  Motivated by a problem asked by Richter and by the long standing Harary-Hill
conjecture, we study the relation between the crossing number of a graph $G$
and the crossing number of its cone $CG$, the graph obtained from $G$ by adding
a new vertex adjacent to all the vertices in $G$. Simple examples show that the
difference $cr(CG)-cr(G)$ can be arbitrarily large for any fixed $k=cr(G)$. In
this work, we are interested in finding the smallest possible difference, that
is, for each non-negative integer $k$, find the smallest $f(k)$ for which there
exists a graph with crossing number at least $k$ and cone with crossing number
$f(k)$. For small values of $k$, we give exact values of $f(k)$ when the
problem is restricted to simple graphs, and show that $f(k)=k+\Theta (\sqrt
{k})$ when multiple edges are allowed.
