An edge e of a graph G is said to be crossing-critical if cr(G e) < cr(G), where cr(G) denotes the crossing number of G on the plane. It is proved that any crossingcritical edge e of a graph G for which cr(G e) ( 1 belongs to a subdivision of K, or K,,,, the Kuratowski subgraphs of G. Further, as regards crossing-critical edges e of G for which cr(G e) > 5, it is shown that the properties of “being a crossingcritical edge of G” and “being contained in a Kuratowski subgraph of G” are independent… CONTINUE READING