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 ff and only if, specifying any k vertices of G, the vertices of G can be represented by points of R ~-a 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 probabilistie (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_ 1/~', and for very dense graphs the Monte Carlo algorithm is faster by a linear factor. The property of k-connectivity of a graph is well-characterized: it is "easy, to exhibit ifa graph is not k-connected and, also, if it is. But there is some asymmetry in this: to exhibit that a graph is not k-connected, it suffices to present a separating set with less than k vertices; to exhibit that it is k-connected, we have to present k openly disjoint paths for each pair of vertices. Is there a more compact ,'proof" of k-connectivity, say, an additional structure whose presence gives a trivially check-able proof of k-connectivity? For k= 1, a spanning tree provides a trivial answer. For k=2, various versions of "ear-decompositions" (see, e.g., [10]) give rise to such "proofs". Another structure characterizing 2-connectivity, closely related to ear-decomposltions, is an s-t numbering for an edge st: :a linear ordering of the nodes, starting with s and ending with t, such that everyother node has a neighbor to its left as well as one to its right. In this paper, we offer some new characterizations of graph k-connectivity, based on geometric and physical intuition. Our main theorem is a geometric characterization of k-vertex connected graphs, generalizing s-t numberings. It says that a graph G is k-connected if and only if G has certain "nondegenerate convex em-beddings" in R ~-x. 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 AMS subject classification (1980): 05 C 40, …