GATE | GATE-CS-2003 | Question 8

Let G be an arbitrary graph with n nodes and k components. If a vertex is removed from G, the number of components in the resultant graph must necessarily lie between
(A) k and n
(B) k – 1 and k + 1
(C) k – 1 and n – 1
(D) k + 1 and n – k


Answer: (C)

Explanation: Minimum: The removed vertex itself is a separate connected component. So removal of a vertex creates k-1 components.

Maximum: It may be possible that the removed vertex disconnects all components. For example the removed vertex is center of a star. So removal creates n-1 components.


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