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 nonnegative 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.
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