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Counting Carambolas

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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 n points in the plane. Configurations of interest include convex polygons, star-shaped polygons and monotone paths. We also consider related problems for directed planar straight-line graphs.

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  1. Throughout this paper, all logarithms are in base 2.

  2. Such paths were inadvertently overlooked in the analysis from [13].


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A. Dumitrescu was supported in part by NSF grant DMS-1001667. M. Löffler was supported by the Netherlands Organization for Scientific Research (NWO) under grant 639.021.123. Research by Tóth was supported in part by NSERC (RGPIN 35586) and NSF (CCF-1423615). This work was initiated at the workshop “Counting and Enumerating Plane Graphs,” which took place at Schloss Dagstuhl in March, 2013.

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Correspondence to Csaba D. Tóth.

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A preliminary version of this paper appeared in the Proceedings of the 25th Canadian Conference on Computational Geometry [13].

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Dumitrescu, A., Löffler, M., Schulz, A. et al. Counting Carambolas. Graphs and Combinatorics 32, 923–942 (2016).

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