  Although topological insulators (TIs) are known to be robust against
non-magnetic perturbations and exhibit edge or surface states as their distinct
feature, experimentally it is known that vacancies often occur in these
materials and impose strong perturbations. Here we investigate effects of
vacancies on the stability of Z$_{2}$ topological order using the Kane-Mele
(KM) model as a prototype of topological insulator. It is shown that even
though a vacancy is not classified as a topological defect in KM model, it
generally induces a pair of degenerate midgap states only in the TI phase. We
show that these midgap states results from edge states that fit into vacancies
and are characterized by the same Z$_{2}$ topological order. Furthermore, in
the presence of many vacancies, an impurity band that is degenerate with edge
states in energy is induced and mixes directly with edge states. However, the
Z$_{2}$ topological order persists and edge states exist between the impurity
band and perturbed bulk bands until a phase transition occurs when Dirac cones
near Dirac points are depleted. Our analyses indicate that the same scenario
holds for point vacancies or line of vacancies in 3D TIs as well.
