  The pivotal quality of proximity graphs is connectivity, i.e. all nodes in
the graph are connected to one another either directly or via intermediate
nodes. These types of graphs are robust, i.e., they are able to function well
even if they are subject to limited removal of elementary building blocks, as
it may occur for random failures or targeted attacks. Here, we study how the
structure of these graphs is affected when nodes get removed successively until
an extensive fraction is removed such that the graphs fragment. We study
different types of proximity graphs for various node removal strategies. We use
different types of observables to monitor the fragmentation process, simple
ones like number and sizes of connected components, and more complex ones like
the hop diameter and the backup capacity, which is needed to make a network N-1
resilient. The actual fragmentation turns out to be described by a second order
phase transition. Using finite-size scaling analyses we numerically assess the
threshold fraction of removed nodes, which is characteristic for the particular
graph type and node deletion scheme, that suffices to decompose the underlying
graphs.
