Before revision:  Then  the crossing edges are incident to a common vertex . Graphs are k-planar if each edge is crossed by at most k edges, and k-gap-planar if each crossing is assigned to an edge involved in the crossing, so that at most k crossings are assigned to each edge .
After revision:  Then if the crossing edges are incident to a common vertex . Graphs are k-planar if each edge is crossed by at most k edges, and k-gap-planar if each crossing is assigned to an edge involved in the crossing, so that at most k crossings are assigned to each edge .