Counting Carambolas
Adrian Dumitrescu, Maarten L\"offler, Andr\'e Schulz, and Csaba D., T\'oth
TL;DR
This paper investigates bounds on the number of specific geometric configurations such as convex polygons, star-shaped polygons, and monotone paths within triangulations of planar point sets, including directed graphs.
Contribution
It provides new upper and lower bounds on the counts of various geometric configurations in planar triangulations and directed graphs.
Findings
Established bounds for convex polygons in triangulations
Derived limits for star-shaped polygons and monotone paths
Extended results to directed planar graphs
Abstract
We give upper and lower bounds on the maximum and minimum number of geometric configurations of various kinds present (as subgraphs) in a triangulation of points in the plane. Configurations of interest include \emph{convex polygons}, \emph{star-shaped polygons} and \emph{monotone paths}. We also consider related problems for \emph{directed} planar straight-line graphs.
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