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.php

<?php
$edges = [[1, 2], [1, 3], [2, 4], [3, 4], [4, 5], [5, 6]];
$adj = [];
foreach ($edges as $e) {
	$u = $e[0];
	$v = $e[1];
	$adj[$u][] = $v;
	$adj[$v][] = $u;
}
ksort($adj);
$parts = [];
foreach ($adj as $v => $nbrs) {
	$parts[] = $v . ": [" . implode(", ", $nbrs) . "]";
}
echo "{" . implode(", ", $parts) . "}\n";

Complexity

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

Implementation notes

PHP: an associative array maps each vertex to a list; ksort orders keys before printing.
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.