## Abstract

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.

Original language | English |
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Pages (from-to) | 753-808 |

Number of pages | 56 |

Journal | SIAM Journal on Computing |

Volume | 30 |

Issue number | 3 |

DOIs | |

State | Published - 1 Jan 2000 |

## Keywords

- Dynamic algorithms
- Edge-connectivity
- Graph structures
- Incremental maintenance
- Minimum cuts