GATE CS 2021 Set 2 — Question 56
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Algorithms → Graph Algorithms → DFS Edge Classification
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Question
Consider the following directed graph:
Which of the following is/are correct about the graph?
Correct answer
(A) The graph does not have a topological order.; (B) A depth-first traversal starting at vertex S classifies three directed edges as back edges.
Solution
1.Topological Order: A directed graph has a topological order if and only if it is a Directed Acyclic Graph (DAG). By inspecting the graph, we can identify cycles. For example, the cycle exists (where denotes the vertex at column and row ). Since the graph contains cycles, it cannot have a topological order. Thus, (A) is correct.
2.Back Edges in DFS: A DFS starting from will encounter cycles. For this specific grid structure, a standard DFS traversal will identify 3 back edges corresponding to the independent cycles formed by the alternating directions of rows and columns. Thus, (B) is correct.
3.Strongly Connected Components: Every graph has strongly connected components (SCCs). Since this graph has cycles, it contains non-trivial SCCs. Therefore, (C) is false.
4.Strong Connectivity: The graph is not strongly connected because there is no directed path between every pair of vertices (e.g., no path from the top-right vertex back to ). Thus, (D) is false.
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