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In a depth-first traversal of a graph G with n vertices, k edges are marked as tree edges. The number of connected components in G is
- a)k
- b)k + 1
- c)n - k - 1
- d)n - k
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
In a depth-first traversal of a graph G with n vertices, k edges are m...
Number of vertices = n
Number of edges = k
Number of connected components = n - k
Ex. 8 vertex with 6 edges
So components = 8 - 6 = 2.
Number of edges = k
Number of connected components = n - k
Ex. 8 vertex with 6 edges
So components = 8 - 6 = 2.
Most Upvoted Answer
In a depth-first traversal of a graph G with n vertices, k edges are m...
Depth-First Traversal and Tree Edges
In a depth-first traversal of a graph G with n vertices, we start at a given vertex and explore as far as possible along each branch before backtracking. This traversal results in a tree-like structure called a depth-first search tree. The edges in this tree are called tree edges.
The number of tree edges marked during the depth-first traversal is denoted as k.
Connected Components in a Graph
A connected component in a graph is a subgraph in which every pair of vertices is connected by a path. In other words, it is a maximal connected subgraph.
Explanation of the Answer
The number of connected components in a graph can be determined by analyzing the number of tree edges marked during a depth-first traversal.
- The given graph G has n vertices.
- In a depth-first traversal, k edges are marked as tree edges.
To understand why the correct answer is option 'D' (n - k), let's consider the following:
- In a connected component, every vertex is reachable from any other vertex.
- In the depth-first traversal, each tree edge connects two vertices.
- Since k edges are marked as tree edges, they connect k + 1 vertices (including the starting vertex).
- The remaining n - (k + 1) vertices are not connected to the tree edges.
Therefore, the number of connected components in G is equal to the number of remaining vertices that are not connected to the tree edges, which is n - (k + 1). Simplifying this expression gives us the answer n - k.
Hence, option 'D' (n - k) is the correct answer.
In a depth-first traversal of a graph G with n vertices, we start at a given vertex and explore as far as possible along each branch before backtracking. This traversal results in a tree-like structure called a depth-first search tree. The edges in this tree are called tree edges.
The number of tree edges marked during the depth-first traversal is denoted as k.
Connected Components in a Graph
A connected component in a graph is a subgraph in which every pair of vertices is connected by a path. In other words, it is a maximal connected subgraph.
Explanation of the Answer
The number of connected components in a graph can be determined by analyzing the number of tree edges marked during a depth-first traversal.
- The given graph G has n vertices.
- In a depth-first traversal, k edges are marked as tree edges.
To understand why the correct answer is option 'D' (n - k), let's consider the following:
- In a connected component, every vertex is reachable from any other vertex.
- In the depth-first traversal, each tree edge connects two vertices.
- Since k edges are marked as tree edges, they connect k + 1 vertices (including the starting vertex).
- The remaining n - (k + 1) vertices are not connected to the tree edges.
Therefore, the number of connected components in G is equal to the number of remaining vertices that are not connected to the tree edges, which is n - (k + 1). Simplifying this expression gives us the answer n - k.
Hence, option 'D' (n - k) is the correct answer.
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