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Let G be an undirected connected graph with distinct edge weights. Let emax be the edge with maximum weight and emin the edge with minimum weight. Which of the following statements is false?
a)
Every minimum spanning tree of G must contain eminb)
No minimum spanning tree contains emax
c)
If emax is in a minimum spanning tree, then its removal must disconnect G
d)
G has a unique minimum spanning treeCorrect answer is option 'B'. Can you explain this answer?
Verified Answer
Let G be an undirected connected graph with distinct edge weights. Let...
All edges are distinct weights.
Removal of emax may not disconnect G, because other edges may cover the vertices incident with emax.
Removal of emax may not disconnect G, because other edges may cover the vertices incident with emax.
Most Upvoted Answer
Let G be an undirected connected graph with distinct edge weights. Let...
Understanding the Statements
In the context of minimum spanning trees (MSTs) in a connected undirected graph with distinct edge weights, the following statements are evaluated:
Statement A: Every minimum spanning tree of G must contain emin
- True. Since emin has the minimum weight, it is always included in any MST; omitting it would lead to a higher total weight.
Statement B: No minimum spanning tree contains emax
- False. This statement is incorrect because there are scenarios where emax can be part of an MST. If removing emax from an MST doesn’t isolate any remaining vertices, it could be included in the structure.
Statement C: If emax is in a minimum spanning tree, then its removal must disconnect G
- True. If emax is included in an MST and removed, the graph would lose connectivity because emax is the maximum weight edge and, by definition, contributes to the overall structure without creating cycles.
Statement D: G has a unique minimum spanning tree
- True. With distinct edge weights, any minimum spanning tree in G is unique. No two edges can have the same weight, ensuring that there is only one way to form the MST.
Conclusion
In summary, the only false statement is B. While all other statements hold true based on the properties of MSTs, statement B incorrectly asserts that no MST can include the maximum weight edge, emax.
In the context of minimum spanning trees (MSTs) in a connected undirected graph with distinct edge weights, the following statements are evaluated:
Statement A: Every minimum spanning tree of G must contain emin
- True. Since emin has the minimum weight, it is always included in any MST; omitting it would lead to a higher total weight.
Statement B: No minimum spanning tree contains emax
- False. This statement is incorrect because there are scenarios where emax can be part of an MST. If removing emax from an MST doesn’t isolate any remaining vertices, it could be included in the structure.
Statement C: If emax is in a minimum spanning tree, then its removal must disconnect G
- True. If emax is included in an MST and removed, the graph would lose connectivity because emax is the maximum weight edge and, by definition, contributes to the overall structure without creating cycles.
Statement D: G has a unique minimum spanning tree
- True. With distinct edge weights, any minimum spanning tree in G is unique. No two edges can have the same weight, ensuring that there is only one way to form the MST.
Conclusion
In summary, the only false statement is B. While all other statements hold true based on the properties of MSTs, statement B incorrectly asserts that no MST can include the maximum weight edge, emax.
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Let G be an undirected connected graph with distinct edge weights. Let emax be the edge with maximum weight and emin the edge with minimum weight. Which of the following statements is false?a) Every minimum spanning tree of G must contain eminb) No minimum spanning tree contains emax c) If emax is in a minimum spanning tree, then its removal must disconnect Gd) G has a unique minimum spanning treeCorrect answer is option 'B'. Can you explain this answer? for Computer Science Engineering (CSE) 2025 is part of Computer Science Engineering (CSE) preparation. The Question and answers have been prepared according to the Computer Science Engineering (CSE) exam syllabus. Information about Let G be an undirected connected graph with distinct edge weights. Let emax be the edge with maximum weight and emin the edge with minimum weight. Which of the following statements is false?a) Every minimum spanning tree of G must contain eminb) No minimum spanning tree contains emax c) If emax is in a minimum spanning tree, then its removal must disconnect Gd) G has a unique minimum spanning treeCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for Computer Science Engineering (CSE) 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let G be an undirected connected graph with distinct edge weights. Let emax be the edge with maximum weight and emin the edge with minimum weight. Which of the following statements is false?a) Every minimum spanning tree of G must contain eminb) No minimum spanning tree contains emax c) If emax is in a minimum spanning tree, then its removal must disconnect Gd) G has a unique minimum spanning treeCorrect answer is option 'B'. Can you explain this answer?.
Let G be an undirected connected graph with distinct edge weights. Let emax be the edge with maximum weight and emin the edge with minimum weight. Which of the following statements is false?a) Every minimum spanning tree of G must contain eminb) No minimum spanning tree contains emax c) If emax is in a minimum spanning tree, then its removal must disconnect Gd) G has a unique minimum spanning treeCorrect answer is option 'B'. Can you explain this answer? for Computer Science Engineering (CSE) 2025 is part of Computer Science Engineering (CSE) preparation. The Question and answers have been prepared according to the Computer Science Engineering (CSE) exam syllabus. Information about Let G be an undirected connected graph with distinct edge weights. Let emax be the edge with maximum weight and emin the edge with minimum weight. Which of the following statements is false?a) Every minimum spanning tree of G must contain eminb) No minimum spanning tree contains emax c) If emax is in a minimum spanning tree, then its removal must disconnect Gd) G has a unique minimum spanning treeCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for Computer Science Engineering (CSE) 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let G be an undirected connected graph with distinct edge weights. Let emax be the edge with maximum weight and emin the edge with minimum weight. Which of the following statements is false?a) Every minimum spanning tree of G must contain eminb) No minimum spanning tree contains emax c) If emax is in a minimum spanning tree, then its removal must disconnect Gd) G has a unique minimum spanning treeCorrect answer is option 'B'. Can you explain this answer?.
Solutions for Let G be an undirected connected graph with distinct edge weights. Let emax be the edge with maximum weight and emin the edge with minimum weight. Which of the following statements is false?a) Every minimum spanning tree of G must contain eminb) No minimum spanning tree contains emax c) If emax is in a minimum spanning tree, then its removal must disconnect Gd) G has a unique minimum spanning treeCorrect answer is option 'B'. Can you explain this answer? in English & in Hindi are available as part of our courses for Computer Science Engineering (CSE).
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