A complete graph is one where every two vertices are connected with an edge. Suppose we run BFS on a complete graph G(V,E) starting from some vertex s in V . What would the BFS layers look like?

Question

A complete graph is one where every two vertices are connected with an edge. Suppose we run BFS on a complete graph G(V,E) starting from some vertex s in V . What would the BFS layers look like?

BlackTom AI · Accepted Answer

In a complete graph Kn every pair of distinct vertices is adjacent, so the distance between any two distinct vertices is 1. That single fact determines the BFS layering from any start vertex s.

1) Relevant knowledge points and concepts
- Complete graph: every vertex pair has an edge.
- BFS layers Li contain vertices at distance i from the source s (L0 = {s}).
- BFS discovers vertices in nondecreasing order of distance from s; once a vertex is discovered it is assigned to the appropriate layer.

2) Step-by-step BFS reasoning
- Start: mark s discovered and put it in L0 and in the BFS queue.
- Dequeue s and examine all its neighbors. In a complete graph, every other vertex v ∈ V \ {s} is a neighbor of s.
- For each such vertex v that is undiscovered, BFS sets distance(v) = distance(s) + 1 = 1, places v in L1, marks discovered, and enqueues v.
- After processing s, every vertex except s is discovered and placed in L1. No undiscovered vertices remain to form L2 or beyond.
- When BFS later dequeues a vertex v ∈ L1 and scans its neighbors, all neighbors are already discovered (either s or other L1 vertices), so no new vertices are added.

3) Intermediate calculation/detail
- If |V| = n, then |L0| = 1 and |L1| = n − 1. Distances: distance(s,s)=0; distance(s,v)=1 for all v ≠ s.

4) Why the correct answer is correct
- Because completeness implies direct adjacency from s to every other vertex, all other vertices are at distance 1; BFS therefore places them all into L1 immediately. There cannot be any vertices at distance ≥2.

5) Why other plausible answers are wrong
- Any answer asserting additional layers (L2, L3, etc.) would imply some vertex is at distance ≥2 from s, contradicting completeness. Any answer putting some other distribution (e.g., multiple layers beyond L1) is impossible.

6) Conclusion
- Therefore the BFS layers are L0 = {s} and L1 = V − {s}, which matches the provided answer.

A complete graph is one where every two vertices are connected with an edge. Suppose we run BFS on a complete graph G(V,E) starting from some vertex s in V . What would the BFS layers look like?单项选择题

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