Breadth-First Search

From a start vertex, explore the graph layer by layer. Use a queue and a "visited" set. Dequeue a vertex, visit it, enqueue all unvisited neighbours.

Algorithm

Canonical adjacency list:

1 -> [2, 3]
2 -> [1, 4]
3 -> [1, 4]
4 -> [2, 3, 5]
5 -> [4, 6]
6 -> [5]

Starting at vertex 1, the BFS visit order is [1, 2, 3, 4, 5, 6].

Basic Implementation

basic.cs

using System;
using System.Collections.Generic;

class Program {
	static void Main() {
		Dictionary<int, List<int>> adj = new Dictionary<int, List<int>>();
		adj[1] = new List<int> { 2, 3 };
		adj[2] = new List<int> { 1, 4 };
		adj[3] = new List<int> { 1, 4 };
		adj[4] = new List<int> { 2, 3, 5 };
		adj[5] = new List<int> { 4, 6 };
		adj[6] = new List<int> { 5 };
		int start = 1;
		HashSet<int> visited = new HashSet<int>();
		visited.Add(start);
		Queue<int> queue = new Queue<int>();
		queue.Enqueue(start);
		List<int> order = new List<int>();
		while (queue.Count > 0) {
			int v = queue.Dequeue();
			order.Add(v);
			List<int> neighbours = adj[v];
			for (int i = 0; i < neighbours.Count; i++) {
				int nb = neighbours[i];
				if (!visited.Contains(nb)) {
					visited.Add(nb);
					queue.Enqueue(nb);
				}
			}
		}
		Console.WriteLine("[" + string.Join(", ", order) + "]");
	}
}

Complexity

Time: O(V + E)
Space: O(V)

Implementation notes

C#: Dictionary<int, List<int>> is the idiomatic adjacency list and keeps the vertex iteration shape visible; the lesson keeps the queue as a plain Queue<int> rather than reaching for Channel<T> or any async wrapper.
A HashSet<int> is the explicit "visited" set; the queue is a plain Queue<int> with Dequeue and Enqueue.
The replay prints the dequeued vertex, the queue, the visited set, and the running visit order each frame.

queue A `Queue<int>` with `Enqueue` / `Dequeue` implements FIFO without hiding the iteration shape.

visited-before-enqueue Mark a vertex visited before pushing it onto the queue. Keeps the queue size bounded by V.