GATE CS 2025 Set 1 — Question 43
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Algorithms → Graph Algorithms → BFS Shortest Path
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Question
Let
G(V, E) be an undirected and unweighted graph with 100 vertices. Let denote the number of edges in a shortest path between vertices and in . Let the maximum value of such that , be 30. Let be any breadth-first-search tree of . Which ONE of the given options is CORRECT for every such graph ?Correct answer
(C) The height of T is at least 15.
Solution
1.Diameter of the graph: The maximum shortest path distance between any two vertices is the diameter of the graph. Here, .
2.BFS Tree Height: A BFS tree rooted at a vertex has a height equal to the maximum distance from to any other vertex in the graph. This is the eccentricity of vertex , denoted as . So, .
3.Relationship between Radius and Diameter: The radius of a graph is the minimum eccentricity among all vertices, . A well-known property in graph theory is that .
4.Calculation: Given , we have . Since the height of any BFS tree is the eccentricity of its root, and the minimum possible eccentricity is , the height of any BFS tree must be at least .
5.Conclusion: Therefore, for any BFS tree of . Option (C) is correct.
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