Problem 32 - Discrete Mathematics with Applications

Question

Using the DFS method, construct a spanning tree for each graph with the given adjacency list.

Step-by-Step Solution

Answer

Starting from vertex 1, perform DFS traversal and visit connected vertices: visit vertex 2 and add edge (1,2) to the spanning tree. Move to vertex 2, visit vertex 3 and add edge (2,3) to the spanning tree. Move to vertex 3, visit vertex 4 and add edge (3,4) to the spanning tree. The final spanning tree is: ``` 1---2 | 3 | 4 ``` With edge set: {(1,2), (2,3), (3,4)}.

Step 1: Choose the starting vertex

Choose any vertex as the starting point for the DFS algorithm. In this example, we will choose vertex 1 as our starting vertex.

Step 2: Start DFS traversal

Starting from vertex 1, we will perform the DFS traversal, visiting each connected vertex following the depths of the graph before backtracking.

Step 3: DFS traversal from vertex 1

Begin from vertex 1, mark it as visited, and visit the next connected unvisited vertex. In our adjacency list, vertex 2 is connected to vertex 1. So we will visit vertex 2 and add the edge (1,2) to our spanning tree.

Step 4: DFS traversal from vertex 2

Then move from vertex 1 to vertex 2, mark it as visited, and visit the next connected unvisited vertex. In this case, vertex 3 is connected to vertex 2. So we will visit vertex 3 and add the edge (2,3) to our spanning tree.

Step 5: DFS traversal from vertex 3

Move from vertex 2 to vertex 3, mark it as visited, and visit the next connected unvisited vertex. Vertex 4 is connected to vertex 3. We will visit vertex 4 and add the edge (3,4) to our spanning tree.

Step 6: Complete DFS traversal

Since we have visited all the vertices, our DFS traversal is complete, and we have obtained the following spanning tree: ``` 1---2 | 3 | 4 ``` The edge set of this spanning tree is {(1,2), (2,3), (3,4)}, and we have successfully constructed a spanning tree for the given graph using the DFS method.