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

Citations: | 9 - 1 self |

### BibTeX

@MISC{Cheriyan_directeds-t,

author = {Joseph Cheriyan and John H. Reif},

title = {Directed s-t Numberings, Rubber Bands, and Testing Digraph k-Vertex Connectivity},

year = {}

}

### OpenURL

### Abstract

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 IX I = k, there is an embedding of G in the (k- I)-dimensional space Rk-l, ~ : V ~Rk-l, such that no hyperplane contains k points of {~(v) \ v G V}, and for each v E V – X, f(v) is in the convex hull of {~(w) I (v, W) G E}. This result generalizes to directed graphs the notion of convex embedding of undirected graphs introduced by Linial, LOV6SZ and Wigderson in ‘Rubber bands, convex embedding and graph connectivity, ” Combinatorics 8 (1988), 91-102. Using this characterization, a directed graph can be tested for k-vertex connectivity by a Monte Carlo algo-rithm in time O((M(n) + nkf(k)). (log n)) with error probability < l/n, and by a Las Vegas algorithm in ex-pected time O((lf(n)+nM(k)).k), where M(n) denotes the number of arithmetic steps for multiplying two n x n matrices (Al(n) = 0(n2.3755)). Our Monte Carlo algo-rithm improves on the best previous deterministic and randomized time complexities for k> no. *9; e.g., for k = @, the factor of improvement is> n0.G2. Both al-gorithms 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 (logn) times the respective sequential time complexi-ties. 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 nulmberiug for any 2-vertex connected directed graph.