The crossing number of a graph @math is the least number of crossings over all possible drawings of @math . We present a structural characterization of graphs with crossing number one.
Our main result includes the following, slightly surprising, fact: a 4-connected nonplanar graph G has crossing number at least 2 if and only if, for every pair e,f of edges having no common incident vertex, there are vertex-disjoint cycles in G with one containing e and the other containing f.