  Complexes of (not) connected graphs, hypergraphs and their homology appear in
the construction of knot invariants given by V. Vassiliev. In this paper we
study the complexes of not $i$-connected $k$-hypergraphs on $n$ vertices. We
show that the complex of not $2$-connected graphs has the homotopy type of a
wedge of $(n-2)!$ spheres of dimension $2n-5$. This answers one of the
questions raised by Vassiliev in connection with knot invariants. For this case
the $S_n$-action on the homology of the complex is also determined. For
complexes of not $2$-connected $k$-hypergraphs we provide a formula for the
generating function of the Euler characteristic, and we introduce certain
lattices of graphs that encode their topology. We also present partial results
for some other cases. In particular, we show that the complex of not
$(n-2)$-connected graphs is Alexander dual to the complex of partial matchings
of the complete graph. For not $(n-3)$-connected graphs we provide a formula
for the generating function of the Euler characteristic.
