A physical interpretation of graph connectivity, and its algorithmic applications

Connectivity is a basic property of graphs, and is related to other important concepts like reliability, communication and flow. Connectivity is also one of the most well studied areas in graph theory. In this paper, we propose a new point of view on graph connectivity, based on geometric and physical intuition. Our main theorem is a geometric characterization of k-vertex connected graphs. It says that a graph G is k-cQnnected if and only if G has a certain "nondegenerate convex embedding" in (Rk-l. 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. Algebraic properties of this equilibrium ensure that the embedding it defines is nondegenerate exactly when the graph is k-connected. As an application of our theorem we give probabilistic algorithms for computing the connectivity of a graph. The first is a Monte Carlo algorithm that runs in time O(n2.5+nk2.5) where n is the number of vertices and k is the vertex connectivity of the input graph. The second is a Las Vegas algorithm (Le., never errs) that runs in expected time O(kn2.5+nk3.5). For comparison, the best known algorithm (which is deterministic!) runs in time k3nl.5+k2n2 [Gl. Observe that our algorithms are faster for all k ~ In, and for very dens grphs the Monte Carlo algorithm is faster by 8 linear factor! Section 1 contains the main theorem and its proof. In Section 2 we describe the algorithmic applications. These include, in addition to the algorithms above, processor-efficient parallel counte~parts.