An articulation point in a connected graph is a vertex such that removing the vertex and its incident edges disconnects the graph into two or more connected components. Let T be a DFS tree obtained by doing DFS in a connected undirected graph G. Which of the following options is/are correct?Note: This kind of question will be helpful in clearing CTS recruitment.
Question
An articulation point in a connected graph is a vertex such that removing the vertex and its incident edges disconnects the graph into two or more connected components. Let T be a DFS tree obtained by doing DFS in a connected undirected graph G. Which of the following options is/are correct?Note: This kind of question will be helpful in clearing CTS recruitment.
Solution
To determine the correct option(s), we need to understand the concept of articulation points and how they relate to a DFS tree.
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An articulation point is a vertex in a connected graph that, when removed along with its incident edges, causes the graph to become disconnected or split into multiple connected components.
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A DFS tree is a tree-like structure that is obtained by performing a Depth-First Search (DFS) traversal on a connected undirected graph. It is used to explore and represent the connectivity of the graph.
Now, let's consider the given options:
Option 1: Every leaf node in T is an articulation point in G. This statement is incorrect. In a DFS tree, leaf nodes are vertices that have only one neighbor. Removing a leaf node will not disconnect the graph, as the remaining vertices will still be connected.
Option 2: Every non-leaf node in T is an articulation point in G. This statement is also incorrect. Non-leaf nodes in a DFS tree are vertices that have more than one neighbor. Removing a non-leaf node may disconnect the subtree rooted at that node, but it does not necessarily disconnect the entire graph.
Option 3: The root of T is always an articulation point in G. This statement is correct. The root of a DFS tree is the starting point of the traversal and is connected to all other vertices in the graph. Removing the root will disconnect the entire graph into separate connected components.
Therefore, the correct option is Option 3: The root of T is always an articulation point in G.
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