Suppose that I have a graph $G=(V,E)$ with a known crossing number, say $n=cr(G)$. I would like to show that the graph $G'=(V,E\cup e')$ obtained by adding an edge $e'$ to $G$ that was not previously in it satisfies the relation:
$$cr(G')\leq cr(G)+1.$$
Is this true in general? Or at least for the family of complete graphs $K_{m,n}$?