Graphs

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

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";

The graph fixture is pinned once, then the adjacency list writes each undirected edge in both directions.

Step 1 - Pinned graph fixture

The six vertices and six undirected edges are the shared fixture for BFS and DFS.

Graph with edges (1,2), (1,3), (2,4), (3,4), (4,5), (5,6).123456

Step 2 - Final adjacency list

Each row lists neighbours in the same insertion order used by the lesson.

Adjacency list after all six undirected edges are inserted.vertexneighbours1[2, 3]2[1, 4]3[1, 4]4[2, 3, 5]5[4, 6]6[5]

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.