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 ( = cgg) if V is the set of vertices of a convex polygon. For n≥1, 0≤e≤(ⁿ₂) and m≥1 let I = I(n,e,m) (I_c=I_c(n,e,m)) be the maximal number such that for every gg (cgg) G with n vertices and e edges there exists a set of m lines whose union intersects at least I (I_c) edges of G. In this paper we determine I_c(n,e,m) precisely for all admissible n, e and m and show that I(n,e,m) = I_c(n,e,m) if 2me≥n² and in many other cases.