Locally orientable graphs, cell structures, and a new algorithm for the incremental maintenance of connectivity carcasses

Let G - [V, E) be an undirected graph, S be a subset of its vertices. The concept of the connectivity carcass of S has been introduced by the authors at STOC'94; it provides a tool for studying i-edge-connectivity. In this paper we: (i) suggest and study a new type of graphs, locally orientable graphs, which generalize digraphs and provide a concise description of the connectivity carcass; (ii) introduce and study the cell structure of the carcass; (iii) suggest a new, cell-oriented algorithm for the incremental maintenance of the carcass. The complexity for an arbitrary sequence of U edge insertions preserving the connectivity κ of S is reduced from O(|V|· |E|+μ α(μ |V|, )) to O(s|E|+ μα(μ|V|)), where S is the number of the (κ + 1)-edge-connectivity classes in S.