GATE CS 2025 Set 2 Q.37: Minimum Spanning Trees

Let $G$ be an edge-weighted undirected graph with positive edge weights. Suppose a positive constant $\alpha$ is added to the weight of every edge.

Which ONE of the following statements is TRUE about the minimum spanning trees (MSTs) and shortest paths (SPs) in $G$ before and after the edge weight update?

Every MST remains an MST, and every SP remains an SP.

MSTs need not remain MSTs, and every SP remains an SP.

Every MST remains an MST, and SPs need not remain SPs.

MSTs need not remain MSTs, and SPs need not remain SPs.

Official answer and solution

Official answer: C
From: Official GATE 2025 answer key

Solution

Compare how adding $\alpha$ to each edge changes the cost of spanning trees (fixed number of edges $|V|-1$ ) versus paths (variable number of edges).

Minimum Spanning Trees: Every spanning tree of a connected graph $G = (V, E)$ has exactly $|V| - 1$ edges. Adding $\alpha > 0$ to every edge increases the total weight of every spanning tree by the same constant $(|V| - 1)\alpha$ (and preserves the relative order of all edge weights). Therefore, every MST remains an MST.
Shortest Paths: Two paths between $u$ and $v$ may contain different numbers of edges. For example, if a 2-edge path has weights $1 + 1 = 2$ and a 1-edge path has weight $3$ , adding $\alpha = 2$ changes their weights to $(1+2)+(1+2) = 6$ and $3+2 = 5$ , changing the shortest path. Thus SPs need not remain SPs.

Final answer

Option (C)

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 2 (CS2)
Question: 37 of 65
Section: Computer Science and Information Technology

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