting every edge exactly once with precisely two vertices of odd degree, ensuring the possibility of an Euler path.
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