Rubber bands, convex embeddings and graph connectivity

We give various characterizations of k-vertex connected graphs by geometric, algebraic, and "physical" properties. As an example, a graph G is k-connected if and only if, specifying any k vertices of G, the vertices of G can be represented by points of ℝ^k-1 so that no k are on a hyper-plane and each vertex is in the convex hull of its neighbors, except for the k specified vertices. The proof of this theorem appeals to physics. The embedding is found by letting the edges of the graph behave like ideal springs and letting its vertices settle in equilibrium. As an algorithmic application of our results we give probabilistic (Monte-Carlo and Las Vegas) algorithms for computing the connectivity of a graph. Our algorithms are faster than the best known (deterministic) connectivity algorithms for all k≧√n, and for very dense graphs the Monte Carlo algorithm is faster by a linear factor.