GATE CS 2017 Set 1 — Question 26
Go beyond PYQs with Success TrackerAI-powered personalised practice and doubt support. Unlimited practice on eligible plans; AI usage limits apply.MCQ+2 / -0.67MediumTrees & Spanning TreesGraph Theory (Math)Engineering MathematicsGraph AlgorithmsAlgorithms
Algorithms → Graph Algorithms → Trees & Spanning Trees
Last updated
Question
Let be any connected undirected edge-weighted graph. The weights of the edges in are positive and distinct. Consider the following statements:(I) Minimum Spanning Tree of is always unique.
(II) Shortest path between any two vertices of is always unique.Which of the above statements is/are necessarily true?
(II) Shortest path between any two vertices of is always unique.Which of the above statements is/are necessarily true?
Correct answer
(A) (I) only
Solution
Statement (I): If all edge weights in a connected undirected graph are distinct, then the Minimum Spanning Tree (MST) is unique. This is a standard theorem in graph theory. Thus, (I) is true.Statement (II): Distinct edge weights do not guarantee unique shortest paths. Consider a triangle graph with vertices A, B, C and edges:
The shortest path from A to B can be the direct edge with cost 5.
Alternatively, the path has cost .
Since there are two paths with the same minimum cost, the shortest path is not unique. Thus, (II) is false.Therefore, only statement (I) is necessarily true.
- with weight 5
- with weight 3
- with weight 2
The shortest path from A to B can be the direct edge with cost 5.
Alternatively, the path has cost .
Since there are two paths with the same minimum cost, the shortest path is not unique. Thus, (II) is false.Therefore, only statement (I) is necessarily true.
Continue learning with Success Tracker
A step still unclear? Work through it with support
Use Success Tracker to ask about the reasoning, then try another GATE CS question to check your understanding.
AI-powered practice· Unlimited practice on eligible plans
- PYQs with solutions
- Attempt available previous-year questions, then compare your reasoning with the worked solution. Coverage varies by stream.
- Practice that adapts
- Choose a topic, work on weaker areas and bookmark questions to revisit. Your attempts feed your progress tracking.
- AI doubt support
- Ask follow-up questions about a step or concept while practising, instead of stopping at the final answer.
Unlimited practice is available on eligible plans. Free practice and AI usage have limits; check the current plan allowances before choosing.
This page stays readable without an account. AI responses can be wrong; check them against the solution and source material.
More questions on Graph Algorithms
2026 Set 2 Q3A day can only be cloudy or sunny. The probability of a day being cloudy is , independent of…2026 Set 2 Q5‘When it is raining, peacocks dance.’ Based only on this sentence, which one of the following…2026 Set 2 Q8Figures (i) and (ii) represent intercity highway systems. The black dots represent cities and the…2026 Set 1 Q10An unbiased six-faced dice whose faces are marked with numbers 1, 2, 3, 4, 5, and 6 is rolled twice…2026 Set 2 Q10An unbiased six-faced dice whose faces are marked with numbers 1, 2, 3, 4, 5, and 6 is rolled twice…