TY - JOUR
T1 - On the intersection of edges of a geometric graph by straight lines
AU - Alon, N.
AU - Perles, M. A.
PY - 1986
Y1 - 1986
N2 - 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) (Ic=Ic(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 (Ic) edges of G. In this paper we determine Ic(n,e,m) precisely for all admissible n, e and m and show that I(n,e,m) = Ic(n,e,m) if 2me≥n2 and in many other cases.
AB - 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) (Ic=Ic(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 (Ic) edges of G. In this paper we determine Ic(n,e,m) precisely for all admissible n, e and m and show that I(n,e,m) = Ic(n,e,m) if 2me≥n2 and in many other cases.
UR - http://www.scopus.com/inward/record.url?scp=38249039472&partnerID=8YFLogxK
U2 - 10.1016/0012-365X(86)90004-X
DO - 10.1016/0012-365X(86)90004-X
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AN - SCOPUS:38249039472
SN - 0012-365X
VL - 60
SP - 75
EP - 90
JO - Discrete Mathematics
JF - Discrete Mathematics
IS - C
ER -