Egawa and Saito proved that every k-connected graph with girth at least 4 has an induced cycle C such that $G-V(C)$ is $(k-3)$-connected, and every edge of C is contractible. This means that we can find not only a nonseparating cycle C but also one that consists of contractible edges. Motivated by this result, we prove that if G is a k-connected graph which does not contain $K_4^{-}$, then G has an induced cycle C such that $G - V(C)$ is $(k-2)$-connected and either every edge of C is k-contractible or C is a triangle. As a corollary of this result, we get the following result: Every k-connected graph with girth at least 4 has an induced cycle C such that $G-V(C)$ is $(k-2)$-connected, and every edge of C is contractible. This theorem is a generalization of some known theorems. In particular, this generalizes the above-mentioned result proved by Egawa and Saito and the result of Egawa which says that a k-connected graph with girth at least 4 has an induced cycle C such that $G-V(C)$ is $(k-2)$-connected.