BFS

Use a queue and mark a vertex visited when it is enqueued, not when it is later dequeued. That prevents the same vertex being inserted many times through different parents. Maintain distance[neighbor] = distance[current] plus one on first discovery. FIFO order means every vertex at distance d is processed before any vertex at distance d plus one, so first discovery gives the shortest number of edges from the source in an unweighted graph. Keep parent pointers if an actual path is needed. For disconnected graphs, either return only source-reachable vertices or run a new BFS from every unvisited vertex, depending on the requested output. Test an isolated source, cycles, self-loops, parallel edges, and disconnected components. With an adjacency list, BFS is O(V plus E) time and O(V) space for visited, distances, and the queue. Do not claim this solves weighted shortest paths unless every edge has equal weight.