We are tasked with finding the shortest path from node 1 to node 6 using the Breadth-First Search (BFS) algorithm. Below is the process step-by-step, along with the explanation of the queue structure and node selection at each step.

Adjacency List of the Graph

• Node 1 → [2, 3]
• Node 2 → [1, 4, 5]
• Node 3 → [1, 6]
• Node 4 → [2]
• Node 5 → [2, 6]
• Node 6 → [3, 5]

Step-by-Step BFS Process

1. Initialization:

   • Start BFS at node 1 (source).
   • Use a queue to maintain the order of nodes to visit.
   • Maintain a visited set to keep track of visited nodes and prevent cycles.
   • Maintain a parent map to reconstruct the shortest path after BFS finishes.

   Queue: [1]
   Visited: {1}
   Parent: {1: None}

2. Step 1 - Process Node 1:

   • Dequeue node 1.
   • Visit its neighbors: 2 and 3.
   • Add unvisited neighbors to the queue and mark them as visited. Update their parent as 1.

   Queue: [2, 3]
   Visited: {1, 2, 3}
   Parent: {1: None, 2: 1, 3: 1}

3. Step 2 - Process Node 2:

   • Dequeue node 2.
   • Visit its neighbors: 1, 4, and 5.
   • Node 1 is already visited, so ignore it.
   • Add unvisited neighbors 4 and 5 to the queue, mark them as visited, and set their parent as 2.

   Queue: [3, 4, 5]
   Visited: {1, 2, 3, 4, 5}
   Parent: {1: None, 2: 1, 3: 1, 4: 2, 5: 2}

4. Step 3 - Process Node 3:

   • Dequeue node 3.
   • Visit its neighbors: 1 and 6.
   • Node 1 is already visited, so ignore it.
   • Add unvisited neighbor 6 to the queue, mark it as visited, and set its parent as 3.

   Queue: [4, 5, 6]
   Visited: {1, 2, 3, 4, 5, 6}
   Parent: {1: None, 2: 1, 3: 1, 4: 2, 5: 2, 6: 3}

5. Step 4 - Process Node 4:

   • Dequeue node 4.
   • Visit its neighbors: 2.
   • Node 2 is already visited, so do nothing.

   Queue: [5, 6]

6. Step 5 - Process Node 5:

   • Dequeue node 5.
   • Visit its neighbors: 2 and 6.
   • Both neighbors are already visited, so do nothing.

   Queue: [6]

7. Step 6 - Process Node 6:

   • Dequeue node 6.
   • Node 6 is the destination, so BFS stops here.

Shortest Path Reconstruction

To reconstruct the shortest path, we use the Parent map and backtrack from node 6 to node 1:

• Start at node 6 → Parent is 3.
• Move to node 3 → Parent is 1.
• Move to node 1 → Source node reached.

Shortest Path: 1 → 3 → 6

Length of the Path

The length of the shortest path is 2 edges.

Summary of BFS Queue Operations

1. Initially, enqueue the source node 1.
2. Repeatedly dequeue a node, visit its neighbors, and enqueue unvisited neighbors.
3. Stop when the destination node (6) is dequeued.

Final BFS State

• Visited: {1, 2, 3, 4, 5, 6}
• Parent Map: {1: None, 2: 1, 3: 1, 4: 2, 5: 2, 6: 3}

Do you need more explanation or any part clarified further? Here are some follow-up questions you might consider:

1. How does BFS guarantee finding the shortest path in an unweighted graph?
2. How does BFS differ from Depth-First Search (DFS)?
3. How can BFS be implemented using a queue in code?
4. What would happen if the graph were weighted? Can BFS still be used?
5. How can you modify BFS to also return all possible paths between two nodes?

Tip: Always use a queue for BFS and a visited set to avoid infinite loops in graphs with cycles.