Build a Graph as an Adjacency List

Represent an undirected graph as a per-vertex list of neighbours. For every edge (u, v), append v to adj[u] and u to adj[v]. Neighbour lists keep insertion order so the graph is a stable, deterministic fixture for the search lessons.

Algorithm

The canonical fixture is 6 vertices [1..6] with undirected edges (1,2), (1,3), (2,4), (3,4), (4,5), (5,6) inserted in that order. The final adjacency list is {1: [2, 3], 2: [1, 4], 3: [1, 4], 4: [2, 3, 5], 5: [4, 6], 6: [5]}. This same graph drives graph-bfs, graph-dfs, and graph-shortest-path-bfs.

Basic Implementation

basic.cs

using System;
using System.Collections.Generic;
class Program {
	static void Main() {
		int[][] edges = { new int[] {1, 2}, new int[] {1, 3}, new int[] {2, 4}, new int[] {3, 4}, new int[] {4, 5}, new int[] {5, 6} };
		Dictionary<int, List<int>> adj = new Dictionary<int, List<int>>();
		foreach (int[] e in edges) {
			if (!adj.ContainsKey(e[0])) adj[e[0]] = new List<int>();
			if (!adj.ContainsKey(e[1])) adj[e[1]] = new List<int>();
			adj[e[0]].Add(e[1]);
			adj[e[1]].Add(e[0]);
		}
		List<int> keys = new List<int>(adj.Keys);
		keys.Sort();
		List<string> parts = new List<string>();
		foreach (int v in keys) {
			parts.Add(v + ": [" + string.Join(", ", adj[v]) + "]");
		}
		Console.WriteLine("{" + string.Join(", ", parts) + "}");
	}
}

Complexity

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

Implementation notes

C#: a Dictionary<int, List<int>> stores neighbours in edge order; keys are sorted before printing because Dictionary iteration order is unspecified.
The replay shows the adjacency list after each edge is added, matching the lesson spec.

adjacency list Each edge adds two directed entries, one in each direction.