A 2-level cactus model for the system of minimum and minimum+1 edge-cuts in a graph and its incremental maintenance

Let A denote the cardinality of a minimum edgecut of a Multigraph G. The known cactus model represents the A-cuts of G in a clear and compact way and is used in related studies. We suggest new tools for modeling connectivity structures, and, using them, generalize the cactus model to represent all the X and (~+ I)-cuts. Representations obtained, different for A odd and even, have properties similar to those of the cactus model. In particular, they provide algorithms for the maintenance of the (A + 2)-connectivity classes of vertices (called also “(J + 2)-components”) in an arbitrary graph undergoing insertions of edges; the complexity, for A odd and even, is the same as achieved for the cases ~ = 1 and 2, respectively. As a metaresult, we give also a simple characterization of families of cuts that can be modeled by a cactus.