  A graph $G$ is weakly $\gamma$-closed if every induced subgraph of $G$
contains one vertex $v$ such that for each non-neighbor $u$ of $v$ it holds
that $|N(u)\cap N(v)|<\gamma$. The weak closure $\gamma(G)$ of a graph,
recently introduced by Fox et al. [SIAM J. Comp. 2020], is the smallest number
such that $G$ is weakly $\gamma$-closed. This graph parameter is never larger
than the degeneracy (plus one) and can be significantly smaller. Extending the
work of Fox et al. [SIAM J. Comp. 2020] on clique enumeration, we show that
several problems related to finding dense subgraphs, such as the enumeration of
bicliques and $s$-plexes, are fixed-parameter tractable with respect to
$\gamma(G)$. Moreover, we show that the problem of determining whether a weakly
$\gamma$-closed graph $G$ has a subgraph on at least $k$ vertices that belongs
to a graph class $\mathcal{G}$ which is closed under taking subgraphs admits a
kernel with at most $\gamma k^2$ vertices. Finally, we provide fixed-parameter
algorithms for Independent Dominating Set and Dominating Clique when
parameterized by $\gamma+k$ where $k$ is the solution size.
