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 $e_6$.

Let me know if you would like further details or have any questions.

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.