@article{9602,
abstract = {An ordered graph is a graph with a linear ordering on its vertex set. We prove that for every positive integer k, there exists a constant ck > 0 such that any ordered graph G on n vertices with the property that neither G nor its complement contains an induced monotone path of size k, has either a clique or an independent set of size at least n^ck . This strengthens a result of Bousquet, Lagoutte, and Thomassé, who proved the analogous result for unordered graphs.
A key idea of the above paper was to show that any unordered graph on n vertices that does not contain an induced path of size k, and whose maximum degree is at most c(k)n for some small c(k) > 0, contains two disjoint linear size subsets with no edge between them. This approach fails for ordered graphs, because the analogous statement is false for k ≥ 3, by a construction of Fox. We provide some further examples showing that this statement also fails for ordered graphs avoiding other ordered trees.},
author = {Pach, János and Tomon, István},
issn = {00958956},
journal = {Journal of Combinatorial Theory. Series B},
pages = {21--37},
publisher = {Elsevier},
title = {{Erdős-Hajnal-type results for monotone paths}},
doi = {10.1016/j.jctb.2021.05.004},
volume = {151},
year = {2021},
}
@inproceedings{9604,
abstract = {Generalizing Lee’s inductive argument for counting the cells of higher order Voronoi tessellations in ℝ² to ℝ³, we get precise relations in terms of Morse theoretic quantities for piecewise constant functions on planar arrangements. Specifically, we prove that for a generic set of n ≥ 5 points in ℝ³, the number of regions in the order-k Voronoi tessellation is N_{k-1} - binom(k,2)n + n, for 1 ≤ k ≤ n-1, in which N_{k-1} is the sum of Euler characteristics of these function’s first k-1 sublevel sets. We get similar expressions for the vertices, edges, and polygons of the order-k Voronoi tessellation.},
author = {Biswas, Ranita and Cultrera di Montesano, Sebastiano and Edelsbrunner, Herbert and Saghafian, Morteza},
booktitle = {Leibniz International Proceedings in Informatics},
isbn = {9783959771849},
issn = {18688969},
location = {Online},
publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
title = {{Counting cells of order-k voronoi tessellations in ℝ^{3} with morse theory}},
doi = {10.4230/LIPIcs.SoCG.2021.16},
volume = {189},
year = {2021},
}
@article{8317,
abstract = {When can a polyomino piece of paper be folded into a unit cube? Prior work studied tree-like polyominoes, but polyominoes with holes remain an intriguing open problem. We present sufficient conditions for a polyomino with one or several holes to fold into a cube, and conditions under which cube folding is impossible. In particular, we show that all but five special “basic” holes guarantee foldability.},
author = {Aichholzer, Oswin and Akitaya, Hugo A. and Cheung, Kenneth C. and Demaine, Erik D. and Demaine, Martin L. and Fekete, Sándor P. and Kleist, Linda and Kostitsyna, Irina and Löffler, Maarten and Masárová, Zuzana and Mundilova, Klara and Schmidt, Christiane},
issn = {09257721},
journal = {Computational Geometry: Theory and Applications},
publisher = {Elsevier},
title = {{Folding polyominoes with holes into a cube}},
doi = {10.1016/j.comgeo.2020.101700},
volume = {93},
year = {2021},
}
@inproceedings{9605,
abstract = {Given a finite set A ⊂ ℝ^d, let Cov_{r,k} denote the set of all points within distance r to at least k points of A. Allowing r and k to vary, we obtain a 2-parameter family of spaces that grow larger when r increases or k decreases, called the multicover bifiltration. Motivated by the problem of computing the homology of this bifiltration, we introduce two closely related combinatorial bifiltrations, one polyhedral and the other simplicial, which are both topologically equivalent to the multicover bifiltration and far smaller than a Čech-based model considered in prior work of Sheehy. Our polyhedral construction is a bifiltration of the rhomboid tiling of Edelsbrunner and Osang, and can be efficiently computed using a variant of an algorithm given by these authors as well. Using an implementation for dimension 2 and 3, we provide experimental results. Our simplicial construction is useful for understanding the polyhedral construction and proving its correctness. },
author = {Corbet, René and Kerber, Michael and Lesnick, Michael and Osang, Georg F},
booktitle = {Leibniz International Proceedings in Informatics},
isbn = {9783959771849},
issn = {18688969},
location = {Online},
publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
title = {{Computing the multicover bifiltration}},
doi = {10.4230/LIPIcs.SoCG.2021.27},
volume = {189},
year = {2021},
}
@article{9649,
abstract = {Isomanifolds are the generalization of isosurfaces to arbitrary dimension and codimension, i.e. manifolds defined as the zero set of some multivariate vector-valued smooth function f : Rd → Rd−n. A natural (and efficient) way to approximate an isomanifold is to consider its Piecewise-Linear (PL) approximation based on a triangulation T of the ambient space Rd. In this paper, we give conditions under which the PL-approximation of an isomanifold is topologically equivalent to the isomanifold. The conditions are easy to satisfy in the sense that they can always be met by taking a sufficiently
fine triangulation T . This contrasts with previous results on the triangulation of manifolds where, in arbitrary dimensions, delicate perturbations are needed to guarantee topological correctness, which leads to strong limitations in practice. We further give a bound on the Fréchet distance between the original isomanifold and its PL-approximation. Finally we show analogous results for the PL-approximation of an isomanifold with boundary.},
author = {Boissonnat, Jean-Daniel and Wintraecken, Mathijs},
journal = {Foundations of Computational Mathematics },
publisher = {Springer Nature},
title = {{The topological correctness of PL approximations of isomanifolds}},
doi = {10.1007/s10208-021-09520-0},
year = {2021},
}
@inproceedings{9441,
abstract = {Isomanifolds are the generalization of isosurfaces to arbitrary dimension and codimension, i.e. submanifolds of ℝ^d defined as the zero set of some multivariate multivalued smooth function f: ℝ^d → ℝ^{d-n}, where n is the intrinsic dimension of the manifold. A natural way to approximate a smooth isomanifold M is to consider its Piecewise-Linear (PL) approximation M̂ based on a triangulation 𝒯 of the ambient space ℝ^d. In this paper, we describe a simple algorithm to trace isomanifolds from a given starting point. The algorithm works for arbitrary dimensions n and d, and any precision D. Our main result is that, when f (or M) has bounded complexity, the complexity of the algorithm is polynomial in d and δ = 1/D (and unavoidably exponential in n). Since it is known that for δ = Ω (d^{2.5}), M̂ is O(D²)-close and isotopic to M, our algorithm produces a faithful PL-approximation of isomanifolds of bounded complexity in time polynomial in d. Combining this algorithm with dimensionality reduction techniques, the dependency on d in the size of M̂ can be completely removed with high probability. We also show that the algorithm can handle isomanifolds with boundary and, more generally, isostratifolds. The algorithm for isomanifolds with boundary has been implemented and experimental results are reported, showing that it is practical and can handle cases that are far ahead of the state-of-the-art. },
author = {Boissonnat, Jean-Daniel and Kachanovich, Siargey and Wintraecken, Mathijs},
booktitle = {37th International Symposium on Computational Geometry (SoCG 2021)},
isbn = {978-3-95977-184-9},
issn = {1868-8969},
location = {Virtual},
pages = {17:1--17:16},
publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
title = {{Tracing isomanifolds in Rd in time polynomial in d using Coxeter-Freudenthal-Kuhn triangulations}},
doi = {10.4230/LIPIcs.SoCG.2021.17},
volume = {189},
year = {2021},
}
@inproceedings{9345,
abstract = {Modeling a crystal as a periodic point set, we present a fingerprint consisting of density functionsthat facilitates the efficient search for new materials and material properties. We prove invarianceunder isometries, continuity, and completeness in the generic case, which are necessary featuresfor the reliable comparison of crystals. The proof of continuity integrates methods from discretegeometry and lattice theory, while the proof of generic completeness combines techniques fromgeometry with analysis. The fingerprint has a fast algorithm based on Brillouin zones and relatedinclusion-exclusion formulae. We have implemented the algorithm and describe its application tocrystal structure prediction.},
author = {Edelsbrunner, Herbert and Heiss, Teresa and Kurlin , Vitaliy and Smith, Philip and Wintraecken, Mathijs},
booktitle = {37th International Symposium on Computational Geometry (SoCG 2021)},
issn = {1868-8969},
location = {Virtual},
pages = {32:1--32:16},
publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
title = {{The density fingerprint of a periodic point set}},
doi = {10.4230/LIPIcs.SoCG.2021.32},
volume = {189},
year = {2021},
}
@inproceedings{9296,
abstract = { matching is compatible to two or more labeled point sets of size n with labels {1,…,n} if its straight-line drawing on each of these point sets is crossing-free. We study the maximum number of edges in a matching compatible to two or more labeled point sets in general position in the plane. We show that for any two labeled convex sets of n points there exists a compatible matching with ⌊2n−−√⌋ edges. More generally, for any ℓ labeled point sets we construct compatible matchings of size Ω(n1/ℓ) . As a corresponding upper bound, we use probabilistic arguments to show that for any ℓ given sets of n points there exists a labeling of each set such that the largest compatible matching has O(n2/(ℓ+1)) edges. Finally, we show that Θ(logn) copies of any set of n points are necessary and sufficient for the existence of a labeling such that any compatible matching consists only of a single edge.},
author = {Aichholzer, Oswin and Arroyo Guevara, Alan M and Masárová, Zuzana and Parada, Irene and Perz, Daniel and Pilz, Alexander and Tkadlec, Josef and Vogtenhuber, Birgit},
booktitle = {15th International Conference on Algorithms and Computation},
isbn = {9783030682101},
issn = {16113349},
location = {Yangon, Myanmar},
pages = {221--233},
publisher = {Springer Nature},
title = {{On compatible matchings}},
doi = {10.1007/978-3-030-68211-8_18},
volume = {12635},
year = {2021},
}
@article{7962,
abstract = {A string graph is the intersection graph of a family of continuous arcs in the plane. The intersection graph of a family of plane convex sets is a string graph, but not all string graphs can be obtained in this way. We prove the following structure theorem conjectured by Janson and Uzzell: The vertex set of almost all string graphs on n vertices can be partitioned into five cliques such that some pair of them is not connected by any edge (n→∞). We also show that every graph with the above property is an intersection graph of plane convex sets. As a corollary, we obtain that almost all string graphs on n vertices are intersection graphs of plane convex sets.},
author = {Pach, János and Reed, Bruce and Yuditsky, Yelena},
issn = {14320444},
journal = {Discrete and Computational Geometry},
number = {4},
pages = {888--917},
publisher = {Springer Nature},
title = {{Almost all string graphs are intersection graphs of plane convex sets}},
doi = {10.1007/s00454-020-00213-z},
volume = {63},
year = {2020},
}
@inproceedings{8135,
abstract = {Discrete Morse theory has recently lead to new developments in the theory of random geometric complexes. This article surveys the methods and results obtained with this new approach, and discusses some of its shortcomings. It uses simulations to illustrate the results and to form conjectures, getting numerical estimates for combinatorial, topological, and geometric properties of weighted and unweighted Delaunay mosaics, their dual Voronoi tessellations, and the Alpha and Wrap complexes contained in the mosaics.},
author = {Edelsbrunner, Herbert and Nikitenko, Anton and Ölsböck, Katharina and Synak, Peter},
booktitle = {Topological Data Analysis},
isbn = {9783030434076},
issn = {21978549},
pages = {181--218},
publisher = {Springer Nature},
title = {{Radius functions on Poisson–Delaunay mosaics and related complexes experimentally}},
doi = {10.1007/978-3-030-43408-3_8},
volume = {15},
year = {2020},
}
@article{8163,
abstract = {Fejes Tóth [3] studied approximations of smooth surfaces in three-space by piecewise flat triangular meshes with a given number of vertices on the surface that are optimal with respect to Hausdorff distance. He proves that this Hausdorff distance decreases inversely proportional with the number of vertices of the approximating mesh if the surface is convex. He also claims that this Hausdorff distance is inversely proportional to the square of the number of vertices for a specific non-convex surface, namely a one-sheeted hyperboloid of revolution bounded by two congruent circles. We refute this claim, and show that the asymptotic behavior of the Hausdorff distance is linear, that is the same as for convex surfaces.},
author = {Vegter, Gert and Wintraecken, Mathijs},
issn = {1588-2896},
journal = {Studia Scientiarum Mathematicarum Hungarica},
number = {2},
pages = {193--199},
publisher = {AKJournals},
title = {{Refutation of a claim made by Fejes Tóth on the accuracy of surface meshes}},
doi = {10.1556/012.2020.57.2.1454},
volume = {57},
year = {2020},
}
@article{8248,
abstract = {We consider the following setting: suppose that we are given a manifold M in Rd with positive reach. Moreover assume that we have an embedded simplical complex A without boundary, whose vertex set lies on the manifold, is sufficiently dense and such that all simplices in A have sufficient quality. We prove that if, locally, interiors of the projection of the simplices onto the tangent space do not intersect, then A is a triangulation of the manifold, that is, they are homeomorphic.},
author = {Boissonnat, Jean-Daniel and Dyer, Ramsay and Ghosh, Arijit and Lieutier, Andre and Wintraecken, Mathijs},
issn = {0179-5376},
journal = {Discrete and Computational Geometry},
publisher = {Springer Nature},
title = {{Local conditions for triangulating submanifolds of Euclidean space}},
doi = {10.1007/s00454-020-00233-9},
year = {2020},
}
@article{8323,
author = {Pach, János},
issn = {14320444},
journal = {Discrete and Computational Geometry},
pages = {571--574},
publisher = {Springer Nature},
title = {{A farewell to Ricky Pollack}},
doi = {10.1007/s00454-020-00237-5},
volume = {64},
year = {2020},
}
@article{8338,
abstract = {Canonical parametrisations of classical confocal coordinate systems are introduced and exploited to construct non-planar analogues of incircular (IC) nets on individual quadrics and systems of confocal quadrics. Intimate connections with classical deformations of quadrics that are isometric along asymptotic lines and circular cross-sections of quadrics are revealed. The existence of octahedral webs of surfaces of Blaschke type generated by asymptotic and characteristic lines that are diagonally related to lines of curvature is proved theoretically and established constructively. Appropriate samplings (grids) of these webs lead to three-dimensional extensions of non-planar IC nets. Three-dimensional octahedral grids composed of planes and spatially extending (checkerboard) IC-nets are shown to arise in connection with systems of confocal quadrics in Minkowski space. In this context, the Laguerre geometric notion of conical octahedral grids of planes is introduced. The latter generalise the octahedral grids derived from systems of confocal quadrics in Minkowski space. An explicit construction of conical octahedral grids is presented. The results are accompanied by various illustrations which are based on the explicit formulae provided by the theory.},
author = {Akopyan, Arseniy and Bobenko, Alexander I. and Schief, Wolfgang K. and Techter, Jan},
issn = {14320444},
journal = {Discrete and Computational Geometry},
publisher = {Springer Nature},
title = {{On mutually diagonal nets on (confocal) quadrics and 3-dimensional webs}},
doi = {10.1007/s00454-020-00240-w},
year = {2020},
}
@article{8538,
abstract = {We prove some recent experimental observations of Dan Reznik concerning periodic billiard orbits in ellipses. For example, the sum of cosines of the angles of a periodic billiard polygon remains constant in the 1-parameter family of such polygons (that exist due to the Poncelet porism). In our proofs, we use geometric and complex analytic methods.},
author = {Akopyan, Arseniy and Schwartz, Richard and Tabachnikov, Serge},
issn = {21996768},
journal = {European Journal of Mathematics},
publisher = {Springer Nature},
title = {{Billiards in ellipses revisited}},
doi = {10.1007/s40879-020-00426-9},
year = {2020},
}
@inproceedings{8580,
abstract = {We evaluate the usefulness of persistent homology in the analysis of heart rate variability. In our approach we extract several topological descriptors characterising datasets of RR-intervals, which are later used in classical machine learning algorithms. By this method we are able to differentiate the group of patients with the history of transient ischemic attack and the group of hypertensive patients.},
author = {Graff, Grzegorz and Graff, Beata and Jablonski, Grzegorz and Narkiewicz, Krzysztof},
booktitle = {11th Conference of the European Study Group on Cardiovascular Oscillations: Computation and Modelling in Physiology: New Challenges and Opportunities, },
isbn = {9781728157511},
location = {Pisa, Italy},
publisher = {IEEE},
title = {{The application of persistent homology in the analysis of heart rate variability}},
doi = {10.1109/ESGCO49734.2020.9158054},
year = {2020},
}
@inproceedings{8703,
abstract = {Even though Delaunay originally introduced his famous triangulations in the case of infinite point sets with translational periodicity, a software that computes such triangulations in the general case is not yet available, to the best of our knowledge. Combining and generalizing previous work, we present a practical algorithm for computing such triangulations. The algorithm has been implemented and experiments show that its performance is as good as the one of the CGAL package, which is restricted to cubic periodicity. },
author = {Osang, Georg F and Rouxel-Labbé, Mael and Teillaud, Monique},
booktitle = {28th Annual European Symposium on Algorithms},
isbn = {9783959771627},
issn = {18688969},
location = {Virtual, Online; Pisa, Italy},
publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
title = {{Generalizing CGAL periodic Delaunay triangulations}},
doi = {10.4230/LIPIcs.ESA.2020.75},
volume = {173},
year = {2020},
}
@inbook{74,
abstract = {We study the Gromov waist in the sense of t-neighborhoods for measures in the Euclidean space, motivated by the famous theorem of Gromov about the waist of radially symmetric Gaussian measures. In particular, it turns our possible to extend Gromov’s original result to the case of not necessarily radially symmetric Gaussian measure. We also provide examples of measures having no t-neighborhood waist property, including a rather wide class
of compactly supported radially symmetric measures and their maps into the Euclidean space of dimension at least 2.
