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G = (V;E) is an undirected simple graph in which each edge has a distinct weight,and e is a particular edgeof 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 everyMST of G excludes e
II. If e is the heaviest edge of some cycle in G, then everyMST of G excludes e
- a)I only
- b)II only
- c)both I and II
- d)neither I nor II
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
G = (V;E) is an undirected simple graph in which each edge has a disti...
I. TRUE
Suppose ‘e’ does not belong to T*
Adding ‘e’ to T* creates a(unique) cycle ‘C’ in T*
Some other edge in cycle ‘C’, say ‘f’ has exactly one end point in ‘s’
T = T* ?{e}–f is also spanning Tree and since Ce < Cf
So,cost(T) < cost(T*)
Which is contridicts minimality of T*.
Suppose ‘e’ does not belong to T*
Adding ‘e’ to T* creates a(unique) cycle ‘C’ in T*
Some other edge in cycle ‘C’, say ‘f’ has exactly one end point in ‘s’
T = T* ?{e}–f is also spanning Tree and since Ce < Cf
So,cost(T) < cost(T*)
Which is contridicts minimality of T*.
II. TRUE
Suppose ‘f’ belongs to T*
Deleting ‘f’ from T* disconnects T* and let ‘s’ be one side of cut.
Suppose ‘f’ belongs to T*
Deleting ‘f’ from T* disconnects T* and let ‘s’ be one side of cut.
Some other edge in cycle ‘C’ say ‘e’ has exactly one end point in S
T = T*
{e} – {f} is also a spanning Tree.
Since Ce < Cf so cost(T) < cost(T*)
Which is contradictory to minimality of T*
T = T*
{e} – {f} is also a spanning Tree.Since Ce < Cf so cost(T) < cost(T*)
Which is contradictory to minimality of T*
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G = (V;E) is an undirected simple graph in which each edge has a distinct weight,and e is a particular edgeof 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 eII. If e is the heaviest edge of some cycle in G, then everyMST of G excludes ea)I onlyb)II onlyc)both I and IId)neither I nor IICorrect answer is option 'B'. Can you explain this answer?
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G = (V;E) is an undirected simple graph in which each edge has a distinct weight,and e is a particular edgeof 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 eII. If e is the heaviest edge of some cycle in G, then everyMST of G excludes ea)I onlyb)II onlyc)both I and IId)neither I nor IICorrect answer is option 'B'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about G = (V;E) is an undirected simple graph in which each edge has a distinct weight,and e is a particular edgeof 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 eII. If e is the heaviest edge of some cycle in G, then everyMST of G excludes ea)I onlyb)II onlyc)both I and IId)neither I nor IICorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for G = (V;E) is an undirected simple graph in which each edge has a distinct weight,and e is a particular edgeof 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 eII. If e is the heaviest edge of some cycle in G, then everyMST of G excludes ea)I onlyb)II onlyc)both I and IId)neither I nor IICorrect answer is option 'B'. Can you explain this answer?.
G = (V;E) is an undirected simple graph in which each edge has a distinct weight,and e is a particular edgeof 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 eII. If e is the heaviest edge of some cycle in G, then everyMST of G excludes ea)I onlyb)II onlyc)both I and IId)neither I nor IICorrect answer is option 'B'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about G = (V;E) is an undirected simple graph in which each edge has a distinct weight,and e is a particular edgeof 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 eII. If e is the heaviest edge of some cycle in G, then everyMST of G excludes ea)I onlyb)II onlyc)both I and IId)neither I nor IICorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for G = (V;E) is an undirected simple graph in which each edge has a distinct weight,and e is a particular edgeof 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 eII. If e is the heaviest edge of some cycle in G, then everyMST of G excludes ea)I onlyb)II onlyc)both I and IId)neither I nor IICorrect answer is option 'B'. Can you explain this answer?.
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