Let's analyze each statement one by one:

1. BFS can be used to identify the vertex which is at the farthest distance from vertex $v$ in any graph, in terms of the number of edges, where vertex $v$ is the starting vertex.

   • True. BFS (Breadth-First Search) explores all vertices at the present depth level before moving on to vertices at the next depth level. This means the last vertex visited during a BFS from a starting vertex $v$ will be the one farthest from $v$ in terms of the number of edges.

2. BFS and DFS identify all the vertices reachable from the starting vertex $v$.

   • True. Both BFS and DFS will visit all vertices that are reachable from the starting vertex $v$. If a vertex is reachable, both searches will eventually visit it.

3. BFS cannot be used to check for cycles in the graph, while DFS can be used to check for cycles in the graph.

   • False. Both BFS and DFS can be used to check for cycles in a graph. In BFS, if a vertex is encountered that has already been visited and is not the immediate parent of the current vertex, it indicates the presence of a cycle. Similarly, in DFS, back edges (edges that point to an ancestor in the DFS tree) indicate cycles.

4. DFS can be used to identify the shortest distance from the starting vertex $v$ to every other vertex in the graph, in terms of the number of edges.

   • False. DFS (Depth-First Search) does not guarantee the shortest path in an unweighted graph because it explores as far as possible along each branch before backtracking. The shortest path in terms of the number of edges is guaranteed by BFS, not DFS.

Conclusion:

• True Statements:

  1. BFS can be used to identify the vertex which is at the farthest distance from vertex $v$.
  2. BFS and DFS identify all the vertices reachable from the starting vertex $v$.

• False Statements:

  1. BFS cannot be used to check for cycles in the graph while DFS can be used to check for cycles in the graph.
  2. DFS can be used to identify the shortest distance from the starting vertex $v$ to every other vertex in the graph, in terms of the number of edges.

Would you like a deeper explanation on any of these points, or have other questions?

Relative Questions:

1. How does BFS ensure finding the shortest path in an unweighted graph?
2. Can you explain how DFS detects cycles in a graph?
3. What are the key differences between BFS and DFS in terms of time complexity?
4. How does BFS handle weighted graphs differently compared to unweighted graphs?
5. What modifications to DFS would be necessary to find the shortest path in a graph?

Tip:

When dealing with unweighted graphs, always use BFS to find the shortest path. DFS might be more efficient for exploring all paths, but it doesn't guarantee the shortest one.