Graphs

Build a Graph as an Adjacency List

Represent an undirected graph as a map from each vertex to its 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.py 
edges = [(1, 2), (1, 3), (2, 4), (3, 4), (4, 5), (5, 6)]
adj = {}
for u, v in edges:
    adj.setdefault(u, []).append(v)
    adj.setdefault(v, []).append(u)
print(adj)

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

  • Python: dict.setdefault(key, []).append(value) builds the hash-of-list in one pass while preserving insertion order.
  • The replay shows the adjacency map after each edge is added, matching the lesson spec.
adjacency list Each edge adds two directed entries, one in each direction.