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
Answer: (D)
Explanation: 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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