GATE CS 2014 Set 1 — Question 13
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Programming & Data Structures → Graphs (Data Structure) → Strongly Connected Components
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Question
Let be a directed graph where is the set of vertices and the set of edges. Then which one of the following graphs has the same strongly connected components as ?
Correct answer
(B) G₂ = (V, E₂) where E₂ = \(u,v) (v,u) ∈ E\
Solution
Two vertices are in the same strongly connected component (SCC) exactly when each can reach the other.(B) Transpose graph . A path in is a path in , and vice versa. Mutual reachability is therefore unchanged, so has exactly the SCCs of . This is the official answer.(A) Complement . Not in general: for with no edges, every vertex is its own SCC while the complement is a complete digraph with one SCC.(D) . Removing isolated vertices removes their singleton SCCs, so the set of SCCs changes whenever has an isolated vertex.(C) . Every edge of is a path of length 1, so ; and every edge of stands for a path in . Hence reaches in iff reaches in , so also has the same SCCs as . Mathematically, (C) is therefore also correct; the official GATE 2014 key accepts only (B), and that key is retained here.Answer: (B) (official key)
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