To find the number of connected components in each subgraph induced by the specified vertex subsets, we examine the connectivity of the vertices within each subset based on the original graph structure.

Solution:

1. Subset: $\{a, b, c, d\}$

   • Edges between vertices in this subset: $a \leftrightarrow b$, $a \leftrightarrow c$, $b \leftrightarrow d$, $c \leftrightarrow d$.
   • These edges connect all vertices, forming a single connected component.
   • Result: 1 connected component
   • Component: $\{a, b, c, d\}$

2. Subset: $\{e, f, g, k\}$

   • Edges between vertices in this subset: $g \leftrightarrow f$, $g \leftrightarrow e$.
   • There is no direct or indirect connection to $k$ from other vertices in this subset.
   • Result: 2 connected components
   • Components: $\{e, g, f\}$ and $\{k\}$

3. Subset: $\{a, b, e, g, h\}$

   • Edges between vertices in this subset: $b \leftrightarrow e$, $g \leftrightarrow h$, $e \leftrightarrow h$.
   • There is no connection from $a$ to other vertices in this subset.
   • Result: 2 connected components
   • Components: $\{b, e, g, h\}$ and $\{a\}$

4. Subset: $\{c, d, f, h\}$

   • Edges between vertices in this subset: $c \leftrightarrow d$, $f \leftrightarrow g$, but $g$ is not part of this subset, and no other connections link all vertices in this subset.
   • Result: 2 connected components
   • Components: $\{c, d\}$ and $\{f, h\}$

5. Subset: $\{a, b, f, g, k\}$

   • Edges between vertices in this subset: $a \leftrightarrow b$, $f \leftrightarrow g$, but no edges link $k$ to any other vertices in this subset.
   • Result: 3 connected components
   • Components: $\{a, b\}$, $\{f, g\}$, and $\{k\}$

Let me know if you'd like further clarification on any of these calculations, or have any questions about connected components in graphs.

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Related Questions

1. How do connected components change if additional edges are added between certain vertices?
2. What is the significance of isolated vertices in a subgraph?
3. Can a subgraph with fewer vertices ever have more connected components than the original graph?
4. How do induced subgraphs differ from general subgraphs in terms of connected components?
5. What algorithm could you use to find connected components in a large graph efficiently?

Tip:

When analyzing connected components, it’s often helpful to start by isolating any vertices that don’t share an edge with others in the subset, as they immediately form separate components.