The tree of cuts and minimal $k$-connected graphs

A cut of a $k$-connected graph $G$ is its $k$-element cutset which contains at least one edge. The tree of cuts of a set $\mathfrak S$, consisting of pairwise independent cuts of a $k$-connected graph is defined as follows. Its vertices are cuts of the set $\mathfrak S$ and parts of the decomposition of $G$ by the cuts of $\mathfrak S$. A part $A$ is adjacent to a cut $S$ if and only if $A$ contains all vertices of $S$ and one end of each edge of $S$. It is proved that the graph described above is a tree and have properties similar to properties of classic tree of blocks and cutpoints.
In the second part of the paper the tree of cuts is applied to study properties of minimal $k$-connected graphs for $k\le5$.