GATE 2026 Algorithms PYQs | GREEDY METHOD

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Question 1

G = (V, E ) is an undirected simple graph in which each edge has a distinct weight, and e is a particular edge of G. Which of the following statements about the minimum spanning trees (MSTs) of G is/are TRUE?

I. If e is the lightest edge of some cycle in G, then every MST of G includes e

II. If e is the heaviest edge of some cycle in G, then every MST of G excludes e
GATE CSE 2016,SET1 - [2Marks] (MCQ)

I only
II only
both I and II
neither I and II

Question 2

Let G be a weighted directed acyclic graph with m edges and n vertices. Given G and a source vertex s in G, which one of the following options gives the worst case time complexity of the fastest algorithm to find the lengths of shortest paths from s to all vertices that are reachable from s in G? [GATE 2026 || SET-2 MCQ || 2-mark]

Θ(m + n)
Θ(m + n log(n))
Θ(nm)
Θ(n³)

Question 3

Let 𝐺(𝑉,𝐸) be a simple, undirected, edge-weighted graph with unique edge weights. Which of the following statements about the minimum spanning trees (MST) of 𝐺 is/are true? [GATE 2026 || SET-1 MSQ || 2-mark]

In every cycle 𝐶 of 𝐺, the edge with the largest weight in 𝐶 is not in any MST
In every cycle 𝐶 of 𝐺, the edge with the smallest weight in 𝐶 is in every MST
For every vertex 𝑣 ∈𝑉, the edge with the largest weight incident on 𝑣 is not in any MST
For every vertex 𝑣 ∈𝑉, the edge with the smallest weight incident on 𝑣 is in every MST

Question 4

[Tex]\begin{aligned} &\text{Let } G(V,E) \text{ be an undirected, edge-weighted graph with integer weights. The weight of a path is the sum of the weights of the edges in that path. The length of a path is the number of edges in that path.Let } s \in V \text{ be a vertex in } G\text{. For every } u \in V \text{ and for every } k \ge 0\text{, let } d_k(u) \text{ denote the weight of a shortest path (in terms of weight) from } s \text{ to } u \text{ of length at most } k\text{. If there is no path from } s \text{ to } u \text{ of length at most } k\text{, then } d_k(u) = \infty\text{.Consider the statements:} \\ &\text{S1: For every } k \ge 0 \text{ and } u \in V,\ d_{k+1}(u) \le d_k(u). \\ &\text{S2: For every } (u,v) \in E\text{, if } (u,v) \text{ is part of a shortest} \\ &\qquad \text{path (in terms of weight) from } s \text{ to } v\text{, then for} \\ &\qquad \text{every } k \ge 0, d_k(u) \le d_k(v). \\ &\text{Which one of the following options is correct?} \end{aligned}[/Tex]
[GATE 2026 || SET-1 MCQ || 2-mark]

Only S1 is true
Only S2 is true
Both S1 and S2 are true
Neither S1 nor S2 is true

Question 5

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 d1(u,v) and d2(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?
GATE CSE 2025,SET1 - [1Marks] (MCQ)

d1(u, v) = d2(u, v)
d1(u, v) ≤ d2(u, v)
d1(u, v) ≥ d2(u, v)
d1(u, v) != d2(u, v)

Question 6

Let G be an edge-weighted undirected graph with positive edge weights. Suppose a positive constant α 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?
GATE CSE 2025,SET2 - [2Marks] (MCQ)

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

Question 7

Consider a simple undirected weighted graph G, all of whose edge weights are distinct.

Which of the following statements about the minimum spanning trees of G is/are TRUE?
GATE CSE 2022 - [2Marks] (MSQ)

The edge with the second smallest weight is always part of any minimum spanning tree of G.
One or both of the edges with the third smallest and the fourth smallest weights are part of any minimum spanning tree of G
Suppose S⊆V be such that S != ϕ and S != V. Consider the edge with the minimum weight such that one of its vertices is in S and the other in V∖S. Such an edge will always be part of any minimum spanning tree of G
G can have multiple minimum spanning trees.

Question 8

Consider the weighted undirected graph with 4 vertices, where the weight of edge {i, j} is given by the entry W ij in the matrix W.

The largest possible integer value of x, for which at least one shortest path between some pair of vertices will contain the edge with weight x is______.
GATE CSE 2016,SET1 - [2Marks] (NAT)

Question 9

Let G be a connected undirected weighted graph. Consider the following two statements.

S 1 : There exists a minimum weight edge in G which is present in every minimum spanning tree of G.

S 2 : If every edge in G has distinct weight, then G has a unique minimum spanning tree.

Which one of the following options is correct?
GATE CSE 2021,SET2 - [2Marks] (MCQ)

S 1 is true and S 2 is false.
S 1 is false and S 2 is true.
Both S 1 and S 2 are true.
Both S 1 and S 2 are false.

Question 10

Which one of the following statements is true?
GATE CSE 2021,SET1 - [2Marks] (MCQ)

⌈ diam (G) / 2⌉ < diam (G ₂ ) < diam(G)
diam(G₂) = diam(G)
diam(G) = diam(G ₂ ) <= 2 diam(G)
diam(G ₂ ) <= ⌈ diam (G) / 2 ⌉

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