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In a depth-first traversal of a graph G with n vertices, k edges are marked as tree edges. The number of connected components in G is
- a)k
- b)k + 1
- c)n - k - 1
- d)n - k
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
In a depth-first traversal of a graph G with n vertices, k edges are m...
Tree edges are the edges that are part of DFS tree. If there are x tree edges in a tree, then x+1 vertices in the tree. The output of DFS is a forest if the graph is disconnected. Let us see below simple example where graph is disconnected.
The above example matches with D option More Examples:
1) All vertices of Graph are connected. k must be n-1. We get number of connected components = n- k = n - (n-1) = 1
2) No vertex is connected. k must be 0. We get number of connected components = n- k = n - 0 = n
Most Upvoted Answer
In a depth-first traversal of a graph G with n vertices, k edges are m...
Tree edges are the edges that are part of DFS tree. If there are x tree edges in a tree, then x+1 vertices in the tree. The output of DFS is a forest if the graph is disconnected. Let us see below simple example where graph is disconnected.
The above example matches with D option More Examples:
1) All vertices of Graph are connected. k must be n-1. We get number of connected components = n- k = n - (n-1) = 1
2) No vertex is connected. k must be 0. We get number of connected components = n- k = n - 0 = n
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In a depth-first traversal of a graph G with n vertices, k edges are m...
Explanation:
Depth-first Traversal in a Graph:
- In a depth-first traversal of a graph, we start from a source vertex and explore as far as possible along each branch before backtracking.
Tree Edges in Depth-first Traversal:
- In a depth-first traversal of a graph with n vertices, if k edges are marked as tree edges, this means that these edges are part of the spanning tree formed during the traversal.
Number of Connected Components:
- In a depth-first traversal, the number of connected components in the graph is equal to the number of times the traversal algorithm starts from a new vertex that has not been visited before.
Calculating Number of Connected Components:
- Initially, the graph has n vertices and is considered as one connected component.
- As the depth-first traversal progresses, each time a new unvisited vertex is encountered, a new connected component is formed.
- Since k edges are marked as tree edges, they form part of the spanning tree connecting vertices in the same connected component.
- Therefore, the number of connected components in the graph after the traversal is n - k.
Correct Answer:
- Hence, the correct answer is option 'd) n - k'.
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