# GATE | GATE-CS-2015 (Set 1) | Question 53

Let G be a connected undirected graph with n vertices and m edges. Which of the following statements is true regarding the minimum number of edges required to create a cycle in G?

(A)

The minimum number of edges required to create a cycle is n.

(B)

The minimum number of edges required to create a cycle is n – 1.

(C)

The minimum number of edges required to create a cycle is m – n + 1.

(D)

The minimum number of edges required to create a cycle is m – n + 2.

Explanation:

In a connected undirected graph, a cycle is formed when we have a closed path where each vertex is visited exactly once, except for the starting and ending vertex, which are the same. To form a cycle, we need at least three vertices and three edges.

Consider the minimum number of vertices required to form a cycle, which is three. In this case, we need three edges to connect the vertices in a cyclic manner. Therefore, the minimum number of edges required to create a cycle is n – 1.

It’s important to note that in a connected graph, each vertex must be connected to at least one other vertex. If we add an additional edge, it will create a cycle since there will be a closed path with each vertex visited exactly once.

Hence, option b) is correct, as the minimum number of edges required to create a cycle in a connected undirected graph with n vertices is n – 1.

Quiz of this Question
Please comment below if you find anything wrong in the above post