  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 $O(\kappa
n)$-space data structure that can quickly answer pairwise
$(\kappa+1)$-connectivity queries in a $\kappa$-connected graph. We also show
how to compute the "closest" $\kappa$-cut to every vertex in near linear
$\tilde{O}(m+poly(\kappa)n)$ time.
