True/False: In a graph G=(V,E) with unit weights (that is, w(e) = 1 for all e in E) all s-t paths have minimum length.

Question

True/False: In a graph G=(V,E) with unit weights (that is, w(e) = 1 for all e in E) all s-t paths have minimum length.

BlackTom AI · Accepted Answer

Unit edge weights mean the cost of a path equals its number of edges, so the shortest s–t paths are those with the fewest edges. That does not imply every s–t path has minimum length.

1) Relevant knowledge points:
- In an unweighted (or unit-weight) graph, the length/cost of a path is the number of edges it uses.
- A shortest s–t path is any path between s and t that minimizes that edge count.
- A graph can have many distinct s–t paths with different edge counts.

2) Step-by-step reasoning with a concrete counterexample:
- Construct a small graph: vertices {s, a, b, c, t} and edges {s–a, a–t, s–b, b–c, c–t}. All edges have weight 1.
- Consider path P1: s → a → t. Number of edges = 2. Cost(P1) = 2 × 1 = 2.
- Consider path P2: s → b → c → t. Number of edges = 3. Cost(P2) = 3 × 1 = 3.
- The minimum possible s–t cost in this graph is 2 (from P1). P2 has cost 3 > 2, so P2 is not a shortest path.

3) Intermediate calculations shown:
- Cost(P1) = edges(P1) × 1 = 2 × 1 = 2.
- Cost(P2) = edges(P2) × 1 = 3 × 1 = 3.
- Minimum cost = min(2, 3) = 2.

4) Why the correct answer is correct:
- The statement “all s–t paths have minimum length” would require every s–t path to have identical, minimal edge count. The counterexample shows distinct s–t paths can—and typically do—have different lengths, so not all are minimum.

5) Why the other option is incorrect:
- Choosing “True” would assert every s–t path is shortest in any unit-weight graph, which is false except in degenerate cases (e.g., when only one s–t path exists or when every s–t path happens to have equal length). Those are special cases, not the general rule.

6) Conclusion:
- Therefore the correct answer is False, matching the provided answer.

True/False: In a graph G=(V,E) with unit weights (that is, w(e) = 1 for all e in E) all s-t paths have minimum length.判断题

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