Q9A. Consider the graph with the following vertices and edges: V = {a, b, c, d, e, f} E = {{a, b}, {a, c}, {a, d}, {a, f}, {b, c}, {b, f}, {c, d}, {c, f}, {d, e}, {d, f}} b a C d e Which of the following is true about Hamiltonian cycles and paths for the graph? The graph has a Hamiltonian path but not a Hamiltonian cycle. The graph has a Hamiltonian cycle but not a Hamiltonian path. The graph has a Hamiltonian cycle and a Hamiltonian path. The graph has neither a Hamiltonian path nor a Hamiltonian cycle.
Q9A. Consider the graph with the following vertices and edges: V = {a, b, c, d, e, f} E = {{a, b}, {a, c}, {a, d}, {a, f}, {b, c}, {b, f}, {c, d}, {c, f}, {d, e}, {d, f}} b a C d e Which of the following is true about Hamiltonian cycles and paths for the graph? The graph has a Hamiltonian path but not a Hamiltonian cycle. The graph has a Hamiltonian cycle but not a Hamiltonian path. The graph has a Hamiltonian cycle and a Hamiltonian path. The graph has neither a Hamiltonian path nor a Hamiltonian cycle.
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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Transcribed Image Text:**Graph Theory Problem: Hamiltonian Cycles and Paths**
**Question 9A:** Consider the graph with the following vertices and edges:
- **Vertices (V):** {a, b, c, d, e, f}
- **Edges (E):** {{a, b}, {a, c}, {a, d}, {a, f}, {b, c}, {b, f}, {c, d}, {c, f}, {d, e}, {d, f}}
The graph is depicted as a set of vertices connected by edges, forming a complex network. The vertices (a, b, c, d, e, f) are connected in such a way that multiple edges exist between these points.
**Graph Description:**
- **Vertices:** Represented by points labeled a, b, c, d, e, f.
- **Edges:** Lines connecting these points, some vertices are directly connected to multiple others.
**Question:**
Which of the following is true about Hamiltonian cycles and paths for the graph?
- ( ) The graph has a Hamiltonian path but not a Hamiltonian cycle.
- ( ) The graph has a Hamiltonian cycle but not a Hamiltonian path.
- ( ) The graph has a Hamiltonian cycle and a Hamiltonian path.
- ( ) The graph has neither a Hamiltonian path nor a Hamiltonian cycle.
**Clarification:**
A **Hamiltonian path** is a path through a graph that visits each vertex exactly once. A **Hamiltonian cycle** is a cycle that visits each vertex exactly once and returns to the starting vertex.
Analyze the graph structure to determine which of the above options correctly describes the Hamiltonian properties of the graph.
Transcribed Image Text:**Q10C. Graph Theory Exercise**
Consider the graph with the following vertices and edges:
- **Vertices (V):** {a, b, c, d, e, f}
- **Edges (E):** {{a, b}, {a, c}, {a, d}, {a, f}, {b, e}, {b, f}, {c, d}, {d, e}, {d, f}}
**Graph Description:**
The image depicts a graph with six vertices labeled a, b, c, d, e, and f. The vertices are interconnected through a series of edges:
- Vertex 'a' is connected to vertices b, c, d, and f.
- Vertex 'b' is connected to vertices a, e, and f.
- Vertex 'c' is connected to vertices a and d.
- Vertex 'd' is connected to vertices a, c, e, and f.
- Vertex 'e' is connected to vertices b and d.
- Vertex 'f' is connected to vertices a, b, and d.
**Exercise: Identifying Circuits**
Which of the following are examples of circuits within the graph? (Select all that apply.)
- \( \langle d, c, a, b, e \rangle \)
- \( \langle a, b, c, d, e, f, a \rangle \)
- \( \langle a, b, f, d, c, a \rangle \)
- \( \langle b, f, a, b \rangle \)
- \( \langle f, b, e \rangle \)
A circuit in graph theory is a path that starts and ends at the same vertex without repeating any edges.
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