Let’s analyze the problem step by step:

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(a) Strongly Connected Components (SCCs) in $G$

A strongly connected component (SCC) in a directed graph is a maximal subgraph where every pair of vertices is mutually reachable.

For a directed acyclic graph (DAG):

• By definition, there are no cycles.
• Consequently, no two vertices can form a strongly connected component with more than one vertex.
• Therefore, each vertex of a DAG forms its own SCC.

Conclusion:

In $G$, the number of strongly connected components is equal to the number of vertices $n$.

Justification:

Since $G$ is a DAG, every vertex is an SCC by itself due to the absence of cycles.

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(b) Strongly Connected Components (SCCs) in $G'$

Now, we modify $G$ by adding a directed edge from every sink (a vertex with no outgoing edges) to every source (a vertex with no incoming edges). Let $G'$ be the resulting graph.

Key Observations:

1. Adding these edges can introduce cycles:

   • A sink vertex connected to a source vertex creates a path that can combine previously independent SCCs.

2. In $G'$, the added edges may cause all vertices of $G$ to become mutually reachable:

   • Any vertex can now potentially reach any other vertex because:
     • Sinks are now connected to sources.
     • DAGs have a topological ordering, and the added edges effectively collapse the structure into a single component.

Conclusion:

In $G'$, the number of strongly connected components is 1.

Justification:

The additional edges ensure that every vertex is reachable from every other vertex, merging all previous SCCs into a single SCC.

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Summary of Answers:

(a) The number of SCCs in $G$ is $n$, as each vertex is its own SCC in a DAG.

(b) The number of SCCs in $G'$ is 1, because the added edges create a single, strongly connected graph.

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Follow-Up Questions:

1. How would the SCC count change if the graph $G$ were not a DAG?
2. What is the role of topological ordering in analyzing DAGs?
3. How do SCCs help in understanding the structure of directed graphs?
4. Can the addition of fewer edges (e.g., only one sink to one source) in $G$ reduce the number of SCCs? Why?
5. How would the result change if the graph were undirected instead of directed?

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Tip:

To analyze strongly connected components efficiently, algorithms like Kosaraju’s algorithm or Tarjan’s algorithm can be used, which run in $O(V + E)$ time for a graph with $V$ vertices and $E$ edges.