Using expander graphs to find vertex connectivity

The (vertex) connectivity /spl kappa/ of a graph is the smallest number of vertices whose deletion separates the graph or makes it trivial. We present the fastest known algorithm for finding /spl kappa/. For a digraph with n vertices, m edges and connectivity /spl kappa/ the time bound is O((n+min(/spl kappa//sup 5/2/,/spl kappa/n/sup 3/4/))m). This improves the previous best bound of O((n+min(/spl kappa//sup 3/,/spl kappa/n))m). For an undirected graph both of these bounds hold with m replaced /spl kappa/n. Our approach uses expander graphs to exploit nesting properties of certain separation triples.