  Given a branching random walk on a graph, we consider two kinds of
truncations: by inhibiting the reproduction outside a subset of vertices and by
allowing at most $m$ particles per site. We investigate the convergence of weak
and strong critical parameters of these truncated branching random walks to the
analogous parameters of the original branching random walk. As a corollary, we
apply our results to the study of the strong critical parameter of a branching
random walk restricted to the cluster of a Bernoulli bond percolation.
