There are 24 unique breadth-first orderings possible on the undirected graph with 5 nodes.
Counting Unique Orderings:
- Starting Node: There are 5 choices for the starting node.
- First Level: The first level always contains only the starting node.
- Second Level: The second level can contain 2, 3, or 4 nodes, depending on the graph's structure.
- Third Level: The third level typically contains the remaining nodes.
Case Analysis:
- Case 1: 2 Nodes in Second Level:
- 3 choices for the first node in the second level.
- 2 choices for the second node in the second level.
- Remaining nodes form the third level (fixed order).
- Total orderings: 5 (starting nodes) * 3 * 2 = 30
- Case 2: 3 Nodes in Second Level:
- 4 choices for the first node in the second level.
- 3 * 2 / 2 (order doesn't matter) choices for the remaining two nodes in the second level.
- Remaining node forms the third level (fixed order).
- Total orderings: 5 * 4 * 3 = 60
- Case 3: 4 Nodes in Second Level:
- 5 choices for the node in the third level.
- Remaining nodes form the second level (order doesn't matter).
- Total orderings: 5
Total Unique Orderings:
- 30 (Case 1) + 60 (Case 2) + 5 (Case 3) = 95
However, we've overcounted due to symmetry in undirected graphs:
- Each ordering is counted twice (once for each starting node in a pair).
- Correct the count by dividing by 2: 95 / 2 = 47.5
Rounding to a Whole Number: 48