GATE 2026 Algorithms PYQs | GRAPH TECHNIQUES

The question states: "Let G(V,E) be a directed graph, where V={1,2,3,4,5} is the set of vertices and E is the set of directed edges, as defined by the following adjacency matrix A."

The adjacency matrix A is defined as:

It then clarifies: "A[i][j]=1 indicates a directed edge from node i to node j."

A directed spanning tree of G, rooted at r∈V, is defined as "a subgraph T of G such that the undirected version of T is a tree, and T contains a directed path from r to every other vertex in V.

The number of such directed spanning trees rooted at vertex 5 is ___.

GATE CSE 2022 - [2Marks] (NAT)

Let G be a directed graph and T a depth first search (DFS) spanning tree in G that is rooted at a vertex v. Suppose T is also a breadth first search (BFS) tree in G, rooted at v. Which of the following statements is/are TRUE for every such graph G and tree T?
GATE CSE 2024,SET1 - [2Marks] (MCQ)

Consider the following algorithm someAlgo that takes an undirected graph G as input.

someAlgo (G) :

1. Let v be any vertex in G. Run BFS on G starting at v. Let u be a vertex in G at maximum distance from v as given by the BFS.
2. Run BFS on G again with u as the starting vertex. Let z be the vertex at maximum distance from u as given by the BFS.
3. Output the distance between u and z in G.

GATE CSE 2025,SET2 - [2Marks] (NAT)

​​​​Which of the following statements regarding Breadth First Search (BFS) and Depth First Search (DFS) on an undirected simple graph G is/are TRUE
GATE CSE 2025,SET2 - [2Marks] (MSQ)

[Tex]\begin{aligned} &\textbf{}\quad\text{Consider the following pseudocode for depth-first} \\ &\quad\text{search (DFS) algorithm which takes a directed} \\ &\quad\text{graph } G(V,E) \text{ as input, where } d[v] \text{ and } f[v] \text{ are the} \\ &\quad\text{discovery time and finishing time, respectively, of} \\ &\quad\text{the vertex } v \in V. \\ \\ &\begin{array}{|l|l|} \hline \text{DFS}(G) : & \text{Explore } (\, G, v, t \,) : \\ \quad\text{unmark all } v \in V & \quad\text{mark } v \\ \quad t \leftarrow 0 & \quad t \leftarrow t + 1 \\ \quad\text{for each } v \in V & \quad d[v] \leftarrow t \\ \quad\quad\text{if } v \text{ is unmarked} & \quad\text{for each } (v, w) \in E \\ \quad\quad\quad t \leftarrow \text{Explore} & \quad\quad\text{if } w \text{ is unmarked} \\ (\, G, v, t \,) & \quad\quad\quad t \leftarrow \text{Explore } (\, G, w, t \,) \\ \quad\quad\text{end if} & \quad\quad\text{end if} \\ \quad\text{end for} & \quad\text{end for} \\ & \quad t \leftarrow t + 1 \\ & \quad f[v] \leftarrow t \\ & \quad\text{return } t \\ \hline \end{array} \\ \\ &\quad\text{Suppose that the input directed graph } G(V,E) \text{ is a} \\ &\quad\text{directed acyclic graph (DAG). For an edge} \\ &\quad(u,v) \in E\text{, which of the following options will} \\ &\quad\text{NEVER be correct?} \end{aligned}[/Tex]
[GATE 2026 || SET-1 MSQ || 2-mark]

An undirected, unweighted, simple graph 𝐺(𝑉,𝐸) is said to be 2-colorable if there exists a function 𝑐:𝑉→{0,1} such that for every (u,v)∈E, c(u)≠c(v)
Which of the following statements about 2-colorable graphs is/are true? [GATE CSE 2026 | Set-1 MSQ | 2 marks]