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- W2054698857 abstract "Let G=(V,E) be an undirected graph, S be a subset of its vertices, ${frak C}_S$ be the set of minimum edge-cuts partitioning S, and $lambda_S$ be the cardinality of such a cut. We suggest a graph structure, called the connectivity carcass of S, that represents both cuts in $frak C_S$ and the partition of V by all these cuts; its size is $O(min{|E|,lambda_S|V|})$. In this paper we present general constructions and study in detail the case $lambda_S$ odd; the specifics of the case $lambda_S$ 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 $lambda_S$ odd, just a tree) representing all distinct partitions of S by cuts in ${frak C}_S$, 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 $lambda_S$, the connectivity carcass can be maintained in time $O(|V|min{|E|,lambda_S|V|}+u)$. For two vertices of G, queries asking whether they are separated by a cut in $frak C_S$ are answered in O(1) worst-case time per query. Another possibility is to maintain the carcass in $O(|S|min{|E|,lambda_S|V|}+u)$ time, but to answer the queries in O(1) time only if at least one of the vertices belongs to S." @default.
- W2054698857 created "2016-06-24" @default.
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- W2054698857 date "2000-01-01" @default.
- W2054698857 modified "2023-10-02" @default.
- W2054698857 title "The General Structure of Edge-Connectivity of a Vertex Subset in a Graph and its Incremental Maintenance. Odd Case" @default.
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- W2054698857 doi "https://doi.org/10.1137/s0097539797330045" @default.
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