  The intersection graph induced by a set $\Disks$ of $n$ disks can be dense.
It is thus natural to try and sparsify it, while preserving connectivity.
Unfortunately, sparse graphs can always be made disconnected by removing a
small number of vertices. In this work, we present a sparsification algorithm
that maintains connectivity between two disks in the computed graph, if the
original graph remains ``well-connected'' even after removing an arbitrary
``attack'' set $\BSet \subseteq \Disks$ from both graphs. Thus, the new sparse
graph has similar reliability to the original disk graph, and can withstand
catastrophic failure of nodes while still providing a connectivity guarantee
for the remaining graph. The new graphs has near linear complexity, and can be
constructed in near linear time.
  The algorithm extends to any collection of shapes in the plane, such that
their union complexity is near linear.
