https://file.aiquickdraw.com/mathbot/file/1728372969555gtl531b6.jpg

The image contains a graph \( G \) with vertices \( a, b, c, d, e \) and labeled edges \( e_1, e_2, \dots, e_{11} \). The problem presents three tasks based on this graph:

### a) Find the induced subgraph \( G \langle V \rangle \) where \( V = \{a, d, e, c\} \).

- **Induced subgraph**: A subgraph formed by a subset of vertices \( V \) and all the edges between those vertices in the original graph.
- For \( V = \{a, d, e, c\} \), the vertices are \( a, d, e, c \). To find the induced subgraph, we only keep edges between these vertices:
  - Between \( a \) and \( d \), there is edge \( e_7 \).
  - Between \( d \) and \( e \), there are edges \( e_9 \) and \( e_{11} \).
  - Between \( c \) and \( e \), there are edges \( e_4 \) and \( e_8 \).
  - Between \( a \) and \( e \), there is a self-loop \( e_6 \).
  
  So, the induced subgraph consists of the vertices \( a, d, e, c \) and the edges \( e_6, e_7, e_8, e_9, e_{11} \).

### b) Find \( G - A \) where \( A = \{e_2, e_6, e_{10}\} \).

- **Removing edges from a graph**: When edges are removed, the vertices remain, but the connections (edges) specified in \( A \) are deleted.
  - Remove edge \( e_2 \) between \( b \) and \( c \).
  - Remove edge \( e_6 \) (the self-loop on \( e \)).
  - Remove edge \( e_{10} \) between \( e \) and \( c \).
  
  After removing these edges, the remaining edges are \( e_1, e_3, e_4, e_5, e_7, e_8, e_9, e_{11} \).

### c) The new graph \( G_1 \) obtained from \( G \) by fusion of vertices \( d \) and \( e \).

- **Fusion of vertices**: When two vertices are fused, they are combined into a single vertex, and the new vertex inherits all edges connected to the original vertices.
  - Fusing vertices \( d \) and \( e \) creates a new vertex, say \( v \), replacing \( d \) and \( e \).
  - The new vertex \( v \) will be connected to:
    - \( a \) (via edges \( e_7 \) and \( e_9 \)),
    - \( b \) (via edge \( e_5 \)),
    - \( c \) (via edges \( e_4, e_8, e_{10} \)).
    - Self-loop inherited from \( e_6 \) on vertex \( e \).

So, in the new graph \( G_1 \), vertex \( v \) will be connected to \( a, b, c \), and there will be a self-loop from \( e_6 \).

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### Related Questions:
1. How do we determine the degree of a vertex in a graph?
2. What are the properties of an induced subgraph?
3. How does fusion of vertices affect the structure of a graph?
4. What are the effects of removing edges on the connectivity of a graph?
5. How would you define a cycle in the context of graph theory?

### Tip:
When fusing vertices in a graph, always remember that the resulting vertex will inherit all edges from both original vertices, and any duplicate edges may become multi-edges if the graph permits them.

Consider the following graph G with vertices {a, b, c, d, e} and edges {e1, e2, ..., e11}. Solve the following:

a) Find the induced subgraph G<V> where V={a, d, e, c}.

b) Find G-A where A={e2, e6, e10}.

c) Find the new graph G1 obtained from G by fusion of vertices d and e.

Solve a graph theory problem involving finding an induced subgraph, removing edges, and fusing vertices. The graph includes vertices {a, b, c, d, e} and edges {e1, e2, ..., e11}. Learn how to apply key concepts like vertex fusion and edge removal.

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