# Algorithms | Graph Traversals | Question 12

• Last Updated : 28 Jun, 2021

Let G be an undirected graph. Consider a depth-first traversal of G, and let T be the resulting depth-first search tree. Let u be a vertex in G and let v be the first new (unvisited) vertex visited after visiting u in the traversal. Which of the following statements is always true? (GATE CS 2000)
(A) {u,v} must be an edge in G, and u is a descendant of v in T
(B) {u,v} must be an edge in G, and v is a descendant of u in T
(C) If {u,v} is not an edge in G then u is a leaf in T
(D) If {u,v} is not an edge in G then u and v must have the same parent in T

Explanation:

```In DFS, if 'v' is visited
after 'u', then one of the following is true.
1) (u, v) is an edge.
u
/   \
v     w
/     / \
x     y   z

2) 'u' is a leaf node.
w
/   \
x     v
/     / \
u     y   z ```

In DFS, after visiting a node, we first recur for all unvisited children. If there are no unvisited children (u is leaf), then control goes back to parent and parent then visits next unvisited children.

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