  We study the geometry and topology of the large-scale structure traced by
galaxy clusters in numerical simulations of a box of side 320 $h^{-1}$ Mpc, and
compare them with available data on real clusters. The simulations we use are
generated by the Zel'dovich approximation, using the same methods as we have
used in the first three papers in this series. We consider the following models
to see if there are measurable differences in the topology and geometry of the
superclustering they produce: (i) the standard CDM model (SCDM); (ii) a CDM
model with $\Omega_0=0.2$ (OCDM); (iii) a CDM model with a `tilted' power
spectrum having $n=0.7$ (TCDM); (iv) a CDM model with a very low Hubble
constant, $h=0.3$ (LOWH); (v) a model with mixed CDM and HDM (CHDM); (vi) a
flat low-density CDM model with $\Omega_0=0.2$ and a non-zero cosmological
$\Lambda$ term ($\Lambda$CDM). We analyse these models using a variety of
statistical tests based on the analysis of: (i) the Euler-Poincar\'{e}
characteristic; (ii) percolation properties; (iii) the Minimal Spanning Tree
construction. Taking all these tests together we find that the best fitting
model is $\Lambda$CDM and, indeed, the others do not appear to be consistent
with the data. Our results demonstrate that despite their biased and extremely
sparse sampling of the cosmological density field, it is possible to use
clusters to probe subtle statistical diagnostics of models which go far beyond
the low-order correlation functions usually applied to study superclustering.
