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Let G be a weighted connected undirected graph with distinct positive edge weights. If every edge weight is increased by the same value, then which of the following statements is/are TRUE?
P: Minimum spanning tree of G does not change
Q: Shortest path between any pair of vertices does not change
- P only
- Q only
- Neither P nor Q
- Both P and Q
Correct answer: P only
Solution
Adding a constant to every edge preserves the relative order of edge weights, so the MST is unchanged (P true). But shortest paths can change because a path with more edges accumulates more added constant, so Q is false. Answer: P only.
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