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.