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Let G be a weighted undirected graph and e be an edge with maximum weight in G. Suppose there is a minimum weight spanning tree in G containing the edge e. Which of the following statements is always TRUE?
- a)There exists a cutset in G having all edges of maximum weight.
- b)There exists a cycle in G having all edges of maximum weight
- c)Edge e cannot be contained in a cycle.
- d)All edges in G have the same weight
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
Let G be a weighted undirected graph and e be an edge with maximum wei...
Background : Given a connected and undirected graph, a spanning tree of that graph is a subgraph that is a tree and connects all the vertices together.
- A spanning tree has exactly V - 1 edges.
- A single graph can have many different spanning trees. A minimum spanning tree (MST) or minimum weight spanning tree for a weighted, connected and undirected graph is a spanning tree with weight less than or equal to the weight of every other spanning tree. The weight of a spanning tree is the sum of weights given to each edge of the spanning tree.
- There can be more that one possible spanning trees of a graph. For example, the graph in this question has 6 possible spanning trees.
- MST has lightest edge of every cutset. Remember Prim's algorithm which basically picks the lightest edge from every cutset.
Choices of this question : a) There exists a cutset in G having all edges of maximum weight : This is correct. If there is a heaviest edge in MST, then there exist a cut with all edges with weight equal to heavies edge. See point 4 discussed in above background.
b) There exists a cycle in G having all edges of maximum weight : Not always TRUE. The cutset of heaviest edge may contain only one edge. In fact there may be overall one edge of heavies weight which is part of MST (Consider a graph with two components which are connected by only one edge and this edge is the heavies) c) Edge e cannot be contained in a cycle. Not Always True. The curset may form cycle with other edges. d) All edges in G have the same weight Not always True. As discussed in option b, there can be only one edge of heaviest weight.
Most Upvoted Answer
Let G be a weighted undirected graph and e be an edge with maximum wei...
Explanation:
To understand why option A is always true, let's consider the scenario where there is a minimum weight spanning tree (MST) containing the edge e.
1. Definition of a Cutset:
A cutset in a graph is a set of edges that, when removed, disconnects the graph into two or more components.
2. Properties of an MST:
- An MST is a connected subgraph that includes all the vertices of the original graph, with the minimum total weight.
- In an MST, there is exactly one path between any two vertices.
- An MST does not contain any cycles.
3. Proof for Option A:
- Suppose there exists a minimum weight spanning tree (MST) containing the edge e.
- If e is the edge with the maximum weight in the graph, then it must also be the edge with the maximum weight in the MST.
- Now, consider removing the edge e from the MST. This will disconnect the MST into two or more components.
- Since e is the edge with the maximum weight, the cutset that disconnects the MST will contain all the edges of maximum weight.
- Therefore, there exists a cutset in G having all edges of maximum weight.
4. Explanation for Other Options:
- Option B: There may or may not exist a cycle in G containing all edges of maximum weight. It depends on the specific graph and its weights. So, this option is not always true.
- Option C: This statement contradicts the assumption that there is a minimum weight spanning tree containing the edge e. So, this option is not always true.
- Option D: This statement is not related to the given scenario and does not provide any information about the existence of a cutset. So, this option is not always true.
Conclusion:
Based on the properties of an MST and the given assumptions, the statement in option A is always true. There will always exist a cutset in G having all edges of maximum weight.
To understand why option A is always true, let's consider the scenario where there is a minimum weight spanning tree (MST) containing the edge e.
1. Definition of a Cutset:
A cutset in a graph is a set of edges that, when removed, disconnects the graph into two or more components.
2. Properties of an MST:
- An MST is a connected subgraph that includes all the vertices of the original graph, with the minimum total weight.
- In an MST, there is exactly one path between any two vertices.
- An MST does not contain any cycles.
3. Proof for Option A:
- Suppose there exists a minimum weight spanning tree (MST) containing the edge e.
- If e is the edge with the maximum weight in the graph, then it must also be the edge with the maximum weight in the MST.
- Now, consider removing the edge e from the MST. This will disconnect the MST into two or more components.
- Since e is the edge with the maximum weight, the cutset that disconnects the MST will contain all the edges of maximum weight.
- Therefore, there exists a cutset in G having all edges of maximum weight.
4. Explanation for Other Options:
- Option B: There may or may not exist a cycle in G containing all edges of maximum weight. It depends on the specific graph and its weights. So, this option is not always true.
- Option C: This statement contradicts the assumption that there is a minimum weight spanning tree containing the edge e. So, this option is not always true.
- Option D: This statement is not related to the given scenario and does not provide any information about the existence of a cutset. So, this option is not always true.
Conclusion:
Based on the properties of an MST and the given assumptions, the statement in option A is always true. There will always exist a cutset in G having all edges of maximum weight.
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Let G be a weighted undirected graph and e be an edge with maximum weight in G. Suppose there is a minimum weight spanning tree in G containing the edge e. Which of the following statements is always TRUE?a)There exists a cutset in G having all edges of maximum weight.b)There exists a cycle in G having all edges of maximum weightc)Edge e cannot be contained in a cycle.d)All edges in G have the same weightCorrect answer is option 'A'. Can you explain this answer?
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