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If all the edge weights of an undirected graph are positive, then any subset of edges that connects all the vertices and has minimum total weight is a
- a)Hamiltonian cycle
- b)grid
- c)hypercube
- d)tree
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
If all the edge weights of an undirected graph are positive, then any ...
As here we want subset of edges that connects all the vertices and has minimum total weight i.e. Minimum Spanning Tree Option A - includes cycle, so may or may not connect all edges. Option B - has no relevance to this question. Option C - includes cycle, so may or may not connect all edges. Related
Most Upvoted Answer
If all the edge weights of an undirected graph are positive, then any ...
Explanation:
In an undirected graph with positive edge weights, the minimum spanning tree (MST) is a subset of edges that connects all the vertices with the minimum total weight. The minimum spanning tree is a tree that spans all the vertices of the graph and has the minimum sum of edge weights.
Properties of Minimum Spanning Tree (MST):
1. A minimum spanning tree is always a tree that connects all the vertices.
2. The total weight of the minimum spanning tree is the minimum among all possible spanning trees.
3. The minimum spanning tree may not be unique, but its weight is unique.
Proof:
To prove that the subset of edges that connects all the vertices with the minimum total weight is a tree, we need to show that it satisfies the properties of a tree.
Connectedness:
Since the subset of edges connects all the vertices, it is a connected graph.
No Cycles:
If there is a cycle in the subset of edges, it means we can remove one edge from the cycle to obtain a new spanning tree with a smaller total weight. This contradicts the assumption that the subset has the minimum total weight. Therefore, the subset of edges does not contain any cycles.
Acyclic and Connected:
A graph that is acyclic and connected is known as a tree. Therefore, the subset of edges that connects all the vertices with the minimum total weight is a tree.
Conclusion:
Since the subset of edges that connects all the vertices with the minimum total weight satisfies the properties of a tree, it is a tree. Hence, the correct answer is option 'D' - tree.
In an undirected graph with positive edge weights, the minimum spanning tree (MST) is a subset of edges that connects all the vertices with the minimum total weight. The minimum spanning tree is a tree that spans all the vertices of the graph and has the minimum sum of edge weights.
Properties of Minimum Spanning Tree (MST):
1. A minimum spanning tree is always a tree that connects all the vertices.
2. The total weight of the minimum spanning tree is the minimum among all possible spanning trees.
3. The minimum spanning tree may not be unique, but its weight is unique.
Proof:
To prove that the subset of edges that connects all the vertices with the minimum total weight is a tree, we need to show that it satisfies the properties of a tree.
Connectedness:
Since the subset of edges connects all the vertices, it is a connected graph.
No Cycles:
If there is a cycle in the subset of edges, it means we can remove one edge from the cycle to obtain a new spanning tree with a smaller total weight. This contradicts the assumption that the subset has the minimum total weight. Therefore, the subset of edges does not contain any cycles.
Acyclic and Connected:
A graph that is acyclic and connected is known as a tree. Therefore, the subset of edges that connects all the vertices with the minimum total weight is a tree.
Conclusion:
Since the subset of edges that connects all the vertices with the minimum total weight satisfies the properties of a tree, it is a tree. Hence, the correct answer is option 'D' - tree.
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