  Let $G$ be a $3$-connected graph. A set $W \subset V(G)$ is contractible if
$G(W)$ is connected and $G - W$ is a $2$-connected graph. In 1994, McCuaig and
Ota formulated the conjecture that, for any $k \in \mathbb{N}$, there exists $m
\in \mathbb{N}$ such that any 3-connected graph $G$ with $v(G) \geqslant m$ has
a $k$-vertex contractible set. In this paper we prove that, for any $k
\geqslant 5$, the assertion of the conjecture holds if $\delta(G) \geqslant
\left[ \frac{2k + 1}{3} \right] + 2$.
