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Let G be an undirected connected graph with distinct edge weights. Let emax be the edge with maximum weight and the edge with minimum weight. Which of the following statements is false?
- a)Every minimum spanning tree of G must contain emin
- b)If emax is in a minimum spanning tree, then its removal must disconnect G
- c)No minimum spanning tree contains emax
- d)G has a unique minimum spanning tree
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
Let G be an undirected connected graph with distinct edge weights. Let...
In kruskal’s algorithm, we pick the edges in ascending order and add them to the forest if no cycle is formed. Option A is True because first edge could never create a cycle.
The only reason for emax to be present in the minimum spanning tree could be that it is the only possible edge to cover a particular vertex in a tree since all vertices have to be present in a spanning tree by definition. Consider the image below
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Most Upvoted Answer
Let G be an undirected connected graph with distinct edge weights. Let...
Statement: G has a unique minimum spanning tree
Explanation:
To understand why statement D is false, we need to have a clear understanding of what a minimum spanning tree (MST) is and how it is constructed.
Minimum Spanning Tree:
A minimum spanning tree of a connected graph G is a subgraph of G that is a tree and connects all the vertices of G with the minimum total edge weight.
Construction of MST:
1. Start with any vertex of the graph.
2. Select the edge with the minimum weight adjacent to the current tree.
3. Add the selected edge to the tree.
4. Repeat steps 2 and 3 until all vertices are included in the tree.
Proof:
To prove statement D false, we need to provide a counterexample where there are multiple minimum spanning trees for the graph G.
Consider the following graph G:
e1 (1)
A ------------ B
| \ / |
e3 (2) \ / |
| \ / |
| X |
| / \ |
| / \ |
e2 (3) e4 (4) |
| / \ |
C ------------ D
e5 (5)
In this graph, the edge weights are given in parentheses.
Minimum Spanning Tree 1:
Let's construct a minimum spanning tree by starting with vertex A.
1. Add edge e1 (weight 1) between vertices A and B.
2. Add edge e3 (weight 2) between vertices A and C.
3. Add edge e4 (weight 4) between vertices B and D.
4. Add edge e5 (weight 5) between vertices C and D.
This gives us the minimum spanning tree 1:
e1 (1)
A ------------ B
|
e4 (4)
|
D
e5 (5)
|
C
e3 (2)
The total weight of this tree is 1 + 2 + 4 + 5 = 12.
Minimum Spanning Tree 2:
Now, let's construct another minimum spanning tree by starting with vertex B.
1. Add edge e1 (weight 1) between vertices A and B.
2. Add edge e4 (weight 4) between vertices B and D.
3. Add edge e5 (weight 5) between vertices C and D.
4. Add edge e3 (weight 2) between vertices A and C.
This gives us the minimum spanning tree 2:
A e3 (2) C
- - - - - - - - - -
| |
e1 (1) e5 (5)
| |
B D
e4 (4)
The total weight of this tree is 1 + 4 + 5 + 2 = 12, which is the same as minimum spanning tree 1.
Conclusion:
From the counterexample above, we can see that the graph G has multiple minimum spanning trees (MST 1 and MST 2). Therefore, statement D is false.
Explanation:
To understand why statement D is false, we need to have a clear understanding of what a minimum spanning tree (MST) is and how it is constructed.
Minimum Spanning Tree:
A minimum spanning tree of a connected graph G is a subgraph of G that is a tree and connects all the vertices of G with the minimum total edge weight.
Construction of MST:
1. Start with any vertex of the graph.
2. Select the edge with the minimum weight adjacent to the current tree.
3. Add the selected edge to the tree.
4. Repeat steps 2 and 3 until all vertices are included in the tree.
Proof:
To prove statement D false, we need to provide a counterexample where there are multiple minimum spanning trees for the graph G.
Consider the following graph G:
e1 (1)
A ------------ B
| \ / |
e3 (2) \ / |
| \ / |
| X |
| / \ |
| / \ |
e2 (3) e4 (4) |
| / \ |
C ------------ D
e5 (5)
In this graph, the edge weights are given in parentheses.
Minimum Spanning Tree 1:
Let's construct a minimum spanning tree by starting with vertex A.
1. Add edge e1 (weight 1) between vertices A and B.
2. Add edge e3 (weight 2) between vertices A and C.
3. Add edge e4 (weight 4) between vertices B and D.
4. Add edge e5 (weight 5) between vertices C and D.
This gives us the minimum spanning tree 1:
e1 (1)
A ------------ B
|
e4 (4)
|
D
e5 (5)
|
C
e3 (2)
The total weight of this tree is 1 + 2 + 4 + 5 = 12.
Minimum Spanning Tree 2:
Now, let's construct another minimum spanning tree by starting with vertex B.
1. Add edge e1 (weight 1) between vertices A and B.
2. Add edge e4 (weight 4) between vertices B and D.
3. Add edge e5 (weight 5) between vertices C and D.
4. Add edge e3 (weight 2) between vertices A and C.
This gives us the minimum spanning tree 2:
A e3 (2) C
- - - - - - - - - -
| |
e1 (1) e5 (5)
| |
B D
e4 (4)
The total weight of this tree is 1 + 4 + 5 + 2 = 12, which is the same as minimum spanning tree 1.
Conclusion:
From the counterexample above, we can see that the graph G has multiple minimum spanning trees (MST 1 and MST 2). Therefore, statement D is false.
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Let G be an undirected connected graph with distinct edge weights. Let emax be the edge with maximum weight and the edge with minimum weight. Which of the following statements is false?a)Every minimum spanning tree of G must contain eminb)If emax is in a minimum spanning tree, then its removal must disconnect Gc)No minimum spanning tree contains emaxd)G has a unique minimum spanning treeCorrect answer is option 'D'. Can you explain this answer?
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Let G be an undirected connected graph with distinct edge weights. Let emax be the edge with maximum weight and the edge with minimum weight. Which of the following statements is false?a)Every minimum spanning tree of G must contain eminb)If emax is in a minimum spanning tree, then its removal must disconnect Gc)No minimum spanning tree contains emaxd)G has a unique minimum spanning treeCorrect answer is option 'D'. 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 an undirected connected graph with distinct edge weights. Let emax be the edge with maximum weight and the edge with minimum weight. Which of the following statements is false?a)Every minimum spanning tree of G must contain eminb)If emax is in a minimum spanning tree, then its removal must disconnect Gc)No minimum spanning tree contains emaxd)G has a unique minimum spanning treeCorrect answer is option 'D'. 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 an undirected connected graph with distinct edge weights. Let emax be the edge with maximum weight and the edge with minimum weight. Which of the following statements is false?a)Every minimum spanning tree of G must contain eminb)If emax is in a minimum spanning tree, then its removal must disconnect Gc)No minimum spanning tree contains emaxd)G has a unique minimum spanning treeCorrect answer is option 'D'. 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 the edge with minimum weight. Which of the following statements is false?a)Every minimum spanning tree of G must contain eminb)If emax is in a minimum spanning tree, then its removal must disconnect Gc)No minimum spanning tree contains emaxd)G has a unique minimum spanning treeCorrect answer is option 'D'. 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 an undirected connected graph with distinct edge weights. Let emax be the edge with maximum weight and the edge with minimum weight. Which of the following statements is false?a)Every minimum spanning tree of G must contain eminb)If emax is in a minimum spanning tree, then its removal must disconnect Gc)No minimum spanning tree contains emaxd)G has a unique minimum spanning treeCorrect answer is option 'D'. 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 an undirected connected graph with distinct edge weights. Let emax be the edge with maximum weight and the edge with minimum weight. Which of the following statements is false?a)Every minimum spanning tree of G must contain eminb)If emax is in a minimum spanning tree, then its removal must disconnect Gc)No minimum spanning tree contains emaxd)G has a unique minimum spanning treeCorrect answer is option 'D'. Can you explain this answer?.
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