To find a topological sort of the directed graph shown in the image using Depth-First Search (DFS), starting from node $d$, we will follow these steps:

Understanding the Graph

The directed graph has the following edges:

• $a \rightarrow b$
• $b \rightarrow c$
• $c \rightarrow e$
• $d \rightarrow b$

Depth-First Search (DFS) Approach

To perform a topological sort using DFS:

1. Start from the given node $d$.
2. Mark the node as visited.
3. Recursively apply DFS to all the unvisited nodes it points to.
4. Once all the nodes reachable from a node are visited, add that node to the topological sort list.
5. Repeat for all nodes until all nodes are visited.

Performing DFS from Node $d$

1. Start DFS from $d$:

   • Mark $d$ as visited.
   • Move to its adjacent node $b$.

2. DFS on $b$:

   • Mark $b$ as visited.
   • Move to its adjacent node $c$.

3. DFS on $c$:

   • Mark $c$ as visited.
   • Move to its adjacent node $e$.

4. DFS on $e$:

   • Mark $e$ as visited.
   • $e$ has no unvisited adjacent nodes.
   • Add $e$ to the topological sort list.

5. Backtrack to $c$:

   • Add $c$ to the topological sort list.

6. Backtrack to $b$:

   • Add $b$ to the topological sort list.

7. Backtrack to $d$:

   • Add $d$ to the topological sort list.

8. Remaining node $a$:

   • Perform DFS on $a$.
   • Mark $a$ as visited.
   • Move to its adjacent node $b$, but $b$ is already visited.
   • Add $a$ to the topological sort list.

Topological Sort Result

By adding the nodes to the list as we backtrack, the topological sort order is: $\text{Topological Sort: } a, d, b, c, e$

Summary

The topological order using DFS starting from node $d$ is $a, d, b, c, e$.

Would you like further details or have any questions?

Further Questions

1. How does topological sorting differ in a directed acyclic graph (DAG) compared to a graph with cycles?
2. What is the time complexity of topological sorting using DFS?
3. Can topological sort be performed on a graph that has cycles? Why or why not?
4. How can we modify the DFS algorithm to detect cycles in a graph?
5. Are there any applications of topological sorting in real-world scenarios?

Tip

When performing DFS for topological sort, it's essential to mark nodes as visited to avoid infinite loops and incorrect results.