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Consider the following set of statements:
S1: In a depth-first traversal of a graph G with n vertices, k edges are marked as tree edges. There are n-k connected components in G.
S2: A depth-first search necessarily finds the shortest path between the starting point and any other reachable node.
S3: The depth-first tree on the simple undirected graph never contains a cross edge.
Which of the given statements is false?
  • a)
    Only S2
  • b)
    Only S3
  • c)
    Both S2 and S3
  • d)
    Both S1 and S2
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
Consider the following set of statements:S1: In a depth-first traversa...
Statement S2 is false. It can be corrected as:
A BFS will find the shortest path between the starting point and any other reachable node. A depth-first search will not necessarily find the shortest path.
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Most Upvoted Answer
Consider the following set of statements:S1: In a depth-first traversa...
Understanding the Statements
The statements provided involve properties of depth-first traversal (DFS) in graph theory. Let's analyze each statement to identify the false one.
Statement S1: Connected Components and Tree Edges
- Meaning: In a DFS traversal of a graph G with n vertices, if k edges are marked as tree edges, then there are n-k connected components.
- Truth Value: This statement is true. The number of connected components in a graph is related to the number of vertices and edges discovered during DFS.
Statement S2: Shortest Path in DFS
- Meaning: A depth-first search necessarily finds the shortest path between the starting point and any other reachable node.
- Truth Value: This statement is false. DFS does not guarantee the shortest path; it explores as far as possible along each branch before backtracking. For shortest path calculations, algorithms like Dijkstra's are used instead.
Statement S3: Cross Edges in DFS Tree
- Meaning: The depth-first tree on a simple undirected graph never contains a cross edge.
- Truth Value: This statement is true. In DFS, cross edges are not present in the tree structure; they connect nodes in different branches of the DFS tree but don't contribute to the structure itself.
Conclusion
- Considering the analysis, the only false statement is S2, making the correct answer option 'A'. DFS does not inherently find the shortest paths between nodes, which is a critical distinction in graph traversal techniques.
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