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Let G = (V, E) be any connected undirected edge-weighted graph. The weights of the edges in E are positive any distinct. Consider the following statements:
I. Minimum Spanning Tree of G is always unique.
II. Shortest path between any two vertices of G is always unique.
Q. Which of the above statements is/are necessarily true?
- a)I only
- b)II only
- c)both I and II
- d)neither I and II
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
Let G = (V, E) be any connected undirected edge-weighted graph. The we...
I. Minimum Spanning Tree of G is always unique - MST will wlways be distinct if the edges are unique so Correct II. Shortest path between any two vertices of G is always unique - Shortest path between any two vertices can be same so incorrect Therefore, option A is correct
Alternate solution: We know that minimum spanning tree of a graph is always unique if all the weight are distinct, so statement 1 is correct. Now statement 2 , this might not be
Alternate solution: We know that minimum spanning tree of a graph is always unique if all the weight are distinct, so statement 1 is correct. Now statement 2 , this might not be
true in all cases. Consider the graph.
There are two shortest paths from a to b (one is direct and other via node c) So statement 2 is false Hence option a is correct.
Most Upvoted Answer
Let G = (V, E) be any connected undirected edge-weighted graph. The we...
Minimum Spanning Tree (MST) is always unique:
- The statement I is necessarily true. In a connected undirected graph, the minimum spanning tree is unique if all edge weights are distinct.
- If there are multiple MSTs possible, it means there are multiple ways to connect all vertices with minimum total edge weight, which contradicts the uniqueness of MST.
Shortest path between any two vertices is not always unique:
- The statement II is not necessarily true. While the shortest path between two vertices may be unique in some cases, it can also be non-unique.
- In cases where there are multiple paths with the same minimum total weight between two vertices, the shortest path is not unique.
- This can happen when there are multiple edges with the same weight between two vertices or when there are alternative paths with different weights but the same total weight.
Therefore, the correct answer is option 'A' (I only) as the uniqueness of MST is guaranteed in a connected undirected graph with distinct edge weights, while the uniqueness of the shortest path between any two vertices is not always guaranteed.
- The statement I is necessarily true. In a connected undirected graph, the minimum spanning tree is unique if all edge weights are distinct.
- If there are multiple MSTs possible, it means there are multiple ways to connect all vertices with minimum total edge weight, which contradicts the uniqueness of MST.
Shortest path between any two vertices is not always unique:
- The statement II is not necessarily true. While the shortest path between two vertices may be unique in some cases, it can also be non-unique.
- In cases where there are multiple paths with the same minimum total weight between two vertices, the shortest path is not unique.
- This can happen when there are multiple edges with the same weight between two vertices or when there are alternative paths with different weights but the same total weight.
Therefore, the correct answer is option 'A' (I only) as the uniqueness of MST is guaranteed in a connected undirected graph with distinct edge weights, while the uniqueness of the shortest path between any two vertices is not always guaranteed.
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Let G = (V, E) be any connected undirected edge-weighted graph. The weights of the edges in E are positive any distinct. Consider the following statements:I. Minimum Spanning Tree of G is always unique.II. Shortest path between any two vertices of G is always unique.Q. Which of the above statements is/are necessarily true?a)I onlyb)II onlyc)both I and IId)neither I and IICorrect answer is option 'A'. Can you explain this answer?
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ample number of questions to practice Let G = (V, E) be any connected undirected edge-weighted graph. The weights of the edges in E are positive any distinct. Consider the following statements:I. Minimum Spanning Tree of G is always unique.II. Shortest path between any two vertices of G is always unique.Q. Which of the above statements is/are necessarily true?a)I onlyb)II onlyc)both I and IId)neither I and IICorrect answer is option 'A'. Can you explain this answer? tests, examples and also practice GATE tests.
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