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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?
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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Community Answer
In a depth-first traversal of a graph G with n vertices, k edges are m...
Explanation:

Connected components in a graph:
- In a graph, connected components are subgraphs in which every pair of vertices is connected to each other by paths.
- Each connected component is a maximally connected subgraph.

Depth-first traversal and tree edges:
- In a depth-first traversal of a graph, when we mark k edges as tree edges, these edges form a spanning tree that connects all vertices in the graph.
- Tree edges are edges that are part of the depth-first search tree.

Number of connected components:
- In a graph with n vertices, if k edges are marked as tree edges in a depth-first traversal, the number of connected components in the graph can be calculated as:
- Number of connected components = n - k
- This is because each tree edge in the depth-first search tree reduces the number of connected components by 1.

Correct answer:
- Therefore, the correct answer is option 'D', which states that the number of connected components in the graph G with n vertices and k tree edges marked in a depth-first traversal is n - k.
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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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