Worked Examples: Graphs, BFS & DFS

These examples are designed to step through the logical process of applying graph traversal concepts.

Example 1: BFS Trace

❓ Problem: Run BFS from A: A→{B,C}, B→{D}, C→{D,E}, D→{}, E→{}

💡 Solution: Queue: [A] → [B,C] → [C,D] → [D,E] → [E]. Order: A, B, C, D, E. Distances: A:0, B:1, C:1, D:2, E:2.

Example 2: DFS Timestamps

❓ Problem: Run DFS from A: A→{B,C}, B→{D}, C→{}, D→{C}. Record discovery/finish times.

💡 Solution: A: d=1, f=8. B: d=2, f=7. D: d=3, f=6. C: d=4, f=5. Note: C discovered via D, not directly from A.

Example 3: Edge Classification

❓ Problem: In the DFS above, what type is edge (D→C)?

💡 Solution: When D visits C, C is WHITE → Tree Edge. If C were GRAY → Back Edge (cycle). If BLACK → Forward/Cross Edge.

Example 4: Bipartite Check

❓ Problem: Is this graph bipartite? A-B, B-C, C-D, D-A, A-C

💡 Solution: Color A=Red, B=Blue, C=Red, D=Blue. Edge A-C: both Red → Not bipartite (odd cycle A-B-C-A).

Example 5: BFS vs DFS Complexity

❓ Problem: What is the time complexity of BFS and DFS?

💡 Solution: Both are $O(V + E)$. Each vertex visited once, each edge examined once.

Example 6: Adjacency Matrix vs List

❓ Problem: When to use adjacency matrix vs adjacency list?

💡 Solution: Matrix: $O(1)$ edge check, $O(n^2)$ space. Best for dense graphs. List: $O(V+E)$ space, $O(\deg(v))$ to iterate neighbors. Best for sparse graphs.