 
Summary: Discrete Mathematics 60 (1986) 7590
NorthHolland
75
ON THE INTERSECTION OF EDGES OF A GEOMETRIC
GRAPH BY STRAIGHT LINES
N. ALON
Department of Mathematics, Massachusetts Institute of Technology, MA, U.S.A.
M.A. PERLES
Institute of Mathematics, Hebrew University of Jerusalem, Jerusalem, Israel
Received 1 October 1984
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.
1. Introduction
A geometric graph ( = gg) is a pair G = (V, E), where V is a finite set of points
