### A New Perspective on Vertex Connectivity [chapter]

Keren Censor-Hillel, Mohsen Ghaffari, Fabian Kuhn
2013 Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms
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-edgeconnected graph contains (λ − 1)/2 edge-disjoint spanning trees. We argue that connected
more » ... connected dominating set partitions and packings are the natural analogues of edgedisjoint 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 edgedisjoint 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-vertexconnected graph with n nodes has CDS packings and partitions with sizes, respectively, Ω(k/ log n) and Ω(k/ log 5 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 connectivityΩ(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-vertex-connected graph are independently sampled with probability p, the graph induced by the sampled vertices has vertex connectivityΩ(kp 2 ). This bound is optimal up to poly-log factors and is proven by building anΩ(kp 2 ) 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).