  Extremal properties of sparse graphs, randomly perturbed by the binomial
random graph are considered. It is known that every $n$-vertex graph $G$
contains a complete minor of order $\Omega(n/\alpha(G))$. We prove that adding
$\xi n$ random edges, where $\xi > 0$ is arbitrarily small yet fixed, to an
$n$-vertex graph $G$ satisfying $\alpha(G) \leq \zeta(\xi)n$ asymptotically
almost surely results in a graph containing a complete minor of order $\tilde
\Omega \left( n/\sqrt{\alpha(G)}\right)$; this result is tight up to the
implicit logarithmic terms.
  For complete topological minors, we prove that there exists a constant $C>0$
such that adding $C n$ random edges to a graph $G$ satisfying $\delta(G) =
\omega(1)$, asymptotically almost surely results in a graph containing a
complete topological minor of order $\tilde
\Omega(\min\{\delta(G),\sqrt{n}\})$; this result is tight up to the implicit
logarithmic terms.
  Finally, extending results of Bohman, Frieze, Krivelevich, and Martin for the
dense case, we analyse the asymptotic behaviour of the vertex-connectivity and
the diameter of randomly perturbed sparse graphs.
