A proof of Vassiliev's conjecture on the planarity of singular links

We prove that a finite 4-valent graph with a cross structure at each vertex cannot be embedded in the plane with respect to this structure if and only if there are two cycles without common edges and with precisely one intersection point that is transversal with respect to the cross structure. This leads to an algorithm for recognizing the planarity of such a graph which is quadratic in the number of vertices.