Problem Breakdown

We are asked to find how many distinct topological sorts exist for a Directed Acyclic Graph (DAG) with 3 vertices and 2 edges.

Key Concepts

1. Topological Sort (Toposort): A topological sort of a DAG is a linear ordering of its vertices such that for every directed edge $u \to v$, vertex $u$ comes before vertex $v$ in the ordering.

2. DAG with 3 vertices and 2 edges: A DAG with 3 vertices means we have 3 distinct vertices. With 2 directed edges, we need to arrange these edges in such a way that no cycles are formed, and the vertices respect the edge directionality.

Step 1: Visualizing Possible Structures of the DAG

We have 3 vertices, say $v_1$, $v_2$, and $v_3$, and 2 directed edges. Since there are no cycles allowed (the graph is acyclic), one possible structure of the DAG could be:

• One of the vertices having directed edges to the other two vertices, which ensures no cycles are formed and the graph remains acyclic.

Let's consider a few possible edge configurations:

Example 1: $v_1 \to v_2$, $v_1 \to v_3$

• This graph means vertex $v_1$ has directed edges to both $v_2$ and $v_3$.
• The order of the vertices must be $v_1$ before both $v_2$ and $v_3$. We can freely choose the order of $v_2$ and $v_3$ because there are no edges between them.

Possible topological sorts:

• $v_1, v_2, v_3$
• $v_1, v_3, v_2$

Example 2: $v_1 \to v_2$, $v_2 \to v_3$

• This graph means vertex $v_1$ has a directed edge to $v_2$, and $v_2$ has a directed edge to $v_3$.
• The order of the vertices must be $v_1$ before $v_2$, and $v_2$ before $v_3$. This gives us only one possible ordering.

Possible topological sort:

• $v_1, v_2, v_3$

Example 3: $v_1 \to v_2$, $v_3 \to v_2$

• This graph means vertex $v_1$ has a directed edge to $v_2$, and vertex $v_3$ also has a directed edge to $v_2$. There are no edges between $v_1$ and $v_3$.
• The vertices $v_1$ and $v_3$ can appear in any order, but $v_2$ must appear after both.

Possible topological sorts:

• $v_1, v_3, v_2$
• $v_3, v_1, v_2$

Step 2: Counting Distinct Topological Sorts

From the examples above, we see that:

• In Example 1, there are 2 distinct topological sorts.
• In Example 2, there is 1 distinct topological sort.
• In Example 3, there are 2 distinct topological sorts.

Thus, the number of distinct topological sorts for a DAG with 3 vertices and 2 edges depends on the specific arrangement of edges. However, the possible configurations allow for a total of 5 distinct topological sorts.

Conclusion

The number of distinct topological sorts for a DAG with 3 vertices and 2 edges is 5.

Would you like to explore how the number of edges influences the number of topological sorts or need further details about DAGs and topological sorting?

Related Questions:

1. How does the number of edges in a DAG influence the number of possible topological sorts?
2. What are the conditions under which a DAG has exactly one topological sort?
3. How can we efficiently find all topological sorts of a DAG using algorithms like Kahn's algorithm?
4. How does adding more edges to a DAG with 3 vertices impact the number of topological sorts?
5. What happens when a DAG is a directed cycle graph? How many topological sorts does it have?

Tip: When counting topological sorts, always make sure to check whether there are any edges constraining the order of vertices. If there are no such constraints (e.g., no edges between some vertices), then those vertices can