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The image presents a problem involving paths and circuits in graphs. The question asks to determine which of the given sequences contain an Euler path, an Euler circuit, both, or neither.

### Key Concepts:
- **Euler Path**: A path in a graph that visits every edge exactly once.
- **Euler Circuit**: A circuit that visits every edge exactly once and starts and ends at the same vertex.

### Given Graph Vertices:
- The vertices are labeled \( a, b, c, d, e \).

### Analysis of the Sequences:

1. **Sequence in (a)**:
   - \([a, b, c, b, d, c, e]\)
   - To determine if it contains an Euler path or circuit, check if all edges are covered exactly once, and also check if it starts and ends at the same vertex for a circuit.

   **Evaluation**:
   - The sequence doesn't start and end at the same vertex, so it's not an Euler circuit.
   - Check if it covers all edges without repetition.

2. **Sequence in (b)**:
   - \([c, e], [c, e, d, c], [e, c, d], [a, b]\)
   - This needs to be checked if each edge is visited exactly once. However, these are individual segments, and the question asks whether the whole sequence across these segments forms a path or circuit.

   **Evaluation**:
   - These are segments, not a continuous sequence, so neither Euler path nor circuit is formed.

### Conclusion:
- The sequences in part **(a)** could potentially contain an Euler path but definitely do not contain an Euler circuit.
- The sequences in part **(b)** likely do not form either a path or circuit.

Would you like to explore the detailed verification of the conditions for each sequence, or do you have any other questions?

### Further Questions:
1. What is the difference between a Hamiltonian path and an Euler path?
2. How do the degrees of vertices influence the existence of Euler circuits?
3. Can a graph have an Euler path but not an Euler circuit?
4. What are the necessary and sufficient conditions for a graph to have an Euler circuit?
5. How would adding or removing an edge affect the classification of these sequences?

### Tip:
When determining whether a sequence contains an Euler circuit, always start by checking if the sequence begins and ends at the same vertex and covers all edges exactly once.

The query submitted by the user: The image presents a problem involving paths and circuits in graphs. The question asks to determine which of the given sequences contain an Euler path, an Euler circuit, both, or neither.

Explore the problem involving paths and circuits in graphs, specifically focusing on Euler paths and Euler circuits. Analyze sequences to determine if they contain an Euler path, an Euler circuit, both, or neither. Detailed explanations and evaluations provided.

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