On contractible 5-vertex subgraphs of a 3-connected graph

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 $11$ vertices with minimal degree at least $4$ has a contractible set on $5$ vertices.