GATE CS 2025 Set 1 Q.18: Minimum Spanning Trees

Let $G$ be any undirected graph with positive edge weights, and $T$ be a minimum spanning tree of $G$ . For any two vertices, $u$ and $v$ , let $d_1(u, v)$ and $d_2(u, v)$ be the shortest distances between $u$ and $v$ in $G$ and $T$ , respectively. Which ONE of the options is CORRECT for all possible $G$ , $T$ , $u$ and $v$ ?

$d_1(u, v) = d_2(u, v)$

$d_1(u, v) \le d_2(u, v)$

$d_1(u, v) \ge d_2(u, v)$

$d_1(u, v) \ne d_2(u, v)$

Official answer and solution

Official answer: B
From: Official GATE 2025 answer key

Solution

Compare the set of paths between $u$ and $v$ in subgraph $T$ with the set of paths between $u$ and $v$ in $G$ .

Since $T$ is a spanning tree (a subgraph) of $G$ , the unique path between $u$ and $v$ in $T$ of weight $d_2(u, v)$ is also a valid path between $u$ and $v$ in $G$ . Because $d_1(u, v)$ is the minimum weight over all paths between $u$ and $v$ in $G$ , we always have:

d_1(u, v) \le d_2(u, v)

(Equality holds when $u$ and $v$ are connected by an MST edge, and strict inequality can hold when a non-tree edge provides a shortcut in $G$ .)

Final answer

Option (B)

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2025 Computer Science and Information Technology - Session 1 (CS1)
Question: 18 of 65
Section: Computer Science and Information Technology

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