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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?
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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...
Understanding Depth-First Traversal
In a depth-first traversal of a graph \( G \), we explore as far as possible along each branch before backtracking. During this traversal, edges that lead to new vertices are classified as tree edges.
Key Components of the Graph
- Vertices (\( n \)): The total number of nodes in the graph.
- Edges (\( k \)): The number of edges that are considered tree edges during the traversal.
- Connected Components: A connected component is a subset of the graph where there is a path between any two vertices in that subset.
Relation Between Edges and Components
When we conduct a depth-first search (DFS):
- Each time we discover a new connected component, we start a new tree from an unvisited vertex.
- For each component, the number of edges \( k \) connects \( n - c \) vertices, where \( c \) is the number of components.
Formula Derivation
1. Tree Edges: The \( k \) tree edges connect \( k + 1 \) vertices (since \( k \) edges connect \( k + 1 \) nodes in a tree structure).
2. Connected Components: The number of connected components \( C \) can be derived from the formula:
\[
C = n - k
\]
Here, \( n \) is the total vertices, and \( k \) is the number of edges in the spanning tree.
3. Conclusion: Therefore, the number of connected components is given by:
\[
\text{Number of Components} = n - k
\]
Final Answer
Thus, the number of connected components in graph \( G \) is \( n - k \), confirming that option 'D' is correct.
In a depth-first traversal of a graph \( G \), we explore as far as possible along each branch before backtracking. During this traversal, edges that lead to new vertices are classified as tree edges.
Key Components of the Graph
- Vertices (\( n \)): The total number of nodes in the graph.
- Edges (\( k \)): The number of edges that are considered tree edges during the traversal.
- Connected Components: A connected component is a subset of the graph where there is a path between any two vertices in that subset.
Relation Between Edges and Components
When we conduct a depth-first search (DFS):
- Each time we discover a new connected component, we start a new tree from an unvisited vertex.
- For each component, the number of edges \( k \) connects \( n - c \) vertices, where \( c \) is the number of components.
Formula Derivation
1. Tree Edges: The \( k \) tree edges connect \( k + 1 \) vertices (since \( k \) edges connect \( k + 1 \) nodes in a tree structure).
2. Connected Components: The number of connected components \( C \) can be derived from the formula:
\[
C = n - k
\]
Here, \( n \) is the total vertices, and \( k \) is the number of edges in the spanning tree.
3. Conclusion: Therefore, the number of connected components is given by:
\[
\text{Number of Components} = n - k
\]
Final Answer
Thus, the number of connected components in graph \( G \) is \( n - k \), confirming that option 'D' is correct.
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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 isa)kb)k + 1c)n - k - 1d)n - kCorrect answer is option 'D'. Can you explain this answer?
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