Directed s-t Bumberings, Rubber Bands, and Testing Digraph k-Vertex Connectivity

Let G = (V, E) be a directed graph and n denote |V|. We show that G is k-vertex connected iff for every subset X of V with |X| = k, there is an embedding of G in the (k-1)-dimensional space Rk-1, f : V →Rk-1, such that no hyperplane contains k points of {f(v) | v e V}, and for each v e V - X, f(v) is in the convex hull of {f(w) | (v, w) e E}. This result generalizes to directed graphs the notion of convex embeddings of undirected graphs introduced by Linial, Lova´sz and Wigderson in “Rubber bands, convex embeddings and graph connectivity,” Combinatorica 8 (1988), 91-102. Using this characterization, a directed graph can be tested for k-vertex connectivity by a Monte Carlo algorithm in time O((M(n) + nM(k)).(log n)) with error probability n0.19; e.g., for k = (n0.5, the factor of improvement is > n0.62. Both algorithms have processor efficient parallel versions that run in O((log n)2) time on the EREW PRAM model of computation, using a number of processors equal to (log n) times the respective sequential time complexities. Our Monte Carlo parallel algorithm improves on the number of processors used by the best previous (Monte Carlo) parallel algorithm by a factor of at least (n2/(log n)3) while having the same running time. Generalizing the notion of s-t numberings, we give a combinatorial construction of a directed s-t numbering for any 2-vertex connected directed graph.