  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 edge-disjoint spanning trees. The famous results of Tutte and
Nash-Williams show that a graph with edge connectivity $\lambda$ contains
$\floor{\lambda/2}$ edge-disjoint spanning trees.
  We present connected dominating set (CDS) partition and packing as tools that
are analogous to edge-disjoint spanning trees and that help us to better grasp
the structure of graphs based on their vertex connectivity. The objective of
the CDS partition problem is to partition the nodes of a graph into as many
connected dominating sets as possible. The CDS packing problem is the
corresponding fractional relaxation, where CDSs are allowed to overlap as long
as this is compensated by assigning appropriate weights. CDS partition and CDS
packing can be viewed as the counterparts of the well-studied edge-disjoint
spanning trees, focusing on vertex disjointedness rather than edge
disjointness.
  We constructively show that every $k$-vertex-connected graph with $n$ nodes
has a CDS packing of size $\Omega(k/\log n)$ and a CDS partition of size
$\Omega(k/\log^5 n)$. We prove that the $\Omega(k/\log n)$ CDS packing bound is
existentially optimal.
  Using CDS packing, we show that if vertices of a $k$-vertex-connected graph
are independently sampled with probability $p$, then the graph induced by the
sampled vertices has vertex connectivity $\tilde{\Omega}(kp^2)$. Moreover,
using our $\Omega(k/\log n)$ CDS packing, we get a store-and-forward broadcast
algorithm with optimal throughput in the networking model where in each round,
each node can send one bounded-size message to all its neighbors.
