GATE CS 2025 Set 1 Q.43: Graph Traversals, Topological Sort and Connectivity

Let $G(V, E)$ be an undirected and unweighted graph with 100 vertices. Let $d(u, v)$ denote the number of edges in a shortest path between vertices $u$ and $v$ in $V$ . Let the maximum value of $d(u, v)$ , $u, v \in V$ such that $u \ne v$ , be $30$ . Let $T$ be any breadth-first-search tree of $G$ . Which ONE of the given options is CORRECT for every such graph $G$ ?

The height of $T$ is exactly 15.

The height of $T$ is exactly 30.

The height of $T$ is at least 15.

The height of $T$ is at least 30.

Official answer and solution

Official answer: C
From: Official GATE 2025 answer key

Solution

Apply the triangle inequality to the BFS root $r$ and a pair of diametrically opposite vertices $x, y$ with $d(x, y) = 30$ .

Let $x, y \in V$ be two vertices that achieve the diameter of $G$ , so $d(x, y) = 30$ . Let $r \in V$ be the root of the breadth-first-search (BFS) tree $T$ . Since BFS from $r$ finds shortest paths in an unweighted graph, the depth of any vertex $v$ in $T$ is $d(r, v)$ , and the height of $T$ is $h(T) = \max_{v \in V} d(r, v)$ .

By the triangle inequality:

30 = d(x, y) \le d(r, x) + d(r, y) \le 2 \max(d(r, x), d(r, y)) \le 2 h(T)

Dividing by 2 gives $h(T) \ge 15$ . (Depending on the choice of root $r$ , $h(T)$ can be any integer from $15$ to $30$ , e.g., $15$ when $r$ is the midpoint of a length-30 path, or $30$ when $r$ is an endpoint.) Thus the height of $T$ is at least 15.

Final answer

Option (C)

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

Source

Paper: GATE 2025 Computer Science and Information Technology - Session 1 (CS1)
Question: 43 of 65
Section: Computer Science and Information Technology

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