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.lua

local edges = {{1, 2}, {1, 3}, {2, 4}, {3, 4}, {4, 5}, {5, 6}}
local adj = {}
for _, e in ipairs(edges) do
	local u = e[1]
	local v = e[2]
	if adj[u] == nil then adj[u] = {} end
	if adj[v] == nil then adj[v] = {} end
	adj[u][#adj[u] + 1] = v
	adj[v][#adj[v] + 1] = u
end
io.write("{")
for v = 1, 6 do
	if v > 1 then io.write(", ") end
	io.write(v .. ": [")
	for k = 1, #adj[v] do
		if k > 1 then io.write(", ") end
		io.write(tostring(adj[v][k]))
	end
	io.write("]")
end
io.write("}\n")

Complexity

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

Implementation notes

Lua: a table maps each vertex to a sequence of neighbours; vertices 1..6 are printed in order.
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.