Time Complexity of Breadth First Search (BFS) on a Graph

Breadth First Search (BFS) is a graph traversal algorithm that explores all the vertices of a graph in breadth-first order. It starts at a given vertex and visits all its neighboring vertices before moving to the next level of vertices. The time complexity of BFS depends on the number of vertices (V) and edges (E) in the graph.

1. BFS Algorithm
The BFS algorithm works by maintaining a queue to store the vertices to be visited. It starts by enqueueing the initial vertex and marks it as visited. Then, it repeatedly dequeues a vertex from the queue, visits its neighbors, and enqueues the unvisited neighbors. This process continues until the queue becomes empty.

2. Visiting a Vertex
When a vertex is dequeued from the queue, it is considered visited. The time complexity of visiting a vertex is O(1), as it involves constant time operations such as marking it as visited and performing any necessary processing.

3. Enqueuing Neighbors
When visiting a vertex, all its unvisited neighbors are enqueued. The number of neighbors of a vertex can vary depending on the graph structure. In the worst case, a vertex can have V-1 neighbors (in a complete graph). Therefore, enqueuing the neighbors takes O(V) time.

4. Overall Time Complexity
In the BFS algorithm, each vertex is visited once, and all its neighbors are enqueued once. Therefore, the total number of visits and enqueues is proportional to the number of vertices (V). Additionally, the total number of edges (E) determines the number of neighbors enqueued for all vertices.

Hence, the time complexity of BFS on a graph with V vertices and E edges is O(V + E).

5. Complexity Analysis
- Visiting each vertex takes O(1) time.
- Enqueuing all neighbors of a vertex takes O(V) time.
- Since each edge is associated with two vertices, the total number of edges is at most 2E.
- Therefore, the time complexity is O(V + 2E) = O(V + E).

Thus, the correct answer is option 'C': O(V + E).