Representing and Enumerating Edge Connectivity Cuts in RNC

An undirected edge-weighted graph can have at most (\(\left( {\begin{array}{*{20}c}n \\2 \\\end{array} } \right)\)) edge connectivity cuts. A succinct and algorithmically useful representation for this set of cuts was given by [DKL], and an efficient sequential algorithm for obtaining it was given by [KT]. In this paper, we present a fast parallel algorithm for obtaining this representation; our algorithm is an RNC algorithm in case the weights are given in unary. We also observe that for a unary weighted graph, the problems of counting and enumerating the connectivity cuts are in RNC.

[1]  Charles J. Colbourn,et al.  Bounding all-terminal reliability in computer networks , 1988, Networks.

[2]  Robert E. Tarjan,et al.  Network Flow and Testing Graph Connectivity , 1975, SIAM J. Comput..

[3]  Eli Upfal,et al.  Constructing a perfect matching is in random NC , 1985, STOC '85.

[4]  Harold N. Gabow,et al.  Applications of a poset representation to edge connectivity and graph rigidity , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[5]  Vijay V. Vazirani,et al.  Matching is as easy as matrix inversion , 1987, STOC.

[6]  E. A. Timofeev,et al.  Efficient algorithm for finding all minimal edge cuts of a nonoriented graph , 1986 .

[7]  S. Louis Hakimi,et al.  On computing the connectivities of graphs and digraphs , 1984, Networks.

[8]  Arkady Kanevsky,et al.  On the number of minimum size separating vertex sets in a graph and how to find all of them , 1987, SODA '90.

[9]  Charles U. Martel,et al.  A fast algorithm for optimally increasing the edge-connectivity , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[10]  J. Picard,et al.  Selected Applications of Minimum Cuts in Networks , 1982 .

[11]  Maurice Queyranne,et al.  On the structure of all minimum cuts in a network and applications , 1982, Math. Program..

[12]  R. Bixby.  The Minimum Number of Edges and Vertices in a Graph with Edge Connectivity n and m n-Bonds , 1974, Networks.

[13]  David W. Matula,et al.  Determining edge connectivity in 0(nm) , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[14]  Harold N. Gabow.  A matroid approach to finding edge connectivity and packing arborescences , 1991, STOC '91.