A depth-first traversal starting at vertex \(S\) classifies three directed edges as back edges.
The graph does not have a strongly connected component.
For each pair of vertices \(u\) and \(v\), there is a directed path from \(u\) to \(v\).
Show Solution
The Correct Option isA, B
Solution and Explanation
To analyze the given directed graph, we will evaluate each option step-by-step:
The graph does not have a topological order.
Explanation: A topological order is possible only in a Directed Acyclic Graph (DAG). If the graph contains a cycle, it cannot have a topological order. Examining the graph visually, note the presence of cycles, indicating it is not a DAG.
A depth-first traversal starting at vertex \( S \) classifies three directed edges as back edges.
Explanation: A back edge in a DFS traversal signifies a cycle. By performing a DFS starting at vertex \( S \), observe the cycles present in the graph, which confirms that at least three back edges are found. This observation supports the statement.
The graph does not have a strongly connected component.
Explanation: A strongly connected component (SCC) is a subgraph where any two vertices are mutually reachable. Given the graph's cycles, each cycle could form a strongly connected component. Thus, this statement is false.
For each pair of vertices \( u \) and \( v \), there is a directed path from \( u \) to \( v \).
Explanation: This statement implies that the graph is strongly connected as a whole. However, visual inspection or testing various vertex pairs will show breaks in connectivity, thus invalidating this statement.
Conclusion: The correct statements about the graph are:
The graph does not have a topological order.
A depth-first traversal starting at vertex \( S \) classifies three directed edges as back edges.
