GATE DA 2025 Set 1 — Question 65
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Programming, Data Structures & Algorithms → Graphs → Graph Traversals (BFS, DFS)
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Question
Consider a directed graph , where and
. Suppose the adjacency list of each vertex is in decreasing order of vertex number, and depth-first search (DFS) is performed at vertex 0. The number of vertices that will be discovered after vertex 50 is ________ (Answer in integer)
. Suppose the adjacency list of each vertex is in decreasing order of vertex number, and depth-first search (DFS) is performed at vertex 0. The number of vertices that will be discovered after vertex 50 is ________ (Answer in integer)
Correct answer
75 to 75
Solution
The graph has edges from to and . The adjacency list is sorted in decreasing order, so for vertex , the neighbors are processed in the order .DFS Traversal:
1.Start at 0. Neighbors: . Visit 2.
2.At 2. Neighbors: . Visit 4.
3.This pattern continues along the even numbers: .
4.Vertex 50 is discovered at step 26 (indices ).
5.The DFS continues to 100. All even vertices () are discovered after 50. There are 25 such vertices.
6.After reaching 100 (no outgoing edges to unvisited nodes), the DFS backtracks.
7.Back at 98, it visits the remaining neighbor 99. (99 is discovered after 50).
8.Back at 96, it visits 97. (97 is discovered after 50).
9....
10.Back at 50, it visits 51. (51 is discovered after 50).
11.Back at 48, it visits 49. Since 50 is already finished, 49 is discovered after 50.
12.This continues down to 0 visiting 1. All odd vertices () are discovered after the main even path reaches their parent and returns. Since 50 is on the main path, any odd vertex branched off from a node (like 49 from 48) is discovered after 50 has been visited. Any odd vertex branched off from a node (like 51 from 50) is also discovered after 50.
Calculation:- Vertices discovered before or at 50: The even numbers . Count = .
- Total vertices = 101.
- Vertices discovered after 50 = Total - (Discovered before or at 50) = .
- Even vertices : (25 vertices).
- Odd vertices: (50 vertices).
- Total = .
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