On vertices of degree $6$ of minimally and contraction critically $6$-connected graph

In this paper, we research vertices of degree $6$ of minimally and contraction critically $6$-connected graph, i.e. a $6$-connected graph that will loose $6$-connectivity after removing or contracting of any edge. We prove the following theorem. If $x$ and $z$ are adjacent vertices of degree $6$ of such a graph and no other vertex of degree 6 is adjacent to $x$ or $z$ then $x$ and $z$ have at least $4$ common neighbors. Moreover, in this case we give a detailed description of the neighborhood of the set $\{x,z\}$. Also, we construct an infinite series of examples of minimally and contraction critically $6$-connected graphs, for which a fraction of vertices of degree $6$ is ${11\over17}$.