In this paper we continue a long line of work on representing the cut structure of graphs. We classify the types minimum vertex cuts, and the possible relationships between multiple minimum vertex cuts. As a consequence of these investigations, we exhibit a simple $O(κn)$-space data structure that can quickly answer pairwise $(κ+1)$-connectivity queries in a $κ$-connected graph. We also show how to compute the "closest" $κ$-cut to every vertex in near linear $\tilde{O}(m+poly(κ)n)$ time.