  In the presented article, statistical properties regarding the topology and
standard percolation on relative neighborhood graphs (RNGs) for planar sets of
points, considering the Euclidean metric, are put under scrutiny. RNGs belong
to the family of ``proximity graphs'', i.e. their edge-set encodes proximity
information regarding the close neighbors for the terminal nodes of a given
edge. Therefore they are, e.g., discussed in the context of the construction of
backbones for wireless ad-hoc networks that guarantee connectedness of all
underlying nodes.
  Here, by means of numerical simulations, we determine the asymptotic degree
and diameter of RNGs and we estimate their bond and site percolation
thresholds, which were previously conjectured to be nontrivial. We compare the
results to regular 2D graphs for which the degree is close to that of the RNG.
Finally, we deduce the common percolation critical exponents from the RNG data
to verify that the associated universality class is that of standard 2D
percolation.
