Algorithmica

, Volume 16, Issue 1, pp 111–117

# Applications of the crossing number

• J. Pach
• F. Shahrokhi
• M. Szegedy
Article

## Abstract

LetG be a graph ofn vertices that can be drawn in the plane by straight-line segments so that nok+1 of them are pairwise crossing. We show thatG has at mostc k nlog2k−2n edges. This gives a partial answer to a dual version of a well-known problem of Avital-Hanani, Erdós, Kupitz, Perles, and others. We also construct two point sets {p1,⋯,p n }, {q1,⋯,q n } in the plane such that any piecewise linear one-to-one mappingfR2R2 withf(pi)=qi (1≤in) is composed of at least Ω(n2) linear pieces. It follows from a recent result of Souvaine and Wenger that this bound is asymptotically tight. Both proofs are based on a relation between the crossing number and the bisection width of a graph.

## Key words

Crossing number Geometric graph Bisection width Triangulation

## References

1. [AAP]
P. K. Agarwal, B. Aronov, J. Pach, R. Pollack, and M. Sharir, Quasi-planar graphs have a linear number of edges,Proc. Graph Drawing '95, Passau (F. J. Brandenburg, ed.), Lecture Notes in Computer Science, Vol. 1207, Springer-Verlag, Berlin, 1996, pp. 1–7.Google Scholar
2. [AE]
N. Alon and P. Erdós, Disjoint edges in geometric graphs,Discrete Comput. Geom.,4 (1989), 287–290.Google Scholar
3. [AS]
N. Alon and J. Spencer,The Probabilistic Method, Wiley, New York, 1992.Google Scholar
4. [ASS]
B. Aronov, R. Seidel, and D. Souvaine, On compatible triangulations of simple polygons,Comput. Geom. Theory Appl.,3 (1993), 27–36.Google Scholar
5. [AH]
S. Avital and H. Hanani, Graphs,Gilyonot Lematematika,3 (1966), 2–8 (in Hebrew).Google Scholar
6. [CP]
V. Capoyleas and J. Pach, A Turán-type theorem on chords of a convex polygon,J. Combin. Theory Ser. B,56 (1992), 9–15.Google Scholar
7. [DDSV]
K. Diks, H. N. Djidjev, O. Sykora, and I. Vřto, Edge separators for planar graphs and their applications,Proc. 13th Symp. on Mathematical Foundations of Computer Science (M. P. Chytil, L. Janiga, V. Koubek, eds.), Lecture Notes in Computer Science, Vol. 324, Springer-Verlag, Berlin, 1988, pp. 280–290.Google Scholar
8. [E]
P. Erdós, On sets of distances ofn points,Amer. Math. Monthly,53 (1946), 248–250.Google Scholar
9. [GM]
H. Gazit and G. L. Miller, Planar separators and the Euclidean norm,Algorithms, Proc. International Symp. SIGAL '90 (T. Asanoet al., eds.), Lecture Notes in Computer Science, Vol. 450, Springer-Verlag, Berlin, 1990, pp. 338–347.Google Scholar
10. [GKK]
W. Goddard, M. Katchalski, and D. J. Kleitman, Forcing disjoint segments in the plane,European J. Combin., to appear.Google Scholar
11. [HP]
H. Hopf and E. Pannwitz, Aufgabe No. 167,Jahresber. Deutsch. Math.-Verein.,43 (1934), 114.Google Scholar
12. [K]
Y. S. Kupitz,Extremal Problems in Combinatorial Geometry, Arhus University Lecture Note Series, No. 53, Arhus University, Arhus, 1979.Google Scholar
13. [L]
F. T. Leighton,Complexity Issues in VLSI, Foundations of Computing Series, MIT Press, Cambridge, MA, 1983.Google Scholar
14. [LT]
R. J. Lipton and R. E. Tarjan, A separator theorem for planar graphs,SIAM J. Appl. Math.,36 (1979), 177–189.Google Scholar
15. [M]
G. L. Miller, Finding small simple cycle separators for 2-connected planar graphs,J. Comput. System Sci.,32 (1986), 265–279.Google Scholar
16. [OP]
P. O'Donnel and M. Perles, Every geometric graph withn vertices and 3.6n—3.4 edges contains three pairwise disjoint edges, Manuscript, Rutgers University, New Brunswick, 1991.Google Scholar
17. [P]
J. Pach, Notes on geometric graph theory, in:Discrete and Computational Geometry. Papers from DIMACS Special Year (J. Goodmanet al., eds.), DIMACS Series, Vol. 6, American Mathematical Society, Providence, RI, 1991, pp. 273–285.Google Scholar
18. [PT]
J. Pach and J. Törócsik, Some geometric applications of Dilworth's theorem,Discrete Comput. Geom.,12 (1994), 1–7.Google Scholar
19. [S]
A. Saalfeld, Joint triangulations and triangulation maps,Proc. 3rd Ann. ACM Symp. on Computational Geometry, 1987, pp. 195–204.Google Scholar
20. [SW]
D. Souvaine and R. Wenger, Constructing piecewise linear homeomorphisms,Comput. Geom. Theory Appl., to appear.Google Scholar
21. [U]
J. P. Ullman,Computational Aspects of VLSI, Computer Science Press, Rockville, MD, 1984.Google Scholar

© Springer-Verlag New York Inc. 1996

## Authors and Affiliations

• J. Pach
• 1
• 2
• F. Shahrokhi
• 3
• M. Szegedy
• 4
1. 1.Department of Computer Science, City CollegeCUNYNew YorkUSA
2. 2.Courant InstituteNYUNew YorkUSA
3. 3.Department of Computer ScienceUniversity of North TexasDentonUSA
4. 4.AT&T Bell LaboratoriesMurray HillUSA