GATE CS 2026 Set 1 Q.50: Graph Traversals, Topological Sort and Connectivity

Consider the following pseudocode for depth-first search (DFS) algorithm which takes a directed graph $G(V, E)$ as input, where $d[v]$ and $𝑓[v]$ are the discovery time and finishing time, respectively, of the vertex $v \in V$ .

DFS(G):                          Explore(G, v, t):
  unmark all v in V                mark v
  t <- 0                           t <- t + 1
  for each v in V                  d[v] <- t
    if v is unmarked               for each (v, w) in E
      t <- Explore(G, v, t)          if w is unmarked
    end if                             t <- Explore(G, w, t)
  end for                            end if
                                   end for
                                   t <- t + 1
                                   f[v] <- t
                                   return t

Suppose that the input directed graph $G(V, E)$ is a directed acyclic graph (DAG).

For an edge $(u, v) \in E$ , which of the following options will NEVER be correct?

$d[u] < d[v] < f[v] < f[u]$

$d[v] < d[u] < f[u] < f[v]$

$d[v] < f[v] < d[u] < f[u]$

$d[u] < d[v] < f[u] < f[v]$

Official answer and solution

Official answer: B and D
From: Official GATE 2026 answer key

Solution

Use the DFS Parenthesis Theorem and the fact that a DAG has no back edges (so $f[u] > f[v]$ for every directed edge $(u, v) \in E$ ).

For any edge $(u, v) \in E$ in a DAG:

Option (A) ( $d[u] < d[v] < f[v] < f[u]$ ) occurs whenever $(u, v)$ is a tree edge or forward edge.
Option (B) ( $d[v] < d[u] < f[u] < f[v]$ ) corresponds to $(u, v)$ being a back edge (where $v$ is an ancestor of $u$ in the DFS forest), which would form a directed cycle and can NEVER occur in a DAG.
Option (C) ( $d[v] < f[v] < d[u] < f[u]$ ) occurs whenever $(u, v)$ is a cross edge.
Option (D) ( $d[u] < d[v] < f[u] < f[v]$ ) violates the Parenthesis Theorem (discovery/finishing intervals $[d[u], f[u]]$ and $[d[v], f[v]]$ can never partially overlap) and can NEVER occur in any DFS.

Thus Options (B) and (D) will never be correct.

Final answer

Options (B), (D)

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2026 Computer Science and Information Technology - Session 1 (CS1)
Question: 50 of 65
Section: Computer Science and Information Technology

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