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**Published in:** LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

We introduce the abstract notion of a chain, which is a sequence of n points in the plane, ordered by x-coordinates, so that the edge between any two consecutive points is unavoidable as far as triangulations are concerned. A general theory of the structural properties of chains is developed, alongside a general understanding of their number of triangulations.
We also describe an intriguing new and concrete configuration, which we call the Koch chain due to its similarities to the Koch curve. A specific construction based on Koch chains is then shown to have Ω(9.08ⁿ) triangulations. This is a significant improvement over the previous and long-standing lower bound of Ω(8.65ⁿ) for the maximum number of triangulations of planar point sets.

Daniel Rutschmann and Manuel Wettstein. Chains, Koch Chains, and Point Sets with Many Triangulations. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 59:1-59:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{rutschmann_et_al:LIPIcs.SoCG.2022.59, author = {Rutschmann, Daniel and Wettstein, Manuel}, title = {{Chains, Koch Chains, and Point Sets with Many Triangulations}}, booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)}, pages = {59:1--59:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-227-3}, ISSN = {1868-8969}, year = {2022}, volume = {224}, editor = {Goaoc, Xavier and Kerber, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.59}, URN = {urn:nbn:de:0030-drops-160678}, doi = {10.4230/LIPIcs.SoCG.2022.59}, annote = {Keywords: Planar Point Set, Chain, Koch Chain, Triangulation, Maximum Number of Triangulations, Lower Bound} }

Document

**Published in:** LIPIcs, Volume 77, 33rd International Symposium on Computational Geometry (SoCG 2017)

A set P = H cup {w} of n+1 points in the plane is called a wheel set if all points but w are extreme. We show that for the purpose of counting crossing-free geometric graphs on P, it suffices to know the so-called frequency vector of P. While there are roughly 2^n distinct order types that correspond to wheel sets, the number of frequency vectors is only about 2^{n/2}.
We give simple formulas in terms of the frequency vector for the number of crossing-free spanning cycles, matchings, w-embracing triangles, and many more. Based on these formulas, the corresponding numbers of graphs can be computed efficiently.
Also in higher dimensions, wheel sets turn out to be a suitable model to approach the problem of computing the simplicial depth of a point w in a set H, i.e., the number of simplices spanned by H that contain w. While the concept of frequency vectors does not generalize easily, we show how to apply similar methods in higher dimensions. The result is an O(n^{d-1}) time algorithm for computing the simplicial depth of a point w in a set H of n d-dimensional points, improving on the previously best bound of O(n^d log n).
Configurations equivalent to wheel sets have already been used by Perles for counting the faces of high-dimensional polytopes with few vertices via the Gale dual. Based on that we can compute the number of facets of the convex hull of n=d+k points in general position in R^d in time O(n^max(omega,k-2)) where omega = 2.373, even though the asymptotic number of facets may be as large as n^k.

Alexander Pilz, Emo Welzl, and Manuel Wettstein. From Crossing-Free Graphs on Wheel Sets to Embracing Simplices and Polytopes with Few Vertices. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 54:1-54:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{pilz_et_al:LIPIcs.SoCG.2017.54, author = {Pilz, Alexander and Welzl, Emo and Wettstein, Manuel}, title = {{From Crossing-Free Graphs on Wheel Sets to Embracing Simplices and Polytopes with Few Vertices}}, booktitle = {33rd International Symposium on Computational Geometry (SoCG 2017)}, pages = {54:1--54:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-038-5}, ISSN = {1868-8969}, year = {2017}, volume = {77}, editor = {Aronov, Boris and Katz, Matthew J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.54}, URN = {urn:nbn:de:0030-drops-72101}, doi = {10.4230/LIPIcs.SoCG.2017.54}, annote = {Keywords: Geometric Graph, Wheel Set, Simplicial Depth, Gale Transform, Polytope} }

Document

**Published in:** LIPIcs, Volume 30, 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)

We show that every triangulation (maximal planar graph) on n\ge 6 vertices can be flipped into a Hamiltonian triangulation using a sequence of less than n/2 combinatorial edge flips. The previously best upper bound uses 4-connectivity as a means to establish Hamiltonicity. But in general about 3n/5 flips are necessary to reach a 4-connected triangulation. Our result improves the upper bound on the diameter of the flip graph of combinatorial triangulations on n vertices from 5.2n-33.6 to 5n-23. We also show that for every triangulation on n vertices there is a simultaneous flip of less than 2n/3 edges to a 4-connected triangulation. The bound on the number of edges is tight, up to an additive constant. As another application we show that every planar graph on n vertices admits an arc diagram with less than n/2 biarcs, that is, after subdividing less than n/2 (of potentially 3n-6) edges the resulting graph admits a 2-page book embedding.

Jean Cardinal, Michael Hoffmann, Vincent Kusters, Csaba D. Tóth, and Manuel Wettstein. Arc Diagrams, Flip Distances, and Hamiltonian Triangulations. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 197-210, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{cardinal_et_al:LIPIcs.STACS.2015.197, author = {Cardinal, Jean and Hoffmann, Michael and Kusters, Vincent and T\'{o}th, Csaba D. and Wettstein, Manuel}, title = {{Arc Diagrams, Flip Distances, and Hamiltonian Triangulations}}, booktitle = {32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)}, pages = {197--210}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-78-1}, ISSN = {1868-8969}, year = {2015}, volume = {30}, editor = {Mayr, Ernst W. and Ollinger, Nicolas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2015.197}, URN = {urn:nbn:de:0030-drops-49141}, doi = {10.4230/LIPIcs.STACS.2015.197}, annote = {Keywords: graph embeddings, edge flips, flip graph, separating triangles} }