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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...
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.
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Community Answer
Let G be a weighted undirected graph and e be an edge with maximum wei...
Explanation:
Minimum Weight Spanning Tree:
- A minimum weight spanning tree is a subgraph that is a tree which connects all the vertices in a graph with the minimum possible total edge weight.
Maximum Weight Edge:
- The edge with the maximum weight in a graph is denoted as 'e'.
Statement:
- The statement says that if there is a minimum weight spanning tree containing the edge 'e', then there exists a cutset in the graph having all edges of maximum weight.
Explanation:
- When the minimum weight spanning tree contains the edge with the maximum weight, it means that all other edges in the tree have weights less than 'e'.
- To disconnect the graph into two components, a cutset needs to be identified, and in this case, the cutset will consist of all edges of maximum weight, including the edge 'e'.
- This guarantees that removing these edges will disconnect the graph and separate it into two components.
Conclusion:
- Therefore, the correct statement is option 'A' which states that there exists a cutset in the graph having all edges of maximum weight when a minimum weight spanning tree containing the edge 'e' exists.
Minimum Weight Spanning Tree:
- A minimum weight spanning tree is a subgraph that is a tree which connects all the vertices in a graph with the minimum possible total edge weight.
Maximum Weight Edge:
- The edge with the maximum weight in a graph is denoted as 'e'.
Statement:
- The statement says that if there is a minimum weight spanning tree containing the edge 'e', then there exists a cutset in the graph having all edges of maximum weight.
Explanation:
- When the minimum weight spanning tree contains the edge with the maximum weight, it means that all other edges in the tree have weights less than 'e'.
- To disconnect the graph into two components, a cutset needs to be identified, and in this case, the cutset will consist of all edges of maximum weight, including the edge 'e'.
- This guarantees that removing these edges will disconnect the graph and separate it into two components.
Conclusion:
- Therefore, the correct statement is option 'A' which states that there exists a cutset in the graph having all edges of maximum weight when a minimum weight spanning tree containing the edge 'e' exists.
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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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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? for Computer Science Engineering (CSE) 2024 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 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? covers all topics & solutions for Computer Science Engineering (CSE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 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?.
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? for Computer Science Engineering (CSE) 2024 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 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? covers all topics & solutions for Computer Science Engineering (CSE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 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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