@inproceedings{22000, abstract = {Simplicial approximation provides a framework for constructing simplicial complexes that are homotopy equivalent to a given manifold, provided a CW structure is explicitly known. However, its conventional implementation quickly becomes intractable on a computer: barycentric subdivision produces poorly shaped simplices, and the star condition introduces many vertices. To address these limitations, this article develops a subdivision scheme based on spherical Delaunay triangulations, which attains better refinement properties than barycentric subdivisions. Moreover, the star condition is reframed as two independent problems, one geometric and the other combinatorial, respectively tackled in the language of locally equiconnected spaces and the list homomorphism problem, allowing an exponential reduction in the number of vertices. Via a prototype implementation, we obtain simplicial complexes homotopy equivalent to Grassmannians and Stiefel manifolds up to dimension 5.}, author = {Tinarrage, Raphaël}, booktitle = {42nd International Symposium on Computational Geometry}, isbn = {9783959774185}, issn = {1868-8969}, keywords = {Triangulation of manifolds, Simplicial approximation, CW complexes, Delaunay complexes, List homomorphism problem, Topological Data Analysis}, location = {New Brunswick, NJ, United States}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{Simplicial approximation to CW complexes with spherical Delaunay triangulations}}, doi = {10.4230/LIPIcs.SoCG.2026.93}, volume = {367}, year = {2026}, } @inproceedings{22002, abstract = {Topological simplification is the process of reducing complexity of a function while maintaining its essential features. Its goal is to find a new filter function, which reorders cells of the input complex in a way which eliminates some persistent homological features, without affecting the rest. We present a new approach to simplification based on the concept of forbidden regions and combinatorial dynamics. It allows us to reorder and cancel critical values, whose cancellation is not possible using existing methods because they are not consecutive in the total order. Each such cancellation takes O(c⋅n) time in the worst case, where c is the number of birth-death pairs and n is the size of the input complex.}, author = {Leśkiewicz, Jakub and Furmanek, Bartosz and Lipiński, Michał and Morozov, Dmitriy}, booktitle = {42nd International Symposium on Computational Geometry}, isbn = {9783959774185}, issn = {1868-8969}, keywords = {persistent homology, topological simplification, depth posets}, location = {New Brunswick, NJ, United States}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{Topological simplification guided by forbidden regions}}, doi = {10.4230/LIPIcs.SoCG.2026.72}, volume = {367}, year = {2026}, } @inproceedings{22003, abstract = {Let G be a finite, connected metric graph and let X be a subset of G. If X is sufficiently dense in G, we show that the Gromov-Hausdorff distance matches the Hausdorff distance, namely d_GH(G,X) = d_H(G,X). When the metric graph is the circle G = S¹ with circumference 2π, a recent study established the equality d_GH(S¹,X) = d_H(S¹,X) whenever d_GH(S¹,X) < π/6. Our results relax this hypothesis to d_GH(S¹,X) < π/3, and furthermore, we show that the constant π/3 is the best possible. We lower bound the Gromov-Hausdorff distance d_GH(G,X) by the Hausdorff distance d_H(G,X) via a simple topological obstruction: the existence of a possibly discontinuous function f: G → X with too small distortion contradicts the connectedness of G.}, author = {Adams, Henry and Majhi, Sushovan and Manin, Fedor and Virk, Ziga and Zava, Nicolò}, booktitle = {42nd International Symposium on Computational Geometry}, isbn = {9783959774185}, issn = {1868-8969}, keywords = {Gromov–Hausdorff distance, distortion, connectedness, Borsuk–Ulam theorem}, location = {New Brunswick, NJ, United States}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{Lower bounding the Gromov–Hausdorff distance in metric graphs}}, doi = {10.4230/LIPIcs.SoCG.2026.3}, volume = {367}, year = {2026}, } @inproceedings{22004, abstract = {Recent research on computing the diameter of geometric intersection graphs has made significant strides, primarily focusing on the 2D case [Duraj et al., 2024; Hsien-Chih Chang et al., 2024; Chan et al., 2025] where truly subquadratic-time algorithms were given for simple objects such as unit-disks and (axis-aligned) squares. However, in three or higher dimensions, there is no known truly subquadratic-time algorithm for any intersection graph of non-trivial objects, even basic ones such as unit balls or (axis-aligned) unit cubes. This was partially explained by the pioneering work of Bringmann et al. [Karl Bringmann et al., 2022] which gave several truly subquadratic lower bounds, notably for unit balls or unit cubes in 3D when the graph diameter Δ is at least Ω(log n), hinting at a pessimistic outlook for the complexity of the diameter problem in higher dimensions. In this paper, we substantially extend the landscape of diameter computation for objects in three and higher dimensions, giving a few positive results. Our highlighted findings include: 1) A truly subquadratic-time algorithm for deciding if the diameter of unit cubes in 3D is at most 3 (Diameter-3 hereafter), the first algorithm of its kind for objects in 3D or higher dimensions. Our algorithm is based on a novel connection to pseudolines, which is of independent interest. 2) A truly subquadratic time lower bound for Diameter-3 of unit balls in 3D under the Orthogonal Vector (OV) hypothesis, giving the first separation between unit balls and unit cubes in the small diameter regime. Previously, computing the diameter for both objects was known to be quadratic hard when the diameter is Ω(log n) [Karl Bringmann et al., 2022]. 3) A near-linear-time algorithm for Diameter-2 of unit cubes in 3D, generalizing the previous result for unit squares in 2D [Karl Bringmann et al., 2022]. 4) A truly subquadratic-time algorithm and lower bound for Diameter-2 and Diameter-3 of rectangular boxes (of arbitrary dimension and sizes), respectively.}, author = {Chan, Timothy M. and Chang, Hsien Chih and Gao, Jie and Kisfaludi-Bak, Sándor and Le, Hung and Zheng, Da Wei}, booktitle = {42nd International Symposium on Computational Geometry}, isbn = {9783959774185}, issn = {1868-8969}, keywords = {Graph Diameter, Geometric Intersection Graphs, Unit Ball Graphs}, location = {New Brunswick, NJ, United States}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{Charting the diameter computation landscape of intersection graphs in 3D and above}}, doi = {10.4230/LIPIcs.SoCG.2026.29}, volume = {367}, year = {2026}, } @inproceedings{22299, abstract = {The depth poset of a filtered Lefschetz complex reflects the dependencies between the cancellations of different shallow birth-death pairs. Using the fast algorithms for computing the depth poset in [Edelsbrunner et al., 2026] and for updating the persistence diagram under transpositions in [Cohen-Steiner et al., 2006], we give a complete case analysis of how transpositions of cells in the filter affect the depth poset. In addition, we present statistics on the depth poset for random point data and its sensitivity to the transpositions that occur in random straight-line homotopies.}, author = {Edelsbrunner, Herbert and Lipiński, Michał and Mrozek, Marian and Soriano Trigueros, Manuel and Zimin, Fedor}, booktitle = {42nd International Symposium on Computational Geometry}, isbn = {9783959774185}, issn = {1868-8969}, keywords = {Algebraic topology, Lefschetz complexes, persistent homology, vines and vineyards, birth-death pairs, shallow pairs, relations, partial orders, transpositions, Theory of computation → Computational geometry}, location = {New Brunswick, NJ, United States}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{The depth poset under transpositions in the filter}}, doi = {10.4230/LIPICS.SOCG.2026.41}, volume = {367}, year = {2026}, }