  We consider two classes of random graphs: $(a)$ Poissonian random graphs in
which the $n$ vertices in the graph have i.i.d.\ weights distributed as $X$,
where $\mathbb{E}(X) = \mu$. Edges are added according to a product measure and
the probability that a vertex of weight $x$ shares and edge with a vertex of
weight $y$ is given by $1-e^{-xy/(\mu n)}$. $(b)$ A thinned configuration model
in which we create a ground-graph in which the $n$ vertices have i.i.d.\
ground-degrees, distributed as $D$, with $\mathbb{E}(D) = \mu$. The graph of
interest is obtained by deleting edges independently with probability $1-p$.
  In both models the fraction of vertices in the largest connected component
converges in probability to a constant $1-q$, where $q$ depends on $X$ or $D$
and $p$.
  We investigate for which distributions $X$ and $D$ with given $\mu$ and $p$,
$1-q$ is maximized. We show that in the class of Poissonian random graphs, $X$
should have all its mass at 0 and one other real, which can be explicitly
determined. For the thinned configuration model $D$ should have all its mass at
0 and two subsequent positive integers.
