How is a breadth-first search implemented in a graph?

A breadth-first search in a graph is implemented by using a queue to visit and explore each vertex's neighbours before moving on to the next level.

Breadth-first search (BFS) is a strategy for searching in a graph when breadth is prioritised over depth. This means that all the vertices of the current level are visited before moving on to the next level. The algorithm starts at the root (or any arbitrary node of a graph, sometimes referred to as a 'search key') and explores the neighbour nodes at the present depth prior to moving on to nodes at the next depth level.

To implement BFS, a queue data structure is used. The algorithm begins by enqueuing the starting vertex and marking it as visited. Then, it enters a loop where it dequeues a vertex and examines it. If the vertex is the goal, the algorithm stops. Otherwise, it enqueues all the unvisited neighbours of the vertex and marks them as visited. This process continues until all vertices have been visited or the goal has been found.

The pseudocode for BFS is as follows:

1. Create an empty queue and enqueue the starting vertex.
2. Mark the starting vertex as visited.
3. While the queue is not empty, repeat the following steps:
a. Dequeue a vertex from the queue. Let's call this vertex V.
b. For each unvisited neighbour of V, enqueue the neighbour and mark it as visited.

The time complexity of BFS is O(V + E), where V is the number of vertices and E is the number of edges in the graph. This is because each vertex and each edge will be explored in the worst case. BFS is useful for finding the shortest path in a graph (in terms of the number of edges) and for checking if a graph is connected or not. It's also used in many algorithms in graph theory, including Dijkstra's algorithm for finding the shortest path in a weighted graph.