Problem Breakdown:

A DAG (Directed Acyclic Graph) is a directed graph without any cycles. A topological sort (toposort) of a DAG is an ordering of its vertices such that for every directed edge $(u, v)$, vertex $u$ appears before vertex $v$ in the ordering.

We need to determine how many distinct topological sorts are possible for a DAG with 3 vertices ($V = 3$) and 2 edges ($E = 2$).

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Proof and Solution:

Step 1: Possible Structures of the DAG

For $V = 3$, let the vertices be $\{ A, B, C \}$. Since there are 2 edges and the graph is acyclic, the edges must form a partial ordering. Several configurations are possible for how the edges are arranged:

1. Case 1: Linear chain (A → B → C)
   Here, the edges are $A → B$ and $B → C$.
   The only valid toposort is $[A, B, C]$.
   Distinct toposorts = 1.

2. Case 2: Fork (A → B, A → C)
   Here, the edges are $A → B$ and $A → C$.
   Both $B$ and $C$ depend on $A$, but $B$ and $C$ are independent of each other.
   Valid toposorts: $[A, B, C]$ and $[A, C, B]$.
   Distinct toposorts = 2.

3. Case 3: Reverse fork (B → A, C → A)
   Here, the edges are $B → A$ and $C → A$.
   Both $B$ and $C$ must appear before $A$, but $B$ and $C$ are independent of each other.
   Valid toposorts: $[B, C, A]$ and $[C, B, A]$.
   Distinct toposorts = 2.

4. Case 4: Separate chains (B → A, A → C)
   Here, the edges are $B → A$ and $A → C$.
   This creates a linear ordering $B → A → C$.
   The only valid toposort is $[B, A, C]$.
   Distinct toposorts = 1.

Step 2: Total Number of Distinct Toposorts

Adding up the counts from all possible cases:
1 (Linear chain) + 2 (Fork) + 2 (Reverse fork) + 1 (Separate chains) = 6 distinct toposorts.

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Final Answer:

There are 6 distinct topological sorts possible for a DAG with 3 vertices and 2 edges.

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Questions for Further Exploration:

1. What happens to the number of distinct toposorts if the number of edges is increased to 3?
2. How does the structure of the graph affect the number of topological sorts?
3. Can you generalize the formula for counting toposorts for a DAG with $n$ vertices and $m$ edges?
4. What is the computational complexity of finding all topological sorts for a general DAG?
5. How does a topological sort relate to critical path analysis in project scheduling?

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Tip:

To enumerate all toposorts for any DAG, consider using algorithms like Kahn's algorithm or DFS-based topological sorting, which systematically process the graph's vertices respecting their dependencies.