# Colloquium - Gabow

Using Expander Graphs to Find Vertex Connectivity
Professor, Department of Computer Science
11/9/2000
3:30pm-4:30pm

The connectivity k is a central notion in graph theory. It indicates how hard it is to break a graph up into disjoint pieces - e.g., how many site failures can disconnect a communications network. This talk is about a new algorithm for finding k. Connectivity seems much harder to compute than its close companion, edge connectivity. Most introductory algorithms courses discuss the simplest case of computing connectivity, k=1, i.e., finding the articulation points of a graph. The game rapidly gets harder with increasing values of k!

The talk will begin with survey-type remarks on graph connectivity, and also on the main tool of our new algorithm, expander graphs (and spectral graph theory). Then we describe the new algorithm. This last part is a dry run of a conference presentation, whose abstract is the following:

The "connectivity" k 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 computing connectivity. The time bound is

O(min { n + k5/2, kn3/4 } m ),

improving the previous best bound of

O(min { n + k3, kn } m).

Here n and m represent the number of vertices and edges of the given graph, respectively. Our approach uses expander graphs to exploit a nesting property of certain separation triples.