GATE | GATE CS 2021 | Set 1 | Question 51
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.
Answer: (B) (D)
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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