On the intersection of edges of a geometric graph by straight lines

A geometric graph ( = gg) is a pair G = (V, E), where V is a finite set of points ( = vertices) in general position in the plane, and E is a set of open straight line segments ( = edges) whose endpoints are in V. G is a convex gg ( = egg) if V is the set of vertices of a convex polygon. For n 3 1, 0 se c (;) and m Z= 1 let 2 = Z(n ,e, m) (I, = I,(n, e, m)) be the maximal number such that for every gg (egg) G with n vertices and e edges there exists a set of m lines whose union intersects at least 2 (I,) edges of G. In this paper we determine Z,(n, e, m) precisely for all admissible n, e and m and show that Z(n, e, m) = Z,(n, e, m) if 2me > n2 and in many other cases.