GATE | GATE CS 2021 | Set 1 | Question 51

• Difficulty Level : Medium
• Last Updated : 21 Dec, 2021

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?
(A) Root of T can never be an articulation point in G.
(B) Root of T is an articulation point in G if and only if it has 2 or more children.
(C) A leaf of T can be an articulation point in G.
(D) If u is an articulation point in G such that x is an ancestor of u in T and y is a descendant of u in T, then all paths from x to y in G must pass through u.

Explanation: How to find all articulation points ?
DFS- based-approach
We can prove following properties:

• The root of a DFS-tree is an articulation point if and only if it has at least two children.

• Leaf of a DFS-tree are never articulation points.

D is not true because, it is not necessary to have a path through articulation point only. There can a be a direct path from x to y as well in a graph G.

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