The connectivity carcass of a vertex subset in a graph and its incremental maintenance

Abstract

Let G = (V, E) be an undirected graph, S be a subset of its vertices, ts be the set of minimum edge-cuts partitioning S. A data structure representing both cuts in I.Z,S and the partition of V by all these cuts is suggested. One can build it in ISI - 1 max-flow computations in G. It can be maintained, for an arbitrary sequence of u edge insertions, in O(min{]V]. Il?l, klV12 +wa(u, IVI)}) time, where k is the size of a cut in C.g. For two vertices of G, queries asking whether they are separated by a cut in C.S are answered in O (a (q, IV t) ) amortized time per query, where q is the number of queries; such a cut itself is shown in O ( IVI ) amortized time. The dag representation of all cuts in C,S separating two given vertices in S is obtained in O(min{lEl, klVl}) amortized time.