Every $3$-connected graph on at least $13$ vertices has a contractible set on $5$ vertices

A subset $H$ of the set of vertices of a $3$-connected finite graph $G$ is called contractible if $G(H)$ is connected and $G - H$ is $2$-connected. We prove that every $3$-connected graph on at least $13$ vertices has a contractible set on $5$ vertices. And there is a $3$-connected graph on $12$ vertices that does not contain a contractible set on $5$ vertices.