A New Perspective on Vertex Connectivity

Edge connectivity and vertex connectivity are two fundamental concepts in graph theory. Although by now there is a good understanding of the structure of graphs based on their edge connectivity, our knowledge in the case of vertex connectivity is much more limited. An essential tool in capturing edge connectivity are the classical results of Tutte and Nash-Williams from 1961 which show that a λ-edge-connected graph contains d(λ−1)/2e edge-disjoint spanning trees. We argue that connected dominating set partitions and packings are the natural analogues of edge-disjoint spanning trees in the context of vertex connectivity and we use them to obtain structural results about vertex connectivity in the spirit of those for edge connectivity. More specifically, connected dominating set (CDS) partitions and packings are counterparts of edge-disjoint spanning trees, focusing on vertex-disjointness rather than edge-disjointness, and their sizes are always upper bounded by the vertex connectivity k. We constructively show that every k-vertex-connected graph with n nodes has CDS packings and partitions with sizes, respectively, Ω(k/ log n) and Ω(k/ log n), and we prove that the former bound is existentially optimal. Beautiful results by Karger show that when edges of a λ-edge-connected graph are independently sampled with probability p, the sampled graph has edge connectivity Ω̃(λp). Obtaining such a result for vertex sampling remained open. We illustrate the strength of our approach by proving that when vertices of a k-vertexconnected graph are independently sampled with probability p, the graph induced by the sampled vertices has vertex connectivity Ω̃(kp). This bound is optimal up to poly-log factors and is proven by building an Ω̃(kp) size CDS packing on the sampled vertices while sampling happens. As an additional important application, we show CDS packings to be tightly related to the throughput of routing-based algorithms and use our new toolbox to yield a routing-based broadcast algorithm with optimal throughput Ω(k/ log n + 1), improving the (previously best-known) trivial throughput of Θ(1).