The Structure of Minimum Vertex Cuts

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