  We show that the independence complexes of generalised Mycielskian of
complete graphs are homotopy equivalent to a wedge sum of spheres, and
determine the number of copies and the dimensions of these spheres. We also
prove that the independence complexes of categorical product of complete graphs
are wedge sum of circles, upto homotopy. Further, we show that if we perturb a
graph $G$ in a certain way, then the independence complex of this new graph is
homotopy equivalent to the suspension of the independence complex of $G$.