We use a simpler form of Gromov’s pancake argument to produce some estimates of t-neighborhoods of (weighted) volume-critical submanifolds in the spirit of the waist theorems, including neighborhoods of algebraic manifolds in the complex projective space. In the appendix of this paper we provide for reader’s convenience a more detailed explanation of the Caffarelli theorem that we use to handle not necessarily radially symmetric Gaussian
measures.},
author = {Akopyan, Arseniy and Karasev, Roman},
booktitle = {Geometric Aspects of Functional Analysis},
editor = {Klartag, Bo'az and Milman, Emanuel},
isbn = {9783030360191},
issn = {16179692},
pages = {1--27},
publisher = {Springer Nature},
title = {{Gromov's waist of non-radial Gaussian measures and radial non-Gaussian measures}},
doi = {10.1007/978-3-030-36020-7_1},
volume = {2256},
year = {2020},
}
@phdthesis{7460,
abstract = {Many methods for the reconstruction of shapes from sets of points produce ordered simplicial complexes, which are collections of vertices, edges, triangles, and their higher-dimensional analogues, called simplices, in which every simplex gets assigned a real value measuring its size. This thesis studies ordered simplicial complexes, with a focus on their topology, which reflects the connectedness of the represented shapes and the presence of holes. We are interested both in understanding better the structure of these complexes, as well as in developing algorithms for applications.
For the Delaunay triangulation, the most popular measure for a simplex is the radius of the smallest empty circumsphere. Based on it, we revisit Alpha and Wrap complexes and experimentally determine their probabilistic properties for random data. Also, we prove the existence of tri-partitions, propose algorithms to open and close holes, and extend the concepts from Euclidean to Bregman geometries.},
author = {Ölsböck, Katharina},
issn = {2663-337X},
keywords = {shape reconstruction, hole manipulation, ordered complexes, Alpha complex, Wrap complex, computational topology, Bregman geometry},
pages = {155},
publisher = {IST Austria},
title = {{The hole system of triangulated shapes}},
doi = {10.15479/AT:ISTA:7460},
year = {2020},
}
@article{7554,
abstract = {Slicing a Voronoi tessellation in ${R}^n$ with a $k$-plane gives a $k$-dimensional weighted Voronoi tessellation, also known as a power diagram or Laguerre tessellation. Mapping every simplex of the dual weighted Delaunay mosaic to the radius of the smallest empty circumscribed sphere whose center lies in the $k$-plane gives a generalized discrete Morse function. Assuming the Voronoi tessellation is generated by a Poisson point process in ${R}^n$, we study the expected number of simplices in the $k$-dimensional weighted Delaunay mosaic as well as the expected number of intervals of the Morse function, both as functions of a radius threshold. As a by-product, we obtain a new proof for the expected number of connected components (clumps) in a line section of a circular Boolean model in ${R}^n$.},
author = {Edelsbrunner, Herbert and Nikitenko, Anton},
issn = {10957219},
journal = {Theory of Probability and its Applications},
number = {4},
pages = {595--614},
publisher = {SIAM},
title = {{Weighted Poisson–Delaunay mosaics}},
doi = {10.1137/S0040585X97T989726},
volume = {64},
year = {2020},
}