The general structure of edge-connectivity of a vertex subset in a graph and its incremental maintenance. Odd case

Let G = (V, E) be an undirected graph, S be a subset of its vertices, C_S be the set of minimum edge-cuts partitioning S, and λ_S be the cardinality of such a cut. We suggest a graph structure, called the connectivity carcass of S, that represents both cuts in C_S and the partition of V by all these cuts; its size is O(min{|E|, λ_S|V|}). In this paper we present general constructions and study in detail the case λ_S odd; the specifics of the case λ_S even are considered elsewhere. For an adequate description of the connectivity carcass we introduce a new type of graph: locally orientable graphs, which generalize digraphs. The connectivity carcass consists of a locally orientable quotient graph of G, a cactus tree (in case λ_S odd, just a tree) representing all distinct partitions of S by cuts in C_S, and a mapping connecting them. One can build it in O(|S|) max-flow computations in G. For an arbitrary sequence of u edge insertions not changing λ_S, the connectivity carcass can be maintained in time O(|V| min{|E|, λ_S|V|} + u). For two vertices of G, queries asking whether they are separated by a cut in C_S are answered in O(1) worst-case time per query. Another possibility is to maintain the carcass in O(|S| min{|E|, λ_S|V|} + u) time, but to answer the queries in O(1) time only if at least one of the vertices belongs to S.