A 2-Level Cactus Tree Model for the System of Minimum and Minimum+1 Edge . . .

3] Ye. Dinitz, \The 3-edge components and the structural description of all 3-edge cuts in a graph", 5] Ye. Dinitz and Z. Nutov, \A 2-level cactus tree model for the minimum and minimum+1 edge cuts in a graph and its incremental maintenance", Proc. the 27th Symposium on Theory of Computing, 1995, 509{518. 6] Ye. Dinitz and Z. Nutov, \A 2-level cactus tree model for the minimum and mini-mum+1 edge cuts in a graph and its incremental maintenance. Part I: the odd case", a manuscript. 7] Ye. Dinitz and Z. Nutov, \Cactus-tree type models for families of bisections of a set", a manuscript. 8] Ye. Dinitz and A. Vainshtein, \The connectivity carcass of a vertex subset in a graph and its incremental maintenance", 37 The worst case time for each query is O(1). The initialization time is polynomial in n, and the space required is O(n). Relying on 17], the complexity of incremental maintaining can be reduced to O((u + q + n)(u + q; n)). Notice that the above complexities of maintenance can be reduced substituting each instance of n by n +2 in the following way. At the preprocessing stage, we can build the quotient graph G 0 by shrinking each of the n 0 +2 (0 + 2)-classes of G into a single supervertex and apply our algorithm to G 0 , with n 0 +2 supervertices, instead of G. In this version, the current (0 + 2)-class of a vertex v of G is found as the current (0 + 2)-class of the supervertex of G 0 corresponding to the initial (0 + 2)-class of v. This is done via two queries, where nding the supervertex can be supported by a static data structure in O(1) time. 5 Concluding remarks 1. Observe that the properties mentioned in Theorem 4.1 are similar to those of the cactus tree model for the minimum cuts, though more complicated. Since the structure of the modeling cuts is explicit and, in a sense, simple, and since the representation is compact, our model seems to be convenient to represent the minimum and minimum+1 cuts of graphs in various applications. 2. It is likely that the 2-level cactus model can be a useful tool for handling edge-augmentation problems when the increase of the connectivity is 2. Another possible direction is to use our representation for eeective maintenance of optimal augmentation sets of an incremental …