GATE CS 2025 Set 1 — Question 18
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Engineering Mathematics → Graph Theory (Math) → Trees & Spanning Trees
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Question
Let be any undirected graph with positive edge weights, and be a minimum spanning tree of . For any two vertices, and , let and be the shortest distances between and in and , respectively. Which ONE of the options is CORRECT for all possible and ?
Correct answer
(B) d₁(u, v) ≤ d₂(u, v)
Solution
A Minimum Spanning Tree (MST) is a subgraph of ().
- The shortest distance in is the minimum weight among all possible paths between and in the entire graph .
- The shortest distance in is the weight of the unique path between and in the tree .
- Since every path in is also a path in , the set of paths available in is a superset of the set of paths available in .
- The minimum weight over a larger set of paths (in ) must be less than or equal to the weight of any specific path from the smaller set (the path in ).
- Therefore, always holds. Option (B) is correct.
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