Abstract
Let P be a set of n points such that each of its elements has a unique weight in {1, …,n}. P is called a wp-set. A non-crossing polygonal line connecting some elements of P in increasing (or decreasing) order of their weights is called a monotonic path of P. A simple polygon with vertices in P is called monotonic if it is formed by a monotonic path and an edge connecting its endpoints. In this paper we study the problem of finding large monotonic polygons and paths in wp-sets. We establish some sharp bounds concerning these problems. We also study extremal problems on the number of monotonic paths and polygons of a wp-set.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Bárány, I., Valtr, P.: Planar point sets with a small number of empty convex polygons. Studia Sci. Math. Hungar. 41, 243–266 (2004)
Chung, F.R.K.: On unimodal subsequences. J. Combin. Theory Ser. A 29, 267–279 (1980)
Czyzowicz, J., Kranakis, E., Krizanc, D., Urrutia, J.: Maximal length common non-intersecting paths. In: Proc. Eighth Canadian Conference on Computational Geometry, Ottawa, pp. 180–189 (August 1996)
Dehnhardt, K.: Leere konvexe Vielecke in ebenen Punktmengen, Dissertation, TU Braunschweig (1987)
Erdős, P.: Some more problems on elementary geometry. Austral. Math. Soc. Gaz. 5, 52–54 (1978)
Erdős, P., Szekeres, G.: A combinatorial problem in geometry. Compositio Math. 2, 463–470 (1935)
Gerken, T.: Empty convex hexagons in planar point sets. Discrete & Computational Geometry 39(1-3), 239–272 (2008)
Horton, J.D.: Sets with no empty convex 7-gon. Canad. Math. Bull. 26, 482–484 (1983)
Katchalski, M., Meir, A.: On empty triangles determined by points in the plane. Acta. Math. Hungar. 51, 323–328 (1988)
Nicolás, C.M.: The empty hexagon theorem. Discrete & Computational Geometry 38, 389–397 (2007)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Sakai, T., Urrutia, J. (2011). Monotonic Polygons and Paths in Weighted Point Sets. In: Akiyama, J., Bo, J., Kano, M., Tan, X. (eds) Computational Geometry, Graphs and Applications. CGGA 2010. Lecture Notes in Computer Science, vol 7033. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-24983-9_17
Download citation
DOI: https://doi.org/10.1007/978-3-642-24983-9_17
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-24982-2
Online ISBN: 978-3-642-24983-9
eBook Packages: Computer ScienceComputer Science (R0)